Combinatorics

Subsection B.2.1 Abraham ibn Ezra (c. 1093—1167)

Abraham ibn Ezra was an 11th-century Jewish scholar whose commentaries on the bible have been highly influential, and who disseminated Arabic scholarly knowledge into Jewish communities. He was born in Tudela, in what is now the Spanish province of Navarre. In the Middle ages, the town of Tudela had one of the oldest and most important Jewish communities in that region. Abraham moved to Córdoba as a young man and spent about half of his adult life there. In the final 27 years of his life he travelled extensively (going as far as Baghdad), after fleeing from the attacks on hitherto tolerant Moorish Iberia by the fanatic Almohads around 1140. He seems to have made his living largely as a poet and author.

Abraham married and had five children, four of whom are believed to have died young. His youngest son Isaac, an influential poet, converted to Islam in 1140; this event was deeply troubling to Abraham, as reflected in some of his own poetry.

Abraham is primarily known for his commentaries on the Torah. He also wanted to spread the (largely Arabic) knowledge he gained in Spain to the many Jewish communities he visited and lived in during his travels. His own writing was in Hebrew, and he also translated other works into Hebrew. Along with his biblical commentaries, he wrote books on Hebrew grammar; mathematics; philosophy; and astrology, as well as writing poetry. The book in which he discusses the problem described in Example 3.2.8 (mentioned again in passing in Example 4.2.7) is one of his books on astrology.

A lunar crater, Abenezra, was named after Abraham. Robert Browning's poem “Rabbi Ben Ezra” is a meditation on his life. It begins with the famous lines “Grow old along with me!/ The best is yet to be.”

Sources: Stanford Encyclopedia of Philosophy, wikipedia, and Encyclopedia Britannica.

Subsection B.2.2 Abū-l'Abbās Ahmad ibn al-Bannā' (1256—1321)

Abū-l'Abbās Ahmad ibn al-Bannā' (al-Marrākushī) is also known as Abū'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi. He was a 13th century scholar and mathematician whose books include the earliest-known surviving examples of some mathematical formulas and notation that we use to this day. Abü lived in the region that is now Morocco. It is not completely clear whether he was born in Marrakesh, but he certainly spent most of his life in that region, so that “al-Marrākushī” is often added to his name. It was in Marrakesh that he studied mathematics, which he went on to teach at the university in Fez for most of his life.

He wrote at least \(\displaystyle 82\) books, and translated Euclid's Elements into Arabic. It is not clear how much (if any) of the mathematical content of Abū's books is original and how much is due to earlier writers whose work has been lost; the writing seems to indicate mostly the latter. In his book Raf al-Hijab (“Lifting the Veil”) Abū presents the general formula that we still use to calculate the binomial coefficient \(\displaystyle \binom{n}{k}\text{.}\) This was also the first known work to use algebraic notation, though there is again some uncertainty as to whether or not this originated with him. Raf al-Hijab was actually a commentary on a previous book of his that may have been too challenging for his readers, so Abū wrote the commentary to provide additional details. This is why he included the calculations required for a variety of operations, including binomial coefficients. Some of his work is described in Example 4.2.5.

A lunar crater, Al-Marrakushi, was named after Abū.

Sources: encyclopedia.com, wikipedia, and St. Andrews' math history web site.

Subsection B.2.3 Atif Aliyan Abueida (1966—)

Atif Aliyan Abueida is an American researcher specialising in combinatorics, who is originally from Palestine. He graduated with a B.Sc. in 1987 from the United Arab Emirates University, in Al Ain (UAE). After working for eight years as a high school teacher, he moved to the United States, where he completed his M.Sc. at East Tennessee State University in 1996. Abueida then moved to Auburn University in Alabama, where he worked on his Ph.D. under the supervision of Chris Rodger. His thesis was entitled “The Full Embedding Problem”.

Immediately after defending his Ph.D. in 2000, Abueida moved to the University of Dayton in Ohio, where he has been working in the Department of Mathematics ever since. His areas of research interest include graph theory, design theory, and complex analysis. He has more than \(\displaystyle 25\) publications in these areas, including the work with David Angus Pike (1968—) mentioned in Section 17.1. He has also been involved in outreach, running workshops for teachers in the Dayton area.

Abueida is fluent in English and Arabic.

Sources: University of Dayton, University of Dayton, and the Math Genealogy Project, as well as MathSciNet for a publication list. Updated and confirmed through personal communication.

Subsection B.2.4 Ahmad ibn Mun'im al-'Abdarī (11??—1228)

Ahmad ibn Mun'im al-'Abdarī was a 12th-century mathematician, doctor, and teacher, whose surviving works include what may be the earliest examples of combinatorial reasoning. Ahmad lived in the area that is now Morocco. Different sources use slightly different versions of Ahmad's name. The most apparently reliable gives it as Ahmad ibn Ibrāhīm ibn 'Ali Ibn Mun'im al-'Abdarī. He was born near Valencia, Spain. He lived in Marrakesh (Morocco) for most of his life, and that is where he died. In addition to being a mathematical scholar and teacher, he learned medicine at the age of thirty, and was a practicing doctor as well as a scholar from that time.

Ahmad apparently wrote extensively about mathematics. Details remain known about only three of his texts, and only one of these, Fiqh al-hisāb (“Science of Calculation”), survives. This was written during the reign of the fourth Almohad caliph (1199—1213), who was a patron of education and science. In this book Ahmad considers a variety of problems involving permutations and combinations, with and without repetition, including the problems of silk threads and of words using the Arabic alphabet that is referred to in Example 4.2.7 and mentioned again in Section 4.3. The method of reasoning he uses to deduce an identity involving binomial coefficients may be the earliest known example of a combinatorial proof.

Abū-l'Abbās Ahmad ibn al-Bannā' (1256—1321) was a student of al-Qādhī ash-Sharīf, who in turn was a student of Ahmad's.

Sources: Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Combinatorics: Ancient and Modern, and muslimheritage.com.

Subsection B.2.5 Michael Owen Albertson (1946—2009)

Michael Owen Albertson was an American mathematician, particularly known for (with Karen Linda Collins (1959—)) introducing the distinguishing number of graphs. He was born in Philadelphia and raised in Abington, Pennsylvania. He began his undergraduate work at Michigan State University in 1964, and went on to complete a Ph.D at the University of Pennsylvania, where his thesis supervisor was Herbert Wilf.

Albertson defended his Ph.D. thesis (entitled “Irreducibility and Coloring Problems”) in 1971, and shortly thereafter was hired to teach at Smith College in Northampton, Massachusetts. Albertson worked and taught at Smith, while raising a family, for more than 35 years until his untimely death in 2009. Albertson had 3 children. In 2004 Albertson was named the L. Clark Seelye Professor of Mathematics at Smith College. He was married to Debra Lynn Boutin (1957—) for the last 15 years of his life.

Albertson's main area of research was graph theory. Over the course of his career he published more than 70 research papers with 33 different collaborators, including both Boutin and Collins. His work with Collins on the distinguishing number is discussed in Section 12.5. He mentored and collaborated with many junior researchers and students; Collins first worked with him during her undergraduate studies.

Sources: memorial article by Joan Hutchinson, obituary, the Math Genealogy Project, and Smith College. Updated and confirmed through personal communication with Debra Boutin.

Subsection B.2.6 Kenneth Ira Appel (1932—2013)

Kenneth Ira Appel was an American algebraist with expertise in computers, famous for his 1976 computer-aided proof of the Four-Colour Theorem (with Wolfgang Haken (1928—)). Appel was born into a Jewish family in Brooklyn (New York), and raised in Queens. He graduated with a B.Sc. from Queens College in 1953. He then worked briefly as an actuary and spent two years in the army, where he served in the state of Georgia as well as overseas in Germany. After his time in the army, Appel studied at the University of Michigan under the supervision of Roger Lyndon. He was awarded a Master's in 1956. His Ph.D thesis (still at the University of Michigan) was entitled “Two Investigations on the Borderline of Logic and Algebra”, and he defended it in 1959.

During his studies Appel spent summers programming computers for Douglas Aircraft, and after his doctorate he went to work for the federal government's Institute for Defense Analyses in Princeton, New Jersey, doing research in cryptography. He moved to Urbana, Illinois in 1961, where he joined the University of Illinois as a professor. It was there that he collaborated with Haken on the Four-Colour Theorem, discussed in Section 15.3. Haken brought the idea of the problem and the suggestion of using a computer to solve it, but had been convinced that this approach was not feasible. Appel had significant background in computing (having used computers in algebraic research as well as in cryptography), and proposed that they attempt this together.

Appel left the University of Illinois in 1993 to become chair of the mathematics department at the University of New Hampshire, which brought him closer to his grandchildren. He retired in 2003. During his career, Appel supervised at least five doctoral students of his own, one of whom (John Koch) assisted in some of the work on the Four-Colour Theorem.

Appel had a lifelong interest in politics. He served on the Urbana City Council; was treasurer of the Strafford County Democratic Party; and served on the Dover School Board for years until his death. His work on the school board also related to an ongoing interest and involvement in math education and outreach.

Appel was married with three children, two of whom became professors themselves (a daughter in biology and a son in computer science at Princeton). He and his wife met in graduate school, and were married after she obtained her Master's and he finished his Ph.D. Appel involved his children in checking calculations and computer output for the proof of the Four-Colour Theorem; they found hundreds of errors, many of which they were able to fix themselves.

Sources: New York Times obituary, St. Andrews' math history web site, the Math Genealogy Project, University of Illinois, and wikipedia.

Subsection B.2.7 László Babai (1950—)

László Babai is a Hungarian-American mathematician and computer scientist, best known for his coinvention of interactive proofs and for his quasipolynomial-time algorithm for the Graph Isomorphism problem. Born and educated in Budapest, Hungary, Babai (known as “Laci” to friends and colleagues) competed three times in the International Mathematical Olympiad for high school students. Representing Hungary, he won silver medals in 1966 and 1967 and a gold medal in 1968. He graduated from Loránd Eötvös University in Budapest in 1973. Mentored by Pál Turán (1910—1976) and Vera Sós (1930—), he earned his Ph.D. in 1975 from the Hungarian Academy of Sciences with his thesis entitled “Gráfok automorfizmuscsoportjai” (“Automorphism Groups of Graphs”).

Babai worked at Eötvös University from 1973 until 1989. In 1984 he received a D.Sc. (doctor of science) from the Hungarian Academy of Sciences. This is a degree that is available in some countries to recognise notable and prolonged contributions to research. In 1984 Babai became a visiting professor at the Computer Science Department at the University of Chicago in the United States, while maintaining his affiliation with Eötvös University, embarking on a period of commuting between the two continents that lasted nearly a decade. He became a permanent professor of computer science at Chicago in 1987 and was additionally appointed a professor of mathematics in 1995. He held the George and Elizabeth Yovovich Professorship at the University of Chicago from 2010—2019, and since 2019 he has been holding the Bruce V. and Diana M. Rauner Distinguished Service Professorship. Meanwhile he continues to maintain strong ties with the mathematical community in Hungary. He was elected as a corresponding member of the Hungarian Academy of Sciences in 1990 and as a full member in 1994.

Babai enjoys mentoring and engaging young people in math and computer science at various stages of their careers, from high school to postdoc. He has helped to lead Research Experience for Undergraduates programs at the University of Chicago every summer from 2001 to 2016 and continues to mentor REU participants. He has supervised 27 Ph.D. students as of 2021. Five of his former mentees, including one from high school, have become invited speakers at the International Congress of Mathematicians. He is one of the founders of the highly acclaimed Budapest Semesters in Mathematics study-abroad program for undergraduate students. At Chicago, he won a prestigious teaching award. Two pieces of advice that he frequently gives to students are: “It is better to learn several proofs of the same central theorem than to learn more theorems,” and, encouraging students to be broad in their interests: “The only kind of math I never used is the math I never learned.”

Babai has worked in combinatorics, group theory, the theory of algorithms, and complexity theory, with special attention to the interactions among these fields. The quasipolynomial graph isomorphism algorithm, which he discovered at the age of 65, is discussed in Section 11.4.

Babai has won numerous prizes and awards, including the Gödel prize (1993) and the Knuth Prize (2015). He was elected a Fellow of the American Academy of Arts and Sciences in 2015, and has been an invited speaker at the International Congress of Mathematicians three times (once as a plenary speaker).

Babai's native tongue is Hungarian. He is fluent in English and also conversant in German and Russian.

In the 2008 episode “First Contact” of the tv series Stargate: Atlantis, the character Dr. Radek Zelenka claims to have used Babai's work in combinatorics to trace a subspace transmission, but says that he cannot “dumb it down” for his superiors.

Sources: University of Chicago, American Academy of Arts and Sciences, the Math Genealogy Project, Gateworld, and wikipedia. Updated and confirmed through personal communication.

Subsection B.2.8 Eric Temple Bell (1883—1960)

Eric Temple Bell was a British-American analytic number theorist who studied diophantine equations. His work with generating functions is particularly noteworthy; he is also well-known for the popular books he wrote on math and the history of math (some of these are still in print). Bell was born in Peterhead, Scotland, but his family moved to California when he was just over a year old. The family returned to Bedford, England when his father died in 1896 (when Bell was 13), but he moved back to North America six years later, as he said “to escape being shoved into Woolwich [Royal Military Academy] or the India Civil Service”.

Bell studied mathematics at Stanford University in California from 1902—1904, where he was able to take all of the available math courses (and nothing else) and graduate with his bachelor's degree in two years. He taught for three years at a private school in San Francisco, leaving after the great fire of 1906. In 1907 Bell entered the Master's program at the University of Washington, graduating in 1908. He then moved to San José, California, where he earned some money by writing a science fiction novel. During some of the breaks in his education, Bell also worked as a ranch hand, surveyor, and mule skinner. Over the course of his career, he wrote at least 16 science fiction novels under the pseudonym “John Taine”. Many of these were written in a few weeks during the summers. They are now largely unknown and out of print.

Bell taught at Yreka High School in northern California for two years from 1909—1911. It was in Yreka that Bell met and married his wife, who also taught at the high school. After leaving Yreka, Bell moved to New York where he entered the Ph.D. program at Columbia University. He completed his Ph.D. in 1912, under the joint supervision of Frank Cole and Cassius Keyser, with a thesis entitled “The Cyclotomic Quinary Quintic”.

Immediately after completing his doctoral work, Bell moved to Seattle, where he taught at the University of Washington from 1912—1926. His only child, a son named Taine, was born during this time. Bell's reputation in research grew significantly during these years, and he was offered professorships at a number of prominent universities. He ultimately accepted an offer from the California Institute of Technology, where he remained from 1926 until his retirement in 1959.

In the context of Bell's work on generating functions, he studied and wrote about the numbers that have become known as “Bell numbers” (see Section 9.3). In addition to his original research (which included more than 250 papers) and his science fiction, Bell published popular books with colourful accounts of mathematical problems and math history. The ongoing popularity of his Men of Mathematics (which includes one woman) may be due in part to his placing more weight on telling a good story than on strict accuracy. Critics describe Bell's accounts as inaccurate and fanciful, legends rather than histories (some use harsher terms).

Bell won the American Math Society (AMS)'s Bôcher Prize in 1921. He was appointed to the Council of the AMS in 1924, and served as its vice-president beginning in 1926. He was elected president of the Mathematical Association of America, a role he filled from 1931—1933.

Sources: St. Andrews' math history web site, encyclopedia.com, the Math Genealogy Project, The Last Problem, and wikipedia.

Subsection B.2.9 Bhāskara II (1114—1185)

Bhāskara was an Indian mathematician and astronomer from the 12th century whose works contain the first surviving systematic use of the decimal system and introduce many principles of calculus. He was born in Bijapur, India. He is known as Bhāskara II to distinguish him from another mathematician and astronomer of the same name, from the 7th century. He is also often known as Bhāskaracharya, meaning “Bhāskara the teacher”. His father Maheśvara (a Brahman) was also a mathematician and astronomer/astrologer, and saw to his son's early training. As an adult, Bhāskara became the leader of an astronomical observatory in Ujjain. This was the centre of mathematical thought in India at that time.

At the age of 36, Bhāskara wrote the work for which he is best known, the Siddhānta Siromani (“Crown of treatises”). This is a four-part book that uses the decimal numbering system. The first part, Līlāvatī (“The Beautiful”; according to a story written by Fyzi, who translated this book into Persian in 1587, it is named for Bhāskara's daughter) is about calculations, including combinations and permutations. It is in this part that the problems referenced in Sections 3.1 (Example 3.1.6) and 4.2 appear. He did author other works as well. As was traditional in India at the time, his books were all written in verse.

Many extremely important results first appeared in various parts of the Siddhānta Siromani. Bhāskara explained a method for solving Pell's equation (\(\displaystyle Nx^2+1=y^2\)) for a number of specific values of \(\displaystyle N\text{.}\) This is a problem that European mathematicians struggled with centuries later, though Archimedes had significant understanding of it much earlier than Bhāskara. He studied trigonometry, and gave several results including the rule \(\displaystyle \sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)\text{.}\) Though methods for finding solutions to quadratic equations geometrically were known before this time, the general quadratic formula first appeared in his work. Bhāskara also developed and explained many of the principles of differential calculus, in his study of the motion of planets and their instantaneous velocities. He formulated some concepts of integral calculus as well, for calculating the volume of a sphere (planet). This was more than 500 years before the work of Leibniz and Newton. His work included one of the well-known visual proofs of the Pythagorean Theorem. Despite all of these remarkable developments that appeared for the first time in his work, Bhāskara didn't get everything right; for example, he struggled with the concept of division by zero, claiming that \(\displaystyle 0a/0=a\) for any \(\displaystyle a\text{.}\)

The Indian Space Research Organisation named one of their satellites after Bhāskara.

Sources: St. Andrews' math history web site, storyofmathematics.com, New World Encyclopedia, and wikipedia.

Subsection B.2.10 Anthony Bonato (1971—)

Anthony Bonato is an Italian-Canadian graph theorist who identifies as a gay man. In addition to his research in graph theory, he is known for his popular writing about math. He graduated with a B.Sc. from McMaster University in Hamilton, Ontario, in 1993. Bonato received a Master's degree (in 1994) and Ph.D. from the University of Waterloo. He worked under the supervision of Ross Willard, and defended his thesis “Colourings, Generics, and Free Amalgams” in 1998.

After a post-doctoral position at Mount Allison University in New Brunswick, Bonato returned to Ontario and in 1999 took a tenure-track faculty position at Wilfrid Laurier University in Waterloo, Ontario. In 2008 he moved to a tenured faculty position at a university in Toronto then known as Ryerson University. (Because of the role of its namesake Egerton Ryerson in the genocidal residential school system into which Indigenous children were forced, pressure has been mounting for the university to change its name.) Bonato also holds adjunct appointments at Laurier and at Dalhousie University in Nova Scotia.

Bonato is open and vocal about his sexual orientation and advocates for the LGBTQ+ community in mathematics. He usually introduces himself as a “gay man and a mathematician”, with the pronouns he/him/his. As Bonato has said, “the layers of our identity are critical to who we are and how our work is received.” Bonato and his husband have been married since 2004. He organised the “LGBTQ+ Math Day” events to raise the profile of queer mathematicians and to encourage young people in math who identify as LGBTQ+. He has been actively involved in committees to improve equity, diversity and inclusion in mathematics.

Bonato has over 130 publications, including four books on mathematics, with a fifth book forthcoming in 2022. His research is highly regarded. He has served on Canada's science granting council (NSERC) selection committees, including chairing the pure math committee and has been an invited speaker at more than 30 international conferences.

Bonato is passionate about mentoring, collaborating, and communicating mathematics. He has collaborated with more than 100 different coauthors. One of his results is discussed in Section 11.5. He has supervised 48 graduate students and post-doctoral fellows, including 7 PhD students as of 2021. He has won awards for his research and graduate supervision. Bonato's book Limitless Minds is a collection of interviews with prominent mathematicians. He also writes a blog, The Intrepid Mathematician, aimed at communicating about mathematics and mathematicians to a broader audience, and has a very active presence on social media. In one tweet, Bonato wrote: “You're doing math right now without realizing it. Mathematicians are people who realize it.”

Bonato's debut young adult, science-fiction novel Patterns will be published in 2022 and centres on a queer sixteen-year-old mathematician grappling with an alien invasion.

Sources: Ryerson University, The Intrepid Mathematician, LSE blogs, the Math Genealogy Project, and twitter. Updated and confirmed through personal communication.

Subsection B.2.11 John Adrian Bondy (1944—)

John Adrian Bondy is a graph theorist whose career began in England, grew to prominence in Canada, and who moved to France (where he still lives) in 1994. In addition to some important research results in graph theory, he is known for his (co-authored) introductory textbook on this subject. Bondy (who goes by “Adrian”) was born in England in 1944. He obtained his D.Phil. from Oxford University in 1969, under the supervision of Dominic Welsh. His thesis was entitled “Some Uniqueness Theorems in Graph Theory.”

Immediately after completing his doctorate, Bondy moved to Canada where he began to work at the University of Waterloo, in the Department of Combinatorics and Optimization. In 1976, his textbook Graph Theory with Applications, co-authored with colleague U.S.R. Murty, was published and quickly became the standard undergraduate textbook for graph theory. This was also the year in which Bondy's research with Václav Chvátal (1946—) about the closure of a graph (discussed in Section 13.2) was published. Bondy was one of the editors-in-chief of the prestigious Journal of Combinatorial Theory, Series B from 1985 until 2004.

In 1991, Bondy developed a relationship with a woman who lived and worked in Paris. In short succession, Bondy's father died, and he became a father himself. His partner and son remained in France. Bondy's increasing absences from Waterloo when he did not have teaching obligations on campus were a source of friction between himself and university administrators that increased when his request to be granted a sabbatical or to cut back to a half-time position were denied. In 1994, he negotiated a temporary leave with reduced pay in order to seek work in France. During this leave, Bondy applied for and was appointed to a position at Université Lyon I, which was still a long commute from his family in Paris. According to documents, he was subsequently dismissed from the University of Waterloo for a “substantial conflict of interest,” apparently not having informed either Waterloo or Lyon that he was employed by both, and drawing salary from both. The dismissal was appealed, but was upheld at arbitration. This was a sad end to Bondy's distinguished career at Waterloo, and led to Pál Erdős (1913—1996) resigning his honorary degree, and to Chvátal resigning his adjunct appointment at Waterloo.

Bondy continued to work at Université Lyon I until his retirement in 2009. He is credited by the Math Genealogy Project for supervising 12 doctoral students (about half of them at Waterloo and half at Lyon), and published almost 100 papers during his career.

Since childhood, Bondy has been a keen amateur photographer. He began taking photographs more seriously in 1982 during a visit to Paris. Throughout the 1980s he exhibited his photographs frequently at the Kitchener-Waterloo Art Gallery and elsewhere. After his retirement in 2010 he founded Mind's Eye, a non-profit association that runs the “Galerie Adrian Bondy” in Paris. The goal of the association is to explore conceptual links between photography and mathematics. Bondy is a regular exhibitor at the gallery.

Sources: wikipedia, Journal of Combinatorial Theory, Waterloo Gazette (via Wayback Machine), the Math Genealogy Project, Letter from Chvátal, Letter from Erdős, and Mind's Eye. Updated and confirmed through personal communication.

Subsection B.2.12 Raj Chandra Bose (1901—1987)

Raj Chandra Bose was an Indian-American mathematician and statistician whose discoveries in coding theory are still being used in industrial and scientific applications. He is also known for his involvement in disproving a conjecture that had been made by Leonhard Euler (1707—1783). Bose was born in Hoshangabad, India, but raised in Rohtak. His father pushed him to excel in school, a task in which Bose's photographic memory was a significant asset. His mother died in 1918 of the influenza pandemic, and his father died within a year, leaving Bose with four younger siblings. He juggled this responsibility with his own desire for education, earning some money by tutoring while working on his bachelor's degree, which he completed in 1922. Bose managed to get a job teaching high school for the next year, but this interfered with his ability to attend classes and attain his Master's degree, and the following year he was only able to earn money by tutoring. Then the brother of one of his students offered to support him in Calcutta (now Kolkata) to study pure mathematics.

Bose moved to Calcutta in 1925, leaving his siblings in Delhi, where his brother had found a steady job. He drew the attention of Shyamadas Mukherjee, a geometer, who gave him a room to live in and found him tutoring work while acting as a mentor. Bose successfully completed his Master's degree in pure mathematics at the University of Calcutta's Rajabazar Science College in 1927.

Bose worked as a research assistant for two years. Jobs were hard to come by in 1930, and although Bose did manage to become a lecturer at Asutosh College in Calcutta, the work paid so poorly that he also needed to tutor. Bose was married in 1932. At the end of 1932 the director of the new Indian Statistical Institute, having seen his work in geometry, offered Bose a part-time position even though Bose had little background in statistics. Bose became one of the chief mathematicians at the institute, moving to full-time status in 1935. After a visit to India by Sir Ronald Aylmer Fisher (1890—1962) in 1938—1939, Bose developed a significant interest in design theory. In 1940 Bose moved to the University of Calcutta, becoming head of the department of statistics there in 1945.

Bose wanted a position as a professor, but did not have a Ph.D. He submitted some of his published papers and was awarded a doctorate in 1947, examined by Fisher. After this he spent a couple of years visiting the United States, taking positions as a visiting professor before returning briefly to Calcutta in 1948. With job offers from Calcutta and a couple of American universities to choose from, Bose joined the University of North Carolina at Chapel Hill as a professor of statistics in 1949. In 1966 he was given the Kenan Chair there. He retired in 1971, but then took a chair at Colorado State University of Fort Collins. He retired again in 1980, was made a professor emeritus at Colorado State, and died in Colorado. Bose and his wife had two children.

Among Bose's more important discoveries were “BCH” codes. Together with his student Dwijendra Kumar Ray-Chaudhuri (1933—) he discovered these independently, at about the same time as their discovery by Alexis Hocquenghem in 1959. These codes are still used in applications such as compact disc players, DVDs, and solid-state drives; the “B” in their name is for Bose, and the “C” for Ray-Chaudhuri. Bose is also known for proving (with his student Sharadchandra Shankar Shrikhande (1917—2020) and Ernest Tilden Parker (1926—1991)) that pairs of orthogonal Latin squares of order \(\displaystyle n\) exist whenever \(\displaystyle n=4k+2\) with \(\displaystyle k \ge 2\text{,}\) disproving a conjecture of Euler's (as discussed in Section 16.2). This work earned them the nickname of “Euler's spoilers”. The techniques that Bose and his co-authors developed for disproving Euler's conjecture played a significant role in the proof by Richard Michael Wilson (1945—) of Wilson's Theorem. Other significant discoveries include Bose-Mesner algebras, as well as the notions of partial geometry and strongly regular graphs. Bose published over 100 papers, and supervised at least 30 doctoral students in the United States, in addition to his influence on young mathematicians in India.

Among the honours Bose received were honorary degrees from the Indian Statistical Institute in 1974 and from Visva-Bharati University in 1979. In 1976 he was elected to the United States National Academy of Sciences.

Bose spoke a number of languages including English, and liked to recite poetry in Arabic, Bengali, Persian, Sanskrit, and Urdu. He never learned to drive; his wife did all the driving for them. He was an enthusiastic gardener. Bose had an excellent sense of humour, and liked to joke that, working modulo 2, “it is equally as good to give as to receive”!

Sources: wikipedia, St. Andrews' math history web site, the Math Genealogy Project, Indian Statistical Institute, Colorado State University, and Obituary in Journal of the Royal Statistical Society.

Subsection B.2.13 Debra Lynn Boutin (1957—)

Debra Lynn Boutin is an American whose research focuses on the area of algebraic graph theory. She was born in 1957, and joined the Navy after completing high school in Chicopee, Massachusetts in 1975. She served on active duty from 1975—1979, and remained in the Naval Reserve for 16 more years. She retired as a Chief Petty Officer in 1995. She also raised a daughter during these years. In 1985 Boutin enrolled in Springfield Technical Community College in Massachusetts. In 1988 she transferred to Smith College in Northampton, Massachusetts, where she completed her bachelor's degree in mathematics in 1991.

Boutin then entered the doctoral program at Cornell University in Ithaca, New York. She studied geometric group theory under the supervision of Karen Lee Vogtmann, and successfully defended her Ph.D. thesis, “Centralizers of Finite Subgroups of Automorphisms and Outer Automorphisms of Free Groups” in 1998. Shortly afterward, Boutin was hired at Hamilton College in Clinton, NY, where she holds the Samuel F. Pratt professorship. Her areas of research interest include graph theory, geometric graph theory, and group theory. Boutin has published over 25 papers on these topics, and is well-known in the field. Her work on distinguishing cost is discussed in Section 12.5.

Boutin was married to Michael Owen Albertson (1946—2009) for the last 15 years of his life.

Sources: Hamilton College, Hamilton College, LinkedIn, and the Math Genealogy Project. Updated and confirmed through personal communication.

Subsection B.2.14 John M. Boyer (1968—)

John M. Boyer is a Canadian computer scientist who has spent his career working in industry. He lives and works in Victoria, British Columbia. He began working for the startup PureEdge Solutions on its first day of operations in 1993. While working for PureEdge, he also undertook graduate studies at the University of Victoria, beginning in 1995. Boyer completed his Ph.D. under the supervision of Wendy Joanne Myrvold (1961—) in 2001, with his thesis “Simplified \(\displaystyle O(n)\) Algorithms for Planar Graph Embedding, Kuratowski Subgraph Isolation, and Related Problems”.

The planarity testing algorithm by Boyer and Myrvold that is mentioned in Section 15.1 was published in 2004, based on ideas that appeared in Boyer's thesis. It is one of the two state-of-the-art algorithms currently in use for planarity testing. Boyer's implementations of the planarity algorithm and extensions for related problems are publicly available, and the planarity algorithm has been implemented in a variety of packages and applications.

In 2005 Boyer became an IBM employee as a result of IBM's acquisition of PureEdge. The products he had worked on for secure interactive data collection became IBM Forms, and Boyer became the Chief Architect of that division. During his time at IBM, Boyer also served as Chief Architect for several teams, including those working on social computing, machine learning, and data science platforms. Boyer has more than 25 published papers, has co-authored several computer industry standards, and holds over 40 patents. He was appointed an IBM Distinguished Engineer in 2010 and a Master Inventor at IBM in 2012.

Sources: wikipedia, ContactOut, LinkedIn, the Math Genealogy Project, and twitter. Updated and confirmed through personal communication.

Subsection B.2.15 Rowland Leonard Brooks (1916—1993)

Rowland Leonard Brooks was a British tax inspector who as an undergraduate student proved a result about graph colouring that appears in many textbooks. Brooks (who was known as “Leonard”) was born in Caistor, Lincolnshire, England, in 1916. He studied at Trinity College of Cambridge University from 1935 to 1940. While there, he developed close friendships with three other students: Smith, Stone, and Tutte, and the four worked closely together on mathematical problems.

Brooks proved Brooks' Theorem as an undergraduate, and published it in the Proceedings of the Cambridge Philosophical Society in 1941. He and his friends also spent a lot of time as students trying to “square the square” (find a square with integral sides that could be decomposed into smaller unequal squares with integral sides). They were ultimately successful in finding the first known examples of this also, and developed related theory.

After leaving Cambridge, Brooks worked as a tax inspector in London and did not pursue mathematics further, though he continued to play with finding ways to “square squares” throughout his life. He died in Croydon, England. Brooks was a very private person and avoided making biographical information public.

Sources: wikipedia, squaring.net, talk by Bjarne Toft, and rootsweb.

Subsection B.2.16 Eugène Charles Catalan (1814—1894)

Eugène Charles Catalan was a French mathematician whose work included important results in number theory, geometry, and combinatorics. He was born in Bruges, which is now part of Belgium; at that time it was governed by France under Napoleon. Catalan considered himself French, even though he was only 1 when the Netherlands took control of Bruges in 1815. On his birth certificate, Catalan is registered as Eugène Charles Bardin; his mother was 17 and unmarried at the time of his birth. His parents were married in 1821 when Catalan was 7; his father acknowledged him and he took his father's surname of Catalan.

Catalan's father was described as a jeweller; he made his living in a variety of ways. Catalan briefly apprenticed as a jeweller at the age of 10. His family moved to Paris shortly thereafter, in about 1825. By this time his father was an architect, and Catalan entered school to learn this profession also.

At school, Catalan showed an aptitude for mathematics. Catalan passed the entrance examinations for the École Polytechnique in 1833. He had strong political opinions but did not take an active part in the many disturbances of the peace in Paris during this time. Nonetheless, along with all of his fellow students he was dismissed from the school and had to apologise for an “act of insubordination” before being readmitted. He graduated in 1835 and took a position teaching in Châlons-sur-Marne. He also published a number of research papers over the next few years.

Catalan wanted to return to Paris but was unsuccessful in applications for positions there. Liouville advised him to obtain additional qualifications. Catalan took Liouville's advice, and obtained a double baccalaureate in 1839 and a doctorate in 1841. His advisor was Joseph Liouville, and Catalan wrote two theses: “Attraction d'un ellipsoïde homogène sur un point extérieur ou sur un point intérieur” (“Attraction of a homogeneous ellipsoid on an exterior point or on an interior point”) in mechanics, and “Sur le mouvement des étoiles doubles” (“On the movement of double stars”) in astronomy.

Catalan took a more prominent role in politics and political unrest during this period, and this interfered with the advancement of his academic career. He took an active role in the revolution of 1848 that led to the Second Republic, and sat in France's Chamber of Deputies. When Louis-Napoléon Bonaparte assumed absolute power in 1851, Catalan refused to swear allegiance and lost the positions he still held. He continued to tutor and publish, but held no permanent positions until 1865, when he was appointed chair of mathematics at the University of Liège (Belgium). He held this position until his retirement in 1884, and remained in Liège until his death in 1894.

In addition to coming up with the Catalan numbers (see Section 9.2), Catalan published many important results in number theory, geometry, and combinatorics, and a surface that he discovered is also named for him. Catalan formulated a famous conjecture in number theory that was not proven until 2002.

Catalan was elected to many national and regional academies of science, which did not include the French Academy despite several attempts. He was awarded the Knight's Cross of the Légion d'Honneur, the Cross of the Knight of the Order of Léopold, and was made an Officer of the Order of Léopold in 1890. The Royal Academies for Science and the Arts of Belgium has named a prize after him that is awarded every five years for important progress in pure mathematics by a citizen of the European Union.

Sources: wikipedia, St. Andrews' math history web site, wikipedia, and the Math Genealogy Project.

Subsection B.2.17 Maria Chudnovsky (1977—)

Maria Chudnovsky is one of the foremost graph theorists of our time. She is particularly famous for her proof, in a monumental paper (with co-authors), of the Strong Perfect Graph Theorem. Chudnovsky was born in the USSR, and lived in Leningrad (now St. Petersburg, Russia) until the age of 13, when her family moved to Israel. She finished school in Haifa, and completed both her bachelor's (in 1996) and Master's (in 1999) degrees at the Technion there. She was also completing her mandatory service in the Israel Defense Force from 1996 to 1999.

After her Master's, Chudnovsky moved to the United States to undertake graduate work at Princeton University in New Jersey. She worked under the supervision of Paul Seymour (1950—), and received an M.A. in 2002 and her Ph.D. in 2003. Her thesis was entitled “Berge Trigraphs and Their Applications”, and as discussed in Section 14.3, it resulted in the proof of the Strong Perfect Graph Theorem.

Chudnovsky held a post-doctoral fellowship at the Clay Institute in Boston, Massachusetts from 2003—2008, also returning to Princeton as a Veblen Research Instructor (with a year at the Institute for Advanced Study) in 2003—2005, and then an Assistant Professor in 2005. She moved to Columbia University in 2006, and held the Liu Family Professorship in Industrial Engineering and Operations Research there in 2014. Chudnovsky returned to Princeton as a professor in 2015.

Chudnovsky was married in 2011, and has a son born in 2013. She has appeared in commercials for TurboTax and Comfortpedic (as herself, a brilliant mathematician). The TurboTax commercial can be found on YouTube.

As a young, prominent female mathematician, Chudnovsky has been in great demand for outreach and interviews as well as to speak at conferences. She is also a proud member of the Jewish scientific community. She has taken all of this seriously, and has served as a model to many. She has supervised 10 Ph.D. students and 6 post-docs as of 2021, as well as supervising research by undergraduate and Master's students. She has given the advice: “Don't let your self-doubt scare you too much. Just accept that everyone has their moments when they feel like a complete misfit. Just keep pushing.”

Chudnovsky's research is in structural graph theory. In addition to her proof of the Strong Perfect Graph Theorem, some of her more important contributions include finding a polynomial-time algorithm to identify perfect graphs, and determining the structure of claw-free graphs. In 2004 Chudnovsky was named in the “Brilliant 10” by Popular Science magazine, and in 2012 she received a “genius award” from the MacArthur Foundation. She received the D.R. Fulkerson Prize in 2009, jointly with George Neil Robertson (1938—), Seymour, and Robin Thomas (1962—2020). She was an invited speaker at the International Congress of Mathematicians in 2014.

Sources: wikipedia, Princeton, Clay Institute, and the Intrepid Mathematician. Updated and confirmed through personal communication.

Subsection B.2.18 Václav Chvátal (1946—)

Václav Chvátal (known as “Vašek”) is a mathematician from the former Czechoslovakia, whose career has been spent in North America and whose work has been influential across a number of branches of combinatorics. He was born in Prague, Czechoslovakia, where he studied mathematics at Charles University. He and his first wife fled Prague shortly after the Soviet invasion in 1968, and moved to Canada. By the time he enrolled as a Ph.D. student at the University of Waterloo in 1969, Chvátal had already published 6 papers, the first at the age of 19.

Chvátal took only a year to complete his doctorate under the supervision of Crispin Nash-Williams, with a thesis entitled “Hypergraphs and Ramseyan Theorems”. He held a series of positions for relatively short periods over the next 8 years, bouncing back and forth between Montreal, Quebec, and Stanford, California. He started at McGill University in 1971; went to Stanford University for 1972; to the Université de Montréal from 1972—1974; back to Stanford from 1974—1977; and to the Université de Montréal from 1977—1978. He then took a position at McGill University again, where he stayed for longer, from 1978—1986.

In 1986 Chvátal moved to New Jersey to take a position at Rutgers University, where he worked from 1986—2004. He finally returned to Montreal in 2004, and held the Canada Research Chair in Combinatorial Optimization at Concordia University from 2004—2011, followed by the Canada Research Chair in Discrete Mathematics from 2011 until his retirement in 2014.

In addition to Chvátal's work on Hamilton cycles (which goes much further than the result mentioned in Section 13.2), he also proved significant results relating to hypergraphs, algorithmic complexity, linear programming, optimisation, and perfect graphs. In addition, Chvátal worked with David Applegate, Bob Bixby, and Bill Cook on the development of the record-breaking computer code Concorde for solving the Travelling Salesman Problem. He found the smallest triangle-free class-one graph in which every vertex has valency 4 (on 12 vertices), and it is named for him.

In a tribute to Claude Berge, whose book introduced him to graph theory and with whom he later developed a close friendship, Chvátal wrote that this experience “took me through the looking glass to enchanted worlds where I found myself.” He has published more than 125 papers and 5 books. He supervised at least 12 doctoral students, and mentored many other young researchers. His talent in writing extends beyond math: in 1971 he wrote a prize-winning short story, Déjà Vu.

Sources: Concordia University, wikipedia, Vašek Chvátal: A Very Short Introduction, In Praise of Claude Berge, and the Math Genealogy Project. Updated and confirmed through personal communication.

Subsection B.2.19 Gilles Civario (1972—)

Gilles Civario is a high performance computing consultant from France, known in combinatorics for his contribution to the study of Sudokus. He was born in 1972, and began his career as a consultant for the CS Group Information Technology company near Paris, where he worked from 1999—2003. He then spent four years at Bull Information Technology, from 2004—2008, in the Grenoble region.

In 2008, Civario moved to Dublin, Ireland. He spent 8 years working as a Senior Software Architect for the Irish Centre for High-End Computing (ICHEC). This is a national service that provides computational resources and expertise for scientific research in Ireland. It was in this role that Civario carried out the computations necessary to determine that no 16-clue sudoku puzzle has a unique solution, as mentioned in Section 16.1. The computation took approximately 7,000,000 core hours and was being worked on for most of 2011.

Since leaving the ICHEC in 2016, Civario has returned to France, where he works for Dell Technologies as a High Performance Computing application specialist.

Sources: LinkedIn, “Sudoku meets Knights Corner”, ICHEC, and the Irish Times. Updated and confirmed through personal communication.

Subsection B.2.20 Karen Linda Collins (1959—)

Karen Linda Collins is an American mathematician best known for her research in graph theory. She obtained her B.A. from Smith College in 1981, where she completed her honours thesis under the direction of Michael Owen Albertson (1946—2009). From there she went to the Massachusetts Institute of Technology (MIT). She worked at MIT under the supervision of Richard Stanley, and obtained her Ph.D. in 1986 with a thesis entitled “Distance Matrices of Graphs”. She participated in the AT&T Bell Labs Research Program for Women during her graduate student summers, under the direction of Ron Graham and Fan Chung-Graham.

Collins started working at Wesleyan University in Connecticut immediately after completing her doctorate, in 1986, and says that she is thrilled to still be there. She enjoys teaching at all levels, and particularly likes to work with students on research projects. She served as chair of the department of Mathematics and Computer Science in 2007—2010, and is currently (in 2021) serving again. In 2018 she became the Edward Van Vleck Professor of Mathematics at Wesleyan. Her husband, Mark Hovey, is a professor of math at Wesleyan, and currently (in 2021) an Associate Provost.

Collins has taken an active role in the combinatorics and graph theory community in the northeastern United States throughout her career, and is currently a co-organiser of the Discrete Math Days in the Northeast. She is best known for her work with Albertson on graph homomorphisms, and the distinguishing number of graphs (mentioned in Section 12.5) and her work with Ann Trenk of Wellesley College on the distinguishing chromatic number of graphs. She has published at least 30 articles and has supervised at least 6 Ph.D. students as well as 9 other graduate students as of 2021. She is the co-author with Trenk of a chapter on split graphs in the book “Topics in Algorithmic Graph Theory”, which is part of the Encyclopedia of Mathematics and its Applications.

Sources: Wesleyan University, wikipedia, the Wesleyan Argus, and the Math Genealogy Project. Updated and confirmed through personal communication.

Subsection B.2.21 Gabriel Andrew Dirac (1925—1984)

Gabriel Andrew Dirac was a mathematician whose contributions to graph theory helped to establish this field of mathematical research. Born in Budapest, Hungary, and educated in the U.K., he spent most of his career in Denmark. At birth his name was Gábor Balázs. When he was 12, his mother married the physicist and Nobel Laureate Paul Dirac, and the family moved to England. Dirac and his sister were formally adopted, and took the surname of their stepfather.

Dirac began his mathematical studies at St. John's College, Cambridge in 1942. He interrupted his education in 1944 to work in the aircraft industry during the war. He obtained his Master's in 1949, and then went to the University of London for his Ph.D. He studied under Richard Rado (1906—1989) there, and received his doctorate in 1951 with a thesis entitled “On the Colouring of Graphs: Combinatorial topology of Linear Complexes”.

Dirac had a somewhat peripatetic career, with appointments at universities in England (London), Canada (Toronto), Austria (Vienna), Germany (Hamburg and Ilmenau), Ireland (Dublin), and Wales (Swansea). His longest affiliation was with the University of Aarhus in Denmark, where he worked briefly in 1966. He returned there in 1970, and remained until his death in 1984 at the age of 59. His appointment in Dublin was to the Erasmus Smith professorship, from 1964—1966.

Dirac began to study graph theory when the field was still very young and not well-respected outside of Hungary (in Hungary there was an active group of researchers). Dirac's work brought broader recognition to the field. In addition to his important research on Hamilton cycles (mentioned in Section 13.2), Dirac made significant contributions to research into colouring and critical graphs. He also published results in number theory and geometry. In addition to his significant role in the development of graph theory, Dirac made notable contributions to the development of mathematical research in Denmark. Dirac was highly influential in the careers of many young researchers.

Sources: wikipedia, tribute by Carsten Thomassen, Aarhus University obituary, and the Math Genealogy Project.

Subsection B.2.22 Pál Erdős (1913—1996)

Pál (Paul) Erdős was one of the most influential mathematicians of the 20th century. It is hard to condense the life of Erdős into a brief biographical sketch. He was incredibly prolific, generous, collaborative, and also eccentric. Erdős was born in Budapest, Hungary to parents who were born Jewish but did not practice the religion. His two sisters both died of scarlet fever while his mother was in the hospital for his birth, leaving him an only child who grew up in the shadow of this tragedy. His parents were both teachers of mathematics, and Erdős showed an early interest in math. During his young childhood, from 1914—1920, his father was absent, as a prisoner of war in Siberia.

Although anti-Jewish laws prevented most students of Jewish descent from entering university in Hungary at that time, Erdős won a national examination and was allowed to enrol in 1930 at the age of 17, at the university of science in Budapest (now named for physicist Loránd Eötvös). He obtained his doctorate there in 1934 (age 21), under Fourier analyst Lipót (Leopold) Fejér, with a thesis entitled “Über die Primzahlen gewisser arithmetischer Reihen” (“On the prime numbers in certain arithmetic progressions”). The situation in Hungary was becoming untenable for people who had Jewish heritage, and Erdős managed to find a post-doctoral fellowship at the University of Manchester, in England. He continued to visit Hungary when he could. Following world events in 1938, he moved to the United States. Most of his remaining close relatives were murdered during the Holocaust. His mother did survive, and he spent time with her often after he became able to visit Hungary without fear of being unable to leave. In fact, later in life at the age of 84, Erdős's mother joined him in his travels. She journeyed with him around the globe for the next 7 years until her death in Calgary, Canada in 1971.

The first mention of Erdős in this book occurs with the Erdős-Szekeres Theorem. This result appeared in the same paper as his work with Szekeres on the Happy Ending Problem (see Section 14.2). Erdős was also jointly responsible with Alfréd Rényi (1921—1970) for one of the random graph models discussed in Section 11.5. In related probabilistic work, the two also showed that almost every graph has no nontrivial automorphisms, as mentioned in Section 12.5.

Erdős held some temporary appointments through the next sixty years. For the most part, he lived as a professional itinerant scholar and collaborator. He travelled constantly, with only a briefcase early on, though later he acquired a suitcase. His focus was entirely on mathematics: many life skills such as cooking, driving, handling finances, and even tying his own shoes he acquired late, or never. He visited people, enthusiastically talked mathematics, and collaborated, right up to the day of his death in 1996. Erdős was welcomed around the world for his kindness, generosity, and humour.

Erdős had remarkable insight and a breadth of knowledge that resulted in his making significant contributions to a number of fields of mathematics, including combinatorics, number theory, set theory, classical analysis, and probability. He also built foundations of entirely new areas including Ramsey theory, transfinite combinatorics, probabilistic number theory, and probabilistic combinatorics. It is no accident that his name comes up repeatedly in this book. He cultivated his talent for asking interesting questions, and for finding people to solve them with. His lack of official positions meant that most of his mentoring was either unofficial, or came in the form of co-authorship. He had more than 500 coauthors, and published about 1500 research papers. It was this prolificacy that led to the idea of the “Erdős number” for mathematicians: how many degrees of separation are there between your collaborators, and Erdős? His co-authors have Erdős number 1; theirs have Erdős number 2, and so on. More than 12,000 people have an Erdős number of 2, and more than 80,000 people have an Erdős number of 5 (mine is 3).

Erdős had virtually no money. He lived very frugally aside from the cost of his travel itself (he generally stayed with mathematicians), and was extraordinarily generous. He often offered monetary prizes (anywhere from $25 to over $1000) for solutions to problems he posed, and gave money away whenever he saw a need. He said, “I never wanted material possessions. There is an old Greek saying that the wise man has nothing he cannot carry in his hands. If you have something beautiful, you have to look out for it, so I would rather give it away.” Erdős had many unusual philosophies and perspectives. He also had his own terminology for a variety of things: for example, children were “epsilons”, as were budding mathematicians. Each time he reunited with friends, he would ask, “Who are the new epsilons?” and would promptly invite the gifted youngsters to lunch. Erdős met and mentored many mathematicians (Béla Bollobás, Lajos Pósa, Attila Máté, Noga Alon, and Imre Ruzsa to name just a few) while they were still in their teens or even younger.

Erdős believed passionately in the beauty of mathematics, and liked to say that “the SF has this transfinite Book that contains the best proofs of all mathematical theorems, proofs that are elegant and perfect…. You don't have to believe in God, but you should believe in the Book.” The “SF” was Erdős's joking way of referring to God as the “Supreme Fascist”, to express his frustration over how closely The Book is guarded, and how few glimpses a mortal is allowed. He believed that a mathematician's goal in life should be to have some of these glimpses.

Erdős was awarded the American Math Society's Cole Prize in 1951, Hungary's Kossuth Prize in 1958, and the Wolf Prize from Israel in 1983/4. He received many honorary degrees. An asteroid was named after him in 2021 (“Erdőspál”). A massive wallpainting entitled “Saints Dancing” wraps around the interior of the rotunda of St. Gregory of Nyssa Episcopal Church in San Francisco, showing about 90 saints. Erdős is one of these, dancing between Gandhi and Luther.

Sources: St. Andrews' math history web site, wikipedia, Math Association of America, New York Times, University of California San Diego, University of Chicago, Huffington Post, American Math Society, Education Resources Information Center, Purdue University, and the Erdős Number Project. Additional information and clarification by personal communication from Laci Babai.

Subsection B.2.23 Euclid (c.325BCE—c.265BCE)

Euclid was a mathematician from the 3rd century BCE, whose writings established mathematics as a deductive science. There is little to say about his life, because little is known. In contrast to other mathematicians who lived at that time and even earlier, there are no contemporary biographies, nor even much in the way of references to him. He seems to have lived most of his life in Alexandria, where he taught and wrote. There are a variety of stories about him, but significant doubt as to the authenticity of any of them. It has even been suggested, not without grounds, that “Euclid” may have been a pseudonym for a team of mathematicians. There is no evidence as to who might have been part of such a team if that were true, and it seems most likely that he was a real person. Whoever he was, he had many students and a school grew from his teachings.

In addition to his most famous work, the Elements (mentioned in Section 18.3), remnants of five works by Euclid survive to this day, and a number of other lost works are attributed to him. The Elements consists of 13 books, and includes rigorous proofs from geometry and number theory. Many of the results in the Elements were known prior to Euclid; practical geometry had received much attention in Greek, Babylonian, and Egyptian culture, and Euclid certainly draws heavily on work known to students of Plato's Academy.

The main contributions of Euclid are the clarity of the writing, and the rigorous proof methods that he insisted upon. Not only are these very well thought through, but this marked a conceptual change in the way in which mathematics is studied and understood as a science. Up to this time, historical mathematics (as studied in Egypt, India, China, and other ancient cultures) was an empirical science like other sciences, full of observations and experimentation. In Euclid's Elements for the first time, mathematics was treated as a deductive science. This is a distinction that still sets the study of mathematics apart from all other branches of intellectual endeavour. Euclid was very careful to explicitly lay out the definitions, axioms, and postulates on which his deductions relied, and to use only these in his proofs. He included as axioms things that might seem obvious, such as: “Things equal to the same thing are also equal to each other”. (There are some subtle and unmentioned assumptions in his work that were not noticed for centuries.)

The European Space Agency has been constructing a space telescope that has been named Euclid in his honour.

Sources: wikipedia, St. Andrews' math history web site, and worldhistory.org.

Subsection B.2.24 Leonhard Euler (1707—1783)

Leonhard Euler was one of the most influential and prolific mathematicians of all time. He was born in 1707 in Basel, Switzerland. His father, a minister, had studied mathematics along with Johann Bernoulli, under the tutelage of Jacob Bernoulli (and in fact living in the Bernoullis' house) as an undergraduate, and passed some of this learning along to Euler. Euler enrolled at the University of Basel in 1720 (not unusually young at the time). He earned a Master's degree in 1723, with a thesis comparing the philosophies of Newton and Descartes, and began to study theology according to his family's wishes.

By this time Euler had brought himself to the attention of Johann Bernoulli, who gave him advice on mathematical books to read, and made himself available on Saturday afternoons to answer Euler's questions. Between his own interest and Bernoulli's influence, Euler eventually stopped studying theology and devoted his attention to mathematics. His studies in Greek and Hebrew remained useful to him in this context. In 1726 he applied for a position in physics at the University of Basel, submitting a thesis on properties of sound (“Dissertatio physica de sono”) with the support and advice of Bernoulli.

Although his application for the position in Basel was unsuccessful, Euler's connection with the Bernoullis stood him in good stead. Two of Bernoulli's sons were working at the Imperial Russian Academy of Sciences in St. Petersburg. When one of them died, the other recommended that the vacant position be offered to Euler. He accepted the position, and moved to St. Petersburg in 1727.

Euler thrived in St. Petersburg, where the Academy emphasised research and assigned limited teaching responsibilities. He learned Russian, and served the navy in addition to his mathematical work until 1730. He married while there; he and his wife Katharina had 13 children, of whom only 5 survived to adulthood. There was considerable political turmoil in Russia during these years, and in 1741 Euler moved his family to Germany, where he had been offered a post as a founding member of the Berlin Academy (the Academy was not fully and formally established until 1746). Euler spent 25 highly productive years there, before being lured back to St. Petersburg under very generous terms in 1766. (The conditions of his position in Berlin had also deteriorated.) He remained there until his death in 1783.

Euler lost the sight in his right eye after an illness in 1738 when he was 31. He began to lose his remaining sight in 1766 and lost it completely in 1771, due to a cataract in his left eye followed by surgery that led to an abscess. His phenomenal memory (he could recite Virgil's Aenaeid, and knew which line started and ended each page of the edition from which he had learned it) enabled him to continue his extraordinary mathematical productivity: for example, he produced more than 50 papers in the year 1775 alone. He was assisted by scribes in this work, including one of his three sons. These scribes worked out the details of many of his ideas.

Euler's mathematical contributions are immense. In addition to mentoring some highly influential students including Joseph Lagrange, Euler published more than 850 works, including numerous books that span many branches of mathematics and physics. Euler's research contributed to the areas of number theory, combinatorics, probability, infinite series, and partial differential equations, among others. He developed or popularised many familiar pieces of mathematical notation, including introducing the concept of a function and the notation \(\displaystyle f(x)\text{;}\) use of \(\displaystyle \Sigma\) for summations; \(\displaystyle \sin(x), \cos(x)\) and related trigonometric notation; \(\displaystyle e\) for the base of the natural logarithm (known as “Euler's number”); and \(\displaystyle i\) for \(\displaystyle \sqrt{-1}\text{.}\) The identity \(\displaystyle e^{i\pi}=-1\) is named after Euler, as its discoverer.

It would be far beyond the scope of this biographical sketch to even outline the most important of Euler's discoveries, but this book should make it clear that he had a fundamental influence on combinatorics. The contributions discussed in this book begin with his solution to the Königsberg bridge problem introduced in Section 11.1 and covered in detail in Section 13.1, and also include Euler's handshaking lemma, Euler's Formula as covered in Section 15.2, and initiating the study of Latin squares as discussed in Section 16.2. One of his most-read works is the compilation of his Letters to a German Princess written on a variety of topics, in the role Euler had been asked to fill as the princess' tutor.

Euler has been commemorated on stamps in Germany, Russia, and Switzerland, and has appeared on Swiss banknotes. In 1977, an asteroid that had been discovered in 1973 was named “2002 Euler” in his honour. The Mathematical Association of America supports an electronic archive with web pages devoted to each of Euler's works, in addition to writings about him.

Sources: St. Andrews' math history web site, wikipedia, wikipedia, biography.com, Story of Mathematics, and Purdue University.

Subsection B.2.25 Gino Fano (1871—1952)

Gino Fano was an Italian geometer who was instrumental in developing the field of finite geometry. He was born in Mantua, Italy, to wealthy Jewish parents. In that same year of 1871, patriotic troops captured Rome to complete the unification of Italy. Fano studied at the University of Turin under the supervision of Corrado Segre, from 1888 until he was awarded his doctorate in 1892. The contents of his dissertation were published in the paper, “Sopra le curve di dato ordine e dei massimi generi in uno spazio qualunque” (“On the curves of a given order and of maximum genus in any space”).

At Segre's urging, Fano translated Felix Klein's “Erlangen Program” into Italian as an undergraduate, so it was natural that after graduating from Turin, Fano visited Klein to do what would probably now be called postdoctoral work with him in Göttingen from 1893 to 1894. In 1894, Fano moved to Rome, where he worked as the assistant of Guido Castelnuovo through 1898. Fano then spent a few years working in Messina, before accepting an appointment as a professor in Turin in 1901. He held this position until 1938 when at the age of 67 it was stripped from him because of his Jewish descent (along with all of his memberships in Italian scientific institutions and academies), by the Fascist regime.

Fano had married in 1911 and had two grown sons by this time. The family fled Italy; Fano and his wife spent the war years in Lausanne, Switzerland, where he taught in Italian refugee camps and maintained an association with the university. His sons both moved to the United States, where one (Robert) became a professor of computer science at MIT and the other (Ugo) an atomic physicist, reflecting the breadth of topics in which Fano engaged their interest from childhood. (Ugo Fano has reported being told about Bohr's theory of the atom, first introduced just 10 years previously, over dinner at the age of 12.) Fano returned to Italy with all status restored after the war in 1946. He visited his sons in the United States frequently for the remaining 6 years before his death in Verona, Italy, in 1952.

Fano published almost 150 works, including many textbooks. His work focused on projective and algebraic geometry, and he was a pioneer in finite geometry, as evidenced by his discovery of the Fano plane (see Example 18.4.2). Fano gave invited talks at the International Congress of Mathematicians in its inaugural meeting in 1897, and again in 1928. He was committed to public education; beginning in 1905 he was a teacher and organiser for the “Evening School for Women Workers of Turin”, and in 1928 he was awarded a gold medal “Benemerito della Pubblica Istruzione” by the Italian government for this and related work.

Sources: St. Andrews' math history web site, wikipedia, encyclopedia.com, and Unione Italiana Matematica.

Subsection B.2.26 Leonardo Pisano (“Fibonacci”) (c.1170—c.1250)

Leonardo Pisano (“of Pisa”) is more often known as “Fibonacci”, short for filius Bonacci (his father's surname was Bonacci). He was a 13th century mathematician who is largely credited with introducing the digit representations 0 through 9 along with decimal notation into Europe. Fibonacci was born in Pisa, Italy, and grew up in Bugia (now known as “Bejaia”), Algeria, where his father was a diplomat representing the merchants of Pisa. It was in Algeria that Fibonacci was educated in mathematics, giving him an understanding of the Hindu-Arabic number system and of Indian and Arabic mathematical knowledge, which were largely unknown in Europe at that time.

After returning to Pisa in approximately 1200, Fibonacci wrote a number of books, four of which have survived. These were part of his efforts, which also involved travelling and teaching in person, to educate Europeans about the mathematics he had learned. He was a particularly keen proponent of the Hindu-Arabic number system, which used place values and made arithmetic much simpler than the Roman numerals that were being used through most of Europe. With his personal teaching and his most significant book Liber Abaci, the “Book of Calculation”, Fibonacci was highly influential in spreading Arabic numerals to Europe. This was very important to the development of banking and accounting throughout that continent. Fibonacci was a talented mathematician and his understanding and teaching went far beyond the number system itself, certainly into the realms of geometry and number theory.

One of the problems posed and solved in the Liber Abaci involves the speed at which a population of rabbits grows under certain idealised conditions. Calculating the answer, generation by generation, produces what has become known as the Fibonacci sequence (see Section 6.1).

An asteroid discovered in 1982 is named “6765 Fibonacci” in his honour.

Sources: St. Andrews' math history web site, wikipedia, Plus Magazine, and NPR.

Subsection B.2.27 Sir Ronald Aylmer Fisher (1890—1962)

Ronald Aylmer Fisher was a British statistician and geneticist, whose ground-breaking achievements in these fields have been tarnished by his vocal support for eugenics. He was born a twin in London, England (his twin was stillborn). Fisher studied at Cambridge University from 1909 until 1912, earning a First in Mathematics.

From 1913 until 1919 Fisher worked as a statistician in London. He also taught physics and math at a number of schools. He was not eligible to serve in the armed forces during World War I due to poor eyesight. In 1919 Fisher moved to Hertfordshire, and from then until 1933, he worked at the Rothamsted Experimental Station. This was an agricultural research centre, where he used his statistical skills to analyse vast quantities of data. He developed and published a great deal of theory in the course of his work, including in the area of design theory (for the design of experiments). Fisher taught at University College London from 1933—1939, and at Cambridge from 1940 until his retirement in 1956, holding endowed chairs at both places. After retirement, he emigrated to Adelaide, Australia, where he held a research fellowship and lived until his death in 1962.

Fisher held a strong and sustained belief in eugenics, at least from the time he was in university, when he assisted in the formation of a Cambidge University Eugenics Society. More precisely, he believed that “groups of mankind differ in their innate capacity for intellectual and emotional development” and spoke against the 1950 UNESCO statement “The Race Question” on this subject. Even after World War II in a testimony on behalf of a Nazi eugenicist Fisher expressed his belief that the Nazis had “sincerely wished to benefit the German racial stock”. His legacy cannot help but be impacted by these unacceptable views, and some of the honours that were accorded to him have more recently been withdrawn. For this reason, this biography will also be briefer than his accomplishments would otherwise merit.

Despite his unacceptable beliefs, Fisher made huge contributions to the sciences of genetics and statistics, and to design theory. (Fisher's Inequality is stated and proved in Section 17.3.) He unified Darwin's theory of natural selection with Mendel's discoveries about genetic inheritance. He laid down fundamental aspects of the theory of statistics. He received numerous awards and honours, including being knighted in 1952, and being an invited speaker at the International Congress of Mathematicians on two occasions. A minor planet, “21451 Fisher”, was named for him.

Sources: St. Andrews' math history web site, wikipedia, the Royal Society, and University of Adelaide.

Subsection B.2.28 Martin Gardner (1914—2010)

Martin Gardner was an American writer, known particularly for his popular columns about mathematics in the Scientific American. He was born in Tulsa, Oklahoma. Gardner enjoyed math and physics, and initially intended to study physics at university. Then he had trouble with calculus, and never took a course in math after high school. He graduated from the University of Chicago in 1936 with a degree in philosophy. He held a variety of jobs over the next 5 years, including reporting, media relations, and social work. During World War II, he served in the Navy for 4 years.

After the war Gardner returned to the University of Chicago briefly for graduate work, but did not complete a degree. From that time on he was a professional writer: sometimes an editor, sometimes a columnist, and sometimes working freelance. For most of these years, he lived in New York, though he moved to North Carolina in retirement. In 2004, several years after the death of his wife, Gardner moved to Norman, Oklahoma where one of his sons lived. He died there in 2010.

Gardner is best known for his column “Mathematical Games” in the Scientific American. He also gained a notable reputation as a magician, and as a skeptic and debunker of pseudoscience. He wrote not only about puzzles and games, but also traditional mathematical topics such as knot theory, transfinite numbers, and the four-colour problem (as discussed in Section 15.3), in addition to contemporary mathematical discoveries. In relation to his ability to write about even deep mathematical topics in an accessible way, Gardner downplayed his own understanding, saying, “If you are writing popularly about math, I think it's good not to know too much math.” He wrote almost 300 columns over the years, which have been collected and republished in book form. He was very careful to give proper credit and attribution for the ideas he presented; it was through Gardner's columns that notables such as Penrose and the artist Escher first became well-known.

Gardner wrote books, columns, reviews, and articles about many other topics also, including magic, linguistics, and psychics. He was an authority on Lewis Carroll, and wrote annotated editions of several of Carroll's books as well as of a few other classics. Gardner also wrote two novels and a number of short stories. At the age of 95, he wrote an autobiography that appeared posthumously. In all, Gardner wrote or edited more than 100 books.

The asteroid “2587 Gardner” is named for him. The Mathematical Association of America has established an annual lecture that carries his name.

Sources: wikipedia, Scientific American, martin-gardner.org, BBC, and American Math Society.

Subsection B.2.29 Edgar Nelson Gilbert (1923—2013)

Edgar Nelson Gilbert was an American mathematician who spent his career doing research in the areas of combinatorics and probability for Bell Labs. He was born in Woodhaven, New York. In 1940 he began studying physics at Queen's College of the City University of New York, obtaining his B.Sc. in 1943. After a brief stint teaching math at the University of Illinois in Urbana-Champaign, he moved to Massachusetts to study at the Massachusetts Institute of Technology (MIT). He obtained his Ph.D. in physics from MIT in 1948, under the supervision of Norman Levinson. His thesis was entitled “Asymptotic Solution of Relaxation Oscillation Problems”. While at MIT he met and married his wife; the couple had three children.

After completing his doctorate, Gilbert moved to Whippany, New Jersey, where he took a job with Bell Laboratories. His arrival at Bell Labs was just 2 years after Richard Wesley Hamming (1915—1998) began working there, so they were longtime colleagues. Hamming described Gilbert as someone whose ideas he always found very stimulating. Gilbert worked at Bell for 48 years, from 1948 until his retirement in 1996.

During his career, Gilbert made a number of interesting discoveries, chiefly in the areas of combinatorics, optimisation, and probability. He is responsible for one of the models of random graphs discussed in Section 11.5. His combinatorial work included coding theory (stemming from Hamming's discovery of error-correcting codes) as well as graph theory. Gilbert published 85 research papers.

Sources: wikipedia, Daily Record obituary, the Math Genealogy Project, transcription of talk by Hamming, and prezi.com.

Subsection B.2.30 Wolfgang Haken (1928—)

Wolfgang Haken is a German-American topologist, best known in graph theory for his 1976 computer-aided proof, with Kenneth Ira Appel (1932—2013) of the Four-Colour Theorem. He was born Wolfgang Rudolf Gunther Haken in Berlin, Germany, but in 1976 dropped his middle names, changing his legal name to Wolfgang Haken. His father was a physicist who had obtained his doctorate under the supervision of Max Planck, and who worked in the German patent office. At the age of 4, Haken decided that counting should start with 0 rather than 1, and unsuccessfully urged his father to patent this idea. He grew up an only child (his brothers having died of scarlet fever), and his mother died when he was 11, just before the start of World War II.

At the age of 15, Haken was drafted into an anti-aircraft battery, which he served in for the remainder of the war. After working briefly as a farm hand and completing his high school equivalency, Haken entered the University of Kiel in 1946. He graduated with a degree in physics and math in 1948, and continued to study in Kiel, under the supervision of Karl-Heinrich Wiese. He obtained his doctorate in 1953, with a thesis entitled “Ein topologischer Satz über die Einbettung \(\displaystyle (d-1)\)-dimensionaler Mannigfaltigkeiten in \(\displaystyle d\)-dimensionale Mannigfaltigeiten” (“A topological theorem about the embedding of \(\displaystyle d-1\)-dimensional manifolds in \(\displaystyle d\)-dimensional manifolds”). In 1953, Haken also got married; his wife was a student at Kiel who also obtained a Ph.D. in math under the supervision of Wiese, in 1959.

From 1953 until 1962, Haken worked in Munich as an electrical engineer designing microwave devices for Siemans. During this time he continued to complete and publish mathematical research and obtained some impressive results on knot theory and topology that brought him to the attention of academics. He was invited to visit the University of Illinois in Urbana-Champaign (UIUC) for a year in 1962. From there he moved to Princeton (New Jersey), where he worked at the Institute for Advanced Studies for two years before returning to the UIUC as a tenured professor in 1965. He remained there until his retirement in 1998. Almost all of Haken's 6 children went to graduate school and many of them are involved in research or academics.

Haken has always been a keen outdoorsman. In 1956 he fell more than 30 feet while mountain climbing in the alps, a near-fatal accident. He was in a coma for several days, and one of his feet was damaged. He remained an avid hiker, and has been an active participant in the UIUC math department's regular Saturday hikes for decades.

Haken was an invited speaker at the International Congress of Mathematicians in 1978, and received the American Math Society's Fulkerson Prize jointly with Appel in 1979, for the Four-Colour Theorem, discussed in Section 15.3.

Sources: wikipedia, University of Illinois, the Math Genealogy Project, and Project Euclid.

Subsection B.2.31 Halayudha (10th century CE)

Halayudha was a 10th-century Indian mathematician, poet, and social reformer. Biographical information on Halayudha is contradictory and incomplete. Some sources write of a poet by this name, and others of a mathematician, with different writings attributed to each, while some assert that there was a single person who wrote these books. I will follow those who attribute everything to one individual, but the most definitive source I found on the poet has very detailed biographical information without any mention of the mathematical work.

Halayudha lived in India in the 900s (CE), originally in Manyakheta, where he wrote his poetry. He later moved to Ujjain, where he wrote a commentary “Mrta-sañjīvanī”, on a mathematical book by the Indian mathematician Pingala. In this commentary Halayudha writes of the arithmetic triangle as representing Mount Meru, the holy mountain, and discusses its use in describing the number of combinations of long and short syllables that can be included in a line with a fixed number of syllables. This work is described in Section 4.2 and mentioned again in Section 4.3. It is not clear to what extent these ideas were present in the earlier work of Pingala. Halayudha was also famous for his efforts as a social reformer.

Sources: wikipedia, Dhanapāla and His Times, p. 228, Ton Smith, wisdomlib.org, Sanskrit discussion group, The Calcutta Review (pp. 168—183), and Mathematical Reasoning (p. 146).

Subsection B.2.32 Philip Hall (1904—1982)

Philip Hall was a British algebraist, whose fundamental discoveries in group theory set the stage for much of the more recent research in this field. He was born in London, England; his parents were not married and his father abandoned the family shortly after his birth. His mother used his father's surname for both herself and her son. Hall was too young to be drafted in World War I, but did serve in the Officer's Training Corps as a youth. Hall was raised to be very careful with money, which was not plentiful for his mother. He was extremely generous to students, young people, and charities later in life, when he had more funds.

Hall entered Cambridge University in 1922, and graduated in 1925. He did some work as a research assistant at University College, London, and in 1927 obtained a fellowship to return to Cambridge by writing a dissertation, “The Isomorphisms of Abelian Groups”. After a series of fellowships he was appointed as a Lecturer in 1933. During the war years from 1941 to 1945, Hall worked at Bletchley Park, the centre of British efforts to break enemy codes and ciphers. His focus was on Italian and Japanese ciphers, and he learned to read Japanese characters as part of this work (he subsequently took great pleasure in Italian and Japanese poetry).

In 1945 Hall returned to Cambridge. He worked his way up through the ranks and was promoted to the Sadleirian Chair when it was vacated in 1953. He served as president of the London Mathematical Society from 1955—1957. Hall was significantly tied down by caring for his elderly mother in the late 1950s and early 1960s. His mother died in 1965 at the age of 93, and Hall retired two years later. In his retirement he engaged in a hobby of studying family connections between people who lived to be at least 90 (like his mother). Another retirement hobby of his was memorising a sonnet each day.

Hall's Theorem, published in 1935, and discussed in Section 16.3 notwithstanding, Hall's work was almost exclusively in the area of group theory, where he made many fundamental discoveries. (His motivation for Hall's Theorem also came from group theory.) Despite the coincidence of surnames, he should not be confused with Marshall Hall, Jr., who wrote one of the best-known reference books on group theory.

Hall's influence as a mentor was also significant. He supervised at least 34 doctoral students. One of the undergraduate students for whom he served as tutor was Alan Turing, with whom he later worked at Bletchley Park. Hall's opinion was highly influential when it came to making decisions about young people who were seeking fellowships, and he favoured the adventurous. One young person was turned down when Hall described him as “somewhat too reasonable” while another was elected after an admiring comment on his “almost repellant originality”.

Hall was elected a Fellow of the Royal Society in 1951, and in 1961 was awarded the Sylvester Medal. He also received the London Math Society's Senior Berwick Prize in 1958, and the Larmor Prize and De Morgan Medal in 1965. Hall was invited to speak at the International Congress of Mathematicians in 1940 and again in 1950. He was unable to attend on either occasion (due to the war in 1940 and for family reasons in 1950).

Sources: wikipedia, St. Andrews' math history web site, the Math Genealogy Project, the Royal Society, and the London Math Society.

Subsection B.2.33 Sir William Rowan Hamilton (1805—1865)

William Rowan Hamilton was a 19th century Irish astronomer and mathematician, best known for his discovery of the quaternions. He was born in Dublin, Ireland. By the age of 3, Hamilton had been sent to his uncle in Trim (a curate who ran a local school) to be educated. His uncle, a linguist, trained Hamilton in languages, and by the age of 13 he had at least a basic understanding of an impressive number. These included French, Italian, Greek, Latin, Hebrew, Persian, Arabic, Hindustani, Sanskrit, Marathi, and Malay.

In 1823 Hamilton enrolled at Trinity College in Dublin. He studied math and the classics, and obtained a bachelor's degree in 1827 and a Master's in 1837. While still an undergraduate student he was appointed Andrews professor of Astronomy, which came with the title of Royal Astronomer of Ireland. From that time he lived at Dunsink Observatory until his death in 1865.

Although Hamilton's right to have his name associated with Hamilton cycles and paths (as discussed in Section 13.2) is questionable, he does deserve credit for many important discoveries in math and physics, including results in optics, dynamics, and the discovery of the quaternions. Quaternions exist in four dimensions, but can be identified with three-dimensional geometry in ways somewhat similar to the identification of complex numbers with the plane, and using this identification results in the fastest known methods for determining the outcome of successive rotations in three-dimensional space. Quaternions are therefore frequently used in computer graphics and in spaceship navigation, among other applications. In the context of this book, it should be noted that Hamilton also invented and studied “Icosian Calculus”, which is really the group of symmetries of the dodecahedron. He used Hamilton cycles on the dodecahedron to help him understand the structure of this group.

Hamilton had a lifelong interest in poetry (first spurred by an early disappointment in love) and became friends with the poet Wordsworth. Wordsworth preferred the poetry of Hamilton's sister Eliza to Hamilton's own, and at one point wrote Hamilton, asking him pointedly “whether the poetical parts of your nature would not find a field more favourable to their nature in the regions of prose, not because those regions are humbler, but because they may be gracefully and profitably trod, with footsteps less careful and in measures less elaborate”.

Hamilton was knighted in 1835, and was elected president of the Royal Irish Academy in 1837, a position he retained until 1846. He has two research institutes named for him, as well as a public lecture. Commemorative Irish stamps and a silver proof coin were also designed and issued in his honour.

Sources: wikipedia, St. Andrews' math history web site, Physics World, New World Encyclopedia, and the Irish Times.

Subsection B.2.34 Richard Wesley Hamming (1915—1998)

Richard Wesley Hamming was an American mathematician whose discovery of error-correcting codes had significant impact on the development of telecommunications and computer engineering. He was born in Chicago, Illinois (in the United States) in 1915. He was applying to universities during the Great Depression, and the only scholarship offer he received was to the University of Chicago. Hamming wanted to study engineering, but Chicago did not have an engineering school, so he completed a degree in science, graduating in 1937.

Hamming went to the University of Nebraska for his Master's degree, graduating in 1939, and returned to Illinois for his Ph.D. studies at the University of Illinois at Urbana-Champaign. His doctoral thesis was “Some Problems in the Boundary Value Theory of Linear Differential Equations”, written under the supervision of Waldemar Trjitzinsky, and he completed it in 1942. He was also married in 1942.

After teaching for a couple of years at the University of Illinois, in 1944 Hamming took a position at the University of Louisville in Kentucky. He left this job in 1945 to work on the Manhattan Project (development of the atomic bomb) at the Los Alamos Laboratory in New Mexico. He had no idea of what he was getting into; a friend who was working there wrote him only that there was “something interesting going on down here. Come down and work.” Hamming's position involved programming the IBM calculating machines to compute solutions to equations for the physicists on the project. His wife also joined the project as a human computer.

Hamming wrote a humorous account about having the gravity of the Manhattan Project brought home to him when he was asked to check calculations for the probability that the next test would ignite earth's entire atmosphere, but this seems not to have deterred him too much. The lesson he took from it was that “Mathematics is not merely an idle art form, it is an essential part of our society.” He did say in an interview that the Los Alamos lab resembled “the mad scientist's laboratory”, and his experiences there brought his keen interest in science fiction to an abrupt end. He remained with the project until 1946, when he took a position at Bell Telephone Labs. In fact, he stayed on at Los Alamos for 6 months after most others had left, feeling a responsibility to create a written record of the work that had been done there.

Hamming continued to work with calculating machines at Bell Labs. At the time, the machines used a parity check-bit for each block of bits in any bit string, so they could determine if an error occurred, but they simply terminated the active process if an error did arise. After experiencing the frustration of setting a computer to run calculations over a weekend only to discover on Monday that the process had terminated in an error, it occurred to Hamming that if the computer could determine which bit is incorrect, then it could correct the error and proceed. He introduced the theory of error-correcting codes (along with the Hamming distance metric) in a landmark 1950 paper. (See Chapter 19 and in particular Section 19.2 for more about Hamming distance and error-correcting codes.) This introduced a new field of study, as well as setting the stage for a quantum leap in the effectiveness of telecommunications and computer technology.

Hamming retired from Bell Labs in 1976, having evaded or avoided management responsibilities throughout his time there. He later saw this as a failure on his part. He held a number of visiting or adjunct appointments at universities during his time at Bell, including positions with Stanford and with Princeton. These positions allowed him to do some teaching in addition to his research. In 1976 Hamming moved to Monterey, California, where he worked in the computer science department at the Naval Postgraduate School. In 1997 he retired from this position and became a professor emeritus. He died soon thereafter, in 1998. Hamming was known to say: “The purpose of computing is insight, not numbers,” and to students: “If you don't work on important problems, it is not likely that you will do important work.”

Hamming acted as president for the Association for Computing Machinery (ACM) from 1958 to 1960. He earned many honours, beginning by winning the ACM's Turing Award in 1968; he became a Fellow of the ACM in 1994. He became a Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 1968, and won the IEEE's Emanuel R. Piore Award in 1979 and its Computer Society Pioneer Award in 1980. In 1980 Hamming was elected a member of the National Academy of Engineering. He won the Harold Pender Award from the University of Pennsylvania in 1981. Fittingly, he was named the first recipient of the IEEE's “Richard W. Hamming Medal” in 1988; this is an annual award (with a prize of $10,000) given for “exceptional contributions to information sciences, systems, and technology”. Hamming won the Rhein Foundation's Basic Research Award in 1996. He was also given the Navy's Distinguished Public Service Award posthumously.

In addition to his foundational contributions to coding theory, Hamming made significant advances in numerical analysis, numerical integration, and numerical filtering, and along with Ruth Weiss developed one of the earliest programming languages. He also worked on many other projects, and published more than 25 research papers. He became convinced that the ways in which we teach mathematics need to change, and wrote at least 8 textbooks using unconventional approaches.

Sources: wikipedia, St. Andrews' math history web site, IEEE, National Academies Press, and American Mathematical Monthly.

Subsection B.2.35 Percy John Heawood (1861—1955)

Percy John Heawood was a British mathematician, best known in graph theory for his contributions to the Four-Colour Conjecture. He was born in 1861 in Newport, England. He completed his studies at Oxford, beginning in 1880 and remaining there until 1887.

Heawood became a Lecturer in Mathematics at Durham University in 1887. He worked there throughout his career, and was appointed to the Chair of Mathematics in 1911. From 1926 until 1928 he was the Vice-Chancellor of the university. Heawood retired in 1939 at the age of 78.

One focus of Heawood's research throughout his life was the Four-Colour Conjecture. In 1890 he pointed out the error in Sir Alfred Bray Kempe (1849—1922)'s proof (which had been accepted for 11 years at that point) and adapted Kempe's proof to show that five colours suffice (the Five-Colour Theorem), as discussed in Section 15.3. He wrote 6 additional papers on the topic over the course of his career, with the last one in 1949. He also wrote papers on continued fractions, quadratic residues, and geometry. There is a graph on 14 vertices that is named after him. (This graph can be embedded on a torus so that 7 colours are required to colour its faces, and Heawood proved that seven colours suffice in general on a torus.)

Heawood lived to the age of 93, and was married for over 60 years (with 2 children). He had a singular and eccentric appearance. Gabriel Andrew Dirac (1925—1984) wrote about Heawood in the Journal of the London Math Society: “He had an immense moustache and a meagre, slightly stooping figure. He usually wore an Inverness cape of strange pattern and manifest antiquity, and carried an ancient handbag. His walk was delicate and hasty, and he was often accompanied by a dog, which was admitted to his lectures.”

Durham Castle required extensive funds in 1928 to save it, as there was a problem with its foundation. The University of Durham tried to fundraise for the restoration of the castle, but only raised about one-fifth of the £150,000(GBP) that were ultimately required. Heawood doggedly continued working to fundraise for this cause for years, and ultimately succeeded. For these efforts he was awarded the Order of the British Empire in 1939.

Sources: wikipedia, St. Andrews' math history web site, Journal of the London Math Society, wikipedia, and the Times obituary.

Subsection B.2.36 Jeannette Janssen (1963—)

Jeannette Janssen is a Dutch-Canadian combinatorist, best known for her work in graph theory. She was born in the Netherlands in 1963. She enrolled at Eindhoven University of Technology in the Netherlands in 1982, and earned her first university degree there, graduating in 1988. She then spent two years lecturing at the Universidad de Guanajuato in Mexico. Janssen moved to the United States for her Ph.D., studying at Lehigh University in Bethlehem, Pennsylvania under the supervision of Edward Assmus, Jr. She completed her doctorate in 1993 with a dissertation entitled “Even and Odd Latin Squares”.

After her Ph.D., Janssen took a postdoctoral position in Montreal, Canada, working jointly at the Université du Québec à Montréal and Concordia University. She also worked for 2 years at the London School of Economics in England, followed by a year at Acadia University in Nova Scotia, Canada, before moving to Dalhousie University in Halifax, Nova Scotia in 1998. Janssen has remained at Dalhousie University, where she became the first female chair of the math department in 2016. She directed the Atlantic Association for Research in the Mathematical Sciences from 2011 to 2016. Janssen was also chair of the Discrete Math Activity Group for the Society of Industrial and Applied Math for 2021—2022.

Janssen's research focuses on graph theory, in particular graphs as models of complex networks, and graph colouring. One of her most notable results can be expressed in graph theoretic terms, but also relates to extending Latin rectangles. Another of her results is discussed in Section 11.5. Janssen has published more than 60 research papers, and has supervised 5 Ph.D. students as of 2021.

Sources: Dalhousie University, wikipedia, LinkedIn, and the Math Genealogy Project. Updated and confirmed through personal communication.

Subsection B.2.37 Peter Keevash (1978—)

Peter Keevash is one of the foremost combinatorists of our time, best known for his resolution of the long-standing existence problem for combinatorial designs. He was born in London, England in 1978. His family subsequently moved to Brighton, and then to Leeds. Keevash showed an early interest in mathematics. He was a member of the team representing England at the International Mathematical Olympiad (a competition for high school students) in 1995, where he earned a bronze medal.

In 1995, Keevash enrolled at Trinity College of the University of Cambridge. He completed his bachelor's degree there in 1998. At this point he moved to the United States for his doctoral studies. In 2004 Keevash completed his Ph.D. at Princeton University in New Jersey, under the supervision of Benjamin Sudakov. His thesis was entitled “The Role of Approximate Structure in Extremal Combinatorics”.

Keevash held a postdoctoral position at the California Institute of Technology in Pasadena, before moving back to England to work at Queen Mary, University of London, initially as a lecturer and then as a professor. In 2013 Keevash accepted a position at Oxford University.

Keevash has made major contributions to multiple areas of combinatorics, including extremal combinatorics, graph theory, Ramsey theory, and design theory. He proved the best known lower bound for the Ramsey number \(\displaystyle R(3,k)\text{.}\) However, he is best known for his 2014 result in design theory, specifically Keevash's Theorem mentioned in Section 18.2, establishing the existence of \(\displaystyle t-(v,k,\lambda)\) designs whenever the necessary conditions are met and \(\displaystyle v\) is sufficiently large. Keevash has more than 70 publications in all.

Keevash won the European Prize in Combinatorics in 2009. In 2018, he gave an invited talk at the International Congress of Mathematicians.

Sources: Oxford University, Oxford University, wikipedia, International Mathematics Olympiad, and the Math Genealogy Project. Updated and confirmed through personal communication.

Subsection B.2.38 Sir Alfred Bray Kempe (1849—1922)

Alfred Bray Kempe was a British barrister, who also carried out noteworthy mathematical research. He is best known for his attempted solution of the Four-Colour Conjecture. He was born in London, England, and studied at Trinity College, Cambridge University. After graduating with a B.A. in 1872, Kempe became a barrister in 1873. His father was a minister, and Kempe specialised in ecclesiastical law, ultimately becoming Chancellor (legal advisor) for several dioceses. He was highly respected in his chosen profession.

Despite his career in law, Kempe maintained an active interest in mathematics. He published papers sporadically from the 1870s through the 1890s, with his first paper appearing in the year he graduated from Cambridge. In 1881 he was elected a fellow of the Royal Society, and served as its treasurer and vice-president for 21 years. While he held this role, the Royal Society developed the National Physical Laboratory from its inception and ultimately handed it over to the government after 16 years. Kempe also served as president of the London Math Society (LMS) from 1892 to 1894 (many biographies list other years for this service, but these are the dates listed by the LMS).

It could be considered a bit sad that Kempe is best known for incorrect proofs. In addition to his “proof” of the Four-Colour Conjecture in 1879 that is mentioned in Section 15.3, in 1876 Kempe published a flawed proof of a result now known as the Kempe Universality Theorem, about a connection between linkages and algebraic curves. This theorem was ultimately proved in 2002, with somewhat different ideas, but a second proof in 2008 was based on Kempe's methods. His techniques of Kempe chains and unavoidable sets were used by Kenneth Ira Appel (1932—2013) and Wolfgang Haken (1928—) in their eventual proof of the Four-Colour Theorem. So for both of these results (as well as for the Five-Colour Theorem), a correct proof ultimately used several of the techniques that Kempe developed for the “proof”s that he published. It is also no mean accomplishment that on a popular topic like the Four-Colour Conjecture, it took researchers 11 years to notice the flaw in Kempe's proof. Kempe also discovered the Petersen graph a decade before Petersen did, but ironically did not notice its relationship to the Four-Colour Conjecture.

Kempe loved the mountains, and travelled to Switzerland frequently to walk and climb in the Alps. He was also devoted to music. He was married twice, and had three children by his second marriage. He received an honorary degree from the University of Durham in 1908, and was knighted in 1912.

Sources: wikipedia, St. Andrews' math history web site, Royal Society obituaries, MIT thesis proving the Universality Theorem with Kempe's ideas, London Math Society, and Nature magazine obituary.

Subsection B.2.39 Reverend Thomas Penyngton Kirkman (1806—1895)

Thomas Penyngton Kirkman was a 19th-century British minister, who also made extensive contributions to combinatorial research, particularly in the areas of graph theory and design theory. He was born in Bolton, England (and baptised “Thomas Pennington”). The local schoolmaster tried to persuade his family to send him to Cambridge, guaranteeing that he would win a scholarship, but his father (a cotton dealer) would not let him go. At the age of 14 he left school to work for his father. In 1829 he rebelled and left his father's office to attend Trinity College Dublin, working as a tutor to cover his expenses. His schooling had not included any math at all, but he picked up the background on his own. He completed his bachelor's degree in 1833 and moved back to England in 1835.

After university, Kirkman entered the church. He was ordained and became curate of Bury, and later of Lymm. In 1839 he was appointed rector at the parish of Croft with Southworth. He remained there until his retirement 52 years later, in 1892 at the age of 86. Kirkman married in 1841 and had 7 children. He earned extra income through tutoring until his wife came into an inheritance that together with his income from the church sufficed to support the family. Upon his retirement, Kirkman moved a few miles away to Bowdon, where he died in 1895. His wife died just 10 days later.

Kirkman's ministerial duties were light and he had plenty of time for mathematics, particularly after he gave up tutoring. His first publication came in 1846, and he continued to work on math problems until his death. Kirkman was no mere dilettante; he made a number of significant contributions to combinatorics (only some of which are mentioned in this book), and also published work on geometry, group theory, and knot theory. Over the course of his life he published more than 60 substantial papers in addition to many minor ones. His work on Hamilton cycles is mentioned in Section 13.2, and his work in design theory is discussed in Section 18.1.

Kirkman was elected a fellow of the Royal Society in 1857. The institute of Combinatorics and its Applications awards a Kirkman medal annually, named in his honour.

Sources: wikipedia, St. Andrews' math history web site, Lecture by Alexander MacFarlane, and Bulletin of the London Math Society.

Subsection B.2.40 Eszter Klein (1910—2005)

Eszter Klein was a Hungarian-Australian mathematics educator. She was born in Budapest, Hungary, to a single mother who was devoted to Judaism (a devotion her daughter did not inherit). Klein is also known by the anglicised version “Esther” of her name, and by her married name of Szekeres, or the hyphenated Szekeres-Klein. During her high school years, Klein was a successful problem solver in the Hungarian High School Mathematical and Physical Monthly which published her photograph in the tableaux of the top 32 problem solvers for each of the years 1926—27 and 1927—28. In each of these years, the tableaux included 30 boys and 2 girls — Klein, and her classmate Márta Wachsberger from the Jewish Girls' High School.

In spite of laws in Hungary discriminating against people of Jewish heritage in university admissions, in 1928 Klein was admitted to the top university of science in the country (now named for physicist Loránd Eötvös (1848—1919)). At the university she became part of a brilliant group of young mathematicians and friends that included Tibor Gallai (who would go on to become a remarkable teacher and an eminent graph theorist, see the biographical sketch of Vera Sós (1930—)), Pál Erdős (1913—1996), György Szekeres (1911—2005), and Pál Turán (1910—1976). They met regularly to talk about math together. Klein, having first solved it in the case of quadrilaterals, posed the “Happy Ending Problem” (see Section 14.2) to this group, and it deepened their relationships. Erdős and Szekeres later published bounds for the general solution to the problem, and in 1937 Klein and Szekeres were married.

Szekeres had been studying chemical engineering, and spent 6 years working in Budapest as a chemist after graduation, to the late 1930s. During this period, Klein and Szekeres were not able to afford to live together, and saw each other only on weekends. By 1939 Hungary was an ally of Nazi Germany, and an increasingly menacing place for people of Jewish descent. Klein moved with Szekeres to Shanghai (which surprisingly and remarkably served as a safe haven for 20,000 Jewish refugees from Europe) to escape the Nazis. On their first arrival in China, exuberant at having escaped from Europe, they decided to start their family, and their oldest child, a son, was born in Shanghai in 1940. However, life in Shanghai was not easy. They lived through the Japanese occupation and the start of the Chinese Communist revolution. For years, they were without work, living on meagre refugee aid. But while back in Hungary Klein's mother was murdered in the Holocaust, the Szekeres-Klein family in Shanghai survived.

In 1948 Szekeres was offered a position at the University of Adelaide, and the family moved to Australia. For several years after moving to Adelaide, they shared an apartment with the family of Klein's high school friend Márta Wachsberger, now Marta Sved. Klein's second and last child, a daughter, was born in 1954. Klein taught high school math and also tutored at the University of Adelaide, while raising the children.

In 1964 the family moved to Sydney, where Szekeres had been offered a professorship at the University of New South Wales. Klein became a lecturer in math at Macquarie University in Sydney. After their retirement, in 1984 Klein and Szekeres established a weekly problem-solving session as well as enrichment lessons for talented high school students, held at Mercy College. Klein supplied geometry problems and attended these sessions weekly until 2002, when it became too difficult for her. This program has expanded to about 30 groups across Australia and New Zealand.

In 2004 Klein and Szekeres moved back to Adelaide. Not long thereafter, Klein had to move into a nursing home, and Szekeres joined her there in late 2005. Seven weeks later, the two died together, within an hour of each other, after almost 70 years of marriage.

In 1990 Macquarie University gave Klein an honorary doctorate. In 1993, she won the B.H. Neumann Award from the Australian Mathematics Trust. This award is given for significant contribution to teaching mathematical problem-solving in Australia.

Sources: wikipedia, Quanta Magazine, Sydney Morning Herald, Women and Math blog, Australian Academy of Science, and Australian Math Society. Additional information and clarification by personal communication from Laci Babai.

Subsection B.2.41 Tamás Kővári (1930—2010)

Tamás Kővári was a Hungarian-British analyst. He was born in Budapest, Hungary. In later years he often used the anglicised version of his name (“Thomas”). He attended József Eötvös College in Baja, Hungary, beginning in 1948. He graduated from Loránd Eötvös University in Budapest and received his Ph.D. from the Hungarian Academy of Sciences. Kővári was also married in Hungary; his wife had a degree in physics and math from Eötvös University.

In October 1956 there was an uprising in Hungary against the Communist rule and Soviet occupation. It was quickly crushed by Soviet tanks. Before the borders were sealed, 200,000 people escaped the country. The refugees included Kővári and his wife. Eventually they reached London, England, where Kővári became a lecturer and later Reader at the University of London, from 1957 to his retirement in 1995. While working at the University of London, Kővári learned that an English qualification would be useful to him. He completed a Ph.D. under the supervision of Walter Hayman in 1961, with a thesis entitled, “On the Borel Exceptional Values of Lacunary Integral Functions”.

Kővári and his wife had two children. The couple divorced in 1972, and Kővári produced very little new mathematics after this time. Kővári enjoyed music, particularly classical and disco. He also loved sweets, and was in a chocolate-tasting club.

It was in 1954, while still in Hungary, that Kővári's joint paper with Sós and Turán was published, giving an upper bound on the number of edges in a bipartite graph with bipartition sets of given size, before it necessarily contains a particular complete bipartite subgraph. This is probably Kővári's best-known work, and is stated more precisely in Section 14.2. He published more than 20 papers during his career; aside from this result in graph theory, all of his other research related to functions of complex variables.

Sources: London Math Society obituary, Stanford University, My Journey Home: Life after the Holocaust, and the Math Genealogy Project, as well as MathSciNet for a publication list. Additional information and clarification by personal communication from Laci Babai.

Subsection B.2.42 Kazimierz Kuratowski (1896—1980)

Kazimierz Kuratowski was a Polish topologist, best known for his fundamental contribution to graph theory, Kuratowski's Theorem characterising planar graphs. He was born in Warsaw, Poland. Poland at that time was partitioned among Russia, Prussia, and Austria, and Warsaw was under the control of the czar of the Russian empire. Russia had turned the University of Warsaw into a Russian-language institution in 1869, and stopped permitting high school students to study in Polish. Although Polish high schools were allowed by the time Kuratowski was in high school, students attending such schools had to compete in examinations as external candidates for places in universities. As a result, many went abroad for university.

Kuratowski wanted to become an engineer. He attended the University of Glasgow in Scotland, which had a strong reputation in this area. Kuratowski began his studies there in 1913. After he returned home at the end of his first year, the outbreak of World War I made it impossible for him to return to Scotland to continue his studies. One year later in 1915, Russia withdrew from Warsaw. Germany took control, and changed the University of Warsaw back to being a Polish-language university. Kuratowski was one of the first students when it reopened in this form. At the end of World War I Poland became independent. Kuratowski graduated in 1919 and began doctoral work under Stefan Mazurkiewicz and Zygmunt Janiszewski, remaining at the University of Warsaw. Although Janiszewski died in 1920 at the age of 31, of the “Spanish flu” pandemic, Kuratowski was able to complete his Ph.D. in 1921.

Kuratowski moved to Lwów (now Lviv, Ukraine) in 1927, where he worked at the Technical University of Lwów. He married in 1929. He maintained a home near Warsaw that he visited each summer, and moved back to work at the University of Warsaw in 1934, having been offered a newly-established chair. While maintaining an active research career, from this time forward he also devoted significant effort to the development of mathematics in Poland.

In 1939, Nazi Germany invaded Poland. Many academics were killed or sent to concentration camps, and universities were closed. Kuratowski was reportedly of Jewish descent, and apparently had some narrow escapes from the Gestapo, but managed to survive in Warsaw. He also further risked his life to teach in one of the illegal universities that the Poles secretly established. Of this time, he wrote: “The importance of clandestine education consisted among others in keeping up the spirit of resistance, as well as optimism and confidence in the future, which was so necessary in the conditions of occupation. The conditions of a scientist's life at that time were truly tragic. Most painful were the human losses.” He was reportedly completely imperturbable in later life, saying that after what he had already lived through during the war, nothing could ever upset him anymore.

After World War II education in Poland had to be rebuilt, and Kuratowski was at the centre of the mathematical part of this work. He served as president of the Polish Math Society for 8 years immediately after the war. He became director of the Mathematical Institute of the Polish Academy of Sciences in 1949, shortly after its establishment, and held this role for 19 years. He also served as a vice president of the Polish Academy of Sciences from 1957 to 1968, and of the International Mathematics Union from 1963 to 1966.

Kuratowski's research was primarily in topology and set theory, but he also discovered one of the central results in graph theory, Kuratowski's Theorem. The notation for complete graphs derives from his name (as mentioned in Section 11.3). He produced more than 150 publications, and his mentoring was broad and very influential. Kuratowski was elected to a number of national academies of science, and also received several honorary degrees. The Polish Mathematical Society has established a prize that is named for him.

Sources: wikipedia, St. Andrews' math history web site, the Math Genealogy Project, Prominent Poles, and obituary in the Polish Review.

Subsection B.2.43 Clement Wing Hong Lam (1949—)

Clement Wing Hong Lam is a Chinese-Canadian mathematician and computer scientist, best known for his computer-aided proof that there is no projective plane of order 10. He was born in Hong Kong, but left in 1968 to study at the California Institute of Technology, where he completed a B.Sc. in 1971 and a Ph.D. in 1974. His Ph.D. thesis, under the supervision of Herbert Ryser, was entitled “Rational \(\displaystyle g\)-circulants satisfying the matrix equation \(\displaystyle A^2=dI+\lambda J\)”. He spent a year as a visiting assistant professor at the University of Calgary in Canada.

Lam moved to Montreal in 1975, where he took up a position in the computer science department at Concordia University. He remained there until retirement, and is now an emeritus professor there.

Lam determined that there is no projective plane of order 10 (equivalently, no affine plane of order 10, and no set of 9 mutually orthogonal latin squares of order 10), as mentioned in Section 16.2. In his expository paper “The Search for a Finite Projective Plane of Order 10”, Lam wrote that his advisor Ryser had warned him against working on the projective plane of order 10 as his thesis topic as it might not lead to a successful thesis. So it was not until 1980 that Lam began working on this problem. It took almost 10 years of theory, algorithm development, letting the computers run, and patching issues before the result was complete in 1989. (Premature publicity came about on a couple of occasions before this.)

In 1992, Lam received the Lester Randolph Ford Award from the Mathematical Association of America for his expository article. He received the Euler Medal in 2006 from the Institute of Combinatorics and its Applications.

Sources: wikipedia, Concordia University, the Math Genealogy Project, and Mathematical Association of America. Updated and confirmed through personal communication.

Subsection B.2.44 Lu Jiaxi (1935—1983)

Lu Jiaxi was a Chinese teacher of physics who worked under very adverse conditions and received little recognition during his lifetime for his mathematical research despite proving impressive results in design theory. Lu was born in Shanghai, the only survivor of his parents' four children. After his father died when he was 14, he began working to help support himself. He taught himself high school material while working, as well as learning Russian, English, and Japanese in order to be able to read research papers.

Lu was admitted to university at Jilin Normal University (now Northeast Normal University) in 1957 to study physics. After graduating in 1961, he became a physics teacher but was also in charge of a school-run factory producing radio components. He had learned of the Kirkman Schoolgirl Problem in 1956 from a popular science book, spent a lot of time considering it while at university, and in 1961 wrote up his proof that solutions exist whenever the obvious necessary conditions are satisfied. Unfortunately, four years were wasted as he submitted his work to journals that were not suitable for high-level original research. Ultimately his work was inappropriately rejected from Acta Mathematica Sinica in 1965 as “not really new”. The proof by Dwijendra Kumar Ray-Chaudhuri (1933—) and Richard Michael Wilson (1945—) was announced in 1968. Lu's proof has since become available in his collected works.

At this point the Cultural Revolution disrupted Lu's ability to conduct research. In 1972 he was married, and he and his wife started a family. Lu heard about the published solution to the Kirkman Schoolgirl Problem when he managed to resume acquiring copies of research papers to study, in the late 1970s. Although he found this discouraging, he moved on to study the problem of large sets of disjoint Steiner Triple Systems. He published a series of papers on this and on resolvable BIBDs in the early 1980s, mostly in international journals.

Lu remained relatively unknown in China despite increasing international recognition. John Adrian Bondy (1944—) had served as a referee for one of Lu's papers and asked to meet him when attending a conference in China in the summer of 1983, but the local mathematicians did not know who he meant. After this Lu began to be invited to conferences. The school where he worked was unsupportive of his research, refusing to assist him in attending conferences and assigning him extra duties since he apparently had time for other work.

Lu died suddenly of a heart attack on October 30, 1983, at the age of 48.

Sources: wikipedia, “Collected Works of Lu Jiaxi on Combinatorial Designs”, “Triple Systems” by Charles Colbourn, p. 8.

Subsection B.2.45 Gary McGuire (1967—)

Gary McGuire is an Irish mathematician, best known for his computer-aided proof that at least 17 clues are required in a uniquely-solvable Sudoku puzzle. He was born in Dublin, Ireland. He obtained his bachelor's and master's degrees from University College Dublin, and then moved to the United States for his doctorate. McGuire completed his Ph.D. in 1995 at the California University of Technology under the supervision of Richard Michael Wilson (1945—). His thesis was entitled “Absolutely Irreducible Curves with Applications to Combinatorics and Coding Theory”.

After completing his doctorate, McGuire held posts at the University of Virginia and the National University of Ireland, Maynooth before returning to University College Dublin, where he is now a Professor of Mathematics. McGuire has produced more than 80 publications, and has supervised 7 Ph.D. students as of 2021.

McGuire was able to show that at least 17 clues are required in a Sudoku that has a unique solution. It was previously known that there are uniquely-solvable puzzles with 17 clues. Although no 16-clue puzzles had been found, it was not certain that one might not exist. Together with his former student Bastian Tugemann who worked with him on this for several years, and with computational assistance from Gilles Civario (1972—) of the Irish Centre for High-End Computing, McGuire proved that no 16-clue puzzle has a unique solution, as mentioned in Section 16.1. This result was officially announced in 2012 and was covered in news stories worldwide, including prominent publications such as Nature and Scientific American.

Sources: the Irish Times, University College Dublin, the Math Genealogy Project, University College Dublin, and Nature. Updated and confirmed through personal communication.

Subsection B.2.46 Wendy Joanne Myrvold (1961—)

Wendy Joanne Myrvold is a Canadian computer scientist specialising in graph algorithms. She was born in Sarnia, Ontario, Canada. In her youth her family moved to Signal Mountain in Tennessee. In 1979 she went to McGill University in Montreal, Canada. After switching out of physiology and physics, she graduated with a degree in computer science in 1983. During the summers, Myrvold worked at an amusement park, maintained the software for memory testing the first PCs manufactured at IBM, and provided computational support for the chemistry labs at Imperial Oil and Syncrude.

Myrvold moved to Waterloo for her Master's degree. She began working in numerical analysis, but fell in love with graph theory during a course she took from John Adrian Bondy (1944—). She completed her master's degree in combinatorics and optimization in 1984. Her doctorate in computer science at Waterloo was under the supervision of Charles Colburn, and her thesis was entitled “The Ally and Adversary Reconstruction Problems”. After completing her Ph.D. in 1988, Myrvold took a position in computer science at the University of Victoria. She was a professor there until her retirement in 2018, and is now a professor emeritus.

Myrvold has done research on graph theory, graph algorithms, network reliability, topological graph theory, Latin squares, and chemical graph theory. She has more than 60 publications, and had supervised 5 doctoral students as of 2021. The edge addition algorithm mentioned in Section 15.1 that she developed with John M. Boyer (1968—) for either determining that a graph is non-planar, or finding a planar embedding, is the state of the art for this problem.

Myrvold is married with 2 children. She fell in love with ice hockey at McGill and it became a lifelong passion. Her hockey highlights include playing varsity hockey at McGill, competing in the Western Shield and the BC Senior games, and playing with Wayne Gretzky at his Fantasy hockey camp. She also is an ardent skier and loves dance fitness classes.

Sources: wikipedia, University of Victoria, the Math Genealogy Project, and LinkedIn. Updated and confirmed through personal communication.

Subsection B.2.47 Ernest Tilden Parker (1926—1991)

Ernest Tilden Parker was an American mathematician best known for his work in disproving a conjecture by Leonhard Euler (1707—1783) about Latin squares. He was born in Detroit, Michigan (in the United States). Parker completed his Ph.D. at Ohio State University in 1957, under the supervision of Marshall Hall, Jr. His thesis was entitled “On Quadruply Transitive Groups”.

After completing his doctorate, Parker took a job at Remington Rand Univac, which was an early computer company. For recreation, he worked on the problem of mutually orthogonal Latin squares, which fascinated him. H.F. MacNeish proved that there are at least \(\displaystyle q-1\) mutually orthogonal Latin squares of order \(\displaystyle n\text{,}\) where \(\displaystyle q\) is the smallest factor in the prime power decomposition of \(\displaystyle n\text{,}\) and conjectured that there are no more than this. Since \(\displaystyle 21=3\cdot 7\text{,}\) by MacNeish's conjecture there should have been no more than \(\displaystyle 3-1=2\) mutually orthogonal Latin square of order 21, but in 1958 Parker found a set of 4 of them.

Although MacNeish's Conjecture would have implied Euler's conjecture, the failure of MacNeish's Conjecture did not imply anything about Euler's conjecture. Parker found this more of a curiosity than anything else, but his result caught the attention of Raj Chandra Bose (1901—1987) and Sharadchandra Shankar Shrikhande (1917—2020) and spurred them on. They managed to build on Parker's ideas to find a pair of orthogonal Latin squares of order 22. This in turn galvanised Parker, who was able to use a different method to find pairs of mutually orthogonal Latin squares for infinitely many orders of the form \(\displaystyle 4k+2\text{,}\) the smallest of which had order 10 (this was the smallest order for which the answer had been unknown). The three mathematicians then joined together to disprove Euler's conjecture in all cases, earning them the nickname of “Euler's spoilers”. All of this work took place in 1958 and 1959, and the final announcement made the front page of the New York Times in April of 1959. (See also Section 16.2.) The techniques that Parker and his co-authors developed for disproving Euler's conjecture played a significant role in the proof by Richard Michael Wilson (1945—) of Wilson's Theorem.

Some reports say that Parker found his order 10 examples on a computer, but all of the publications use mathematical methods of construction, and Parker's sister stated that a computer was not used. Parker later became a professor at the University of Illinois at Urbana-Champaign. He published at least 50 papers during his career. The other result for which he is best known is disproving a conjecture by Pál Erdős (1913—1996) and Leo Moser about tournaments.

Parker died in Urbana-Champaign, Illinois, either on December 31, 1990, or in 1991 (accounts differ). A memorial fund providing fellowships for graduate students at the University of Illinois was established in his name.

Sources: wikipedia, University of Illinois, the Math Genealogy Project, Family Search, Bhāvanā mathematics magazine, and Mississippi State University.

Subsection B.2.48 Blaise Pascal (1623—1662)

Blaise Pascal was a 17th-century French mathematician, philosopher, and physicist, whose work has had lasting influence on all three fields. He was born in Clermont-Ferrand, France. His mother died when he was only 3 and his family moved to Paris in 1631 when Pascal was 8. Pascal suffered from ill health for most of his life, beginning at the age of 2, and his father educated him at home. A talented mathematician himself, Pascal's father initially refused to teach him any math, wanting to ground him in languages and classics. Pascal was drawn to the forbidden and began studying geometry on his own at the age of 12. After he explained to his father some of the things he had discovered on his own, his father gave him a copy of the Elements of Euclid (c.325BCE—c.265BCE).

When Pascal was 15, his father also began to bring Pascal along with him to the regular meetings of a group organised by Mersenne for discussion of mathematics. At the age of 16, Pascal presented his own work to this group, “Essai pour les coniques” (“Essay on conics”), containing a remarkable result in projective geometry that is now known as Pascal's Theorem. This was published in 1640.

Pascal's father had to flee Paris in 1638 due to his opposition to some of Cardinal Richelieu's policies; he left his children in the care of a neighbour. A year later, he was forgiven and appointed a tax collector in Rouen, where the children joined him. At the age of 18 Pascal invented a mechanical calculator, in part to help his father with the endless addition and subtraction associated with his tax work. The machine was expensive and cumbersome, and not widely distributed, although Pascal continued to make design improvements over the next decade. The design was complicated by the fact that the machine calculated amounts of money, and the French currency did not use a decimal system (for example, there were 12 deniers in a sol). This was the second known mechanical calculator (the first having been manufactured in 1624). Pascal's father died in 1651, and Pascal's sister Jacqueline, who had been Pascal's primary caregiver, moved to a convent shortly thereafter.

Pascal, working with Fermat, laid the basis for probability theory, including the concept of expected value. In 1654 he wrote a treatise on the arithmetic triangle (as mentioned in Section 4.3), which included the first explicit statement of the principle of mathematical induction that we know of. Pascal also made significant contributions to physics, particularly in the areas of fluid mechanics and pressure. After having both a serious carriage accident and a religious experience in 1654, Pascal largely abandoned mathematical work in favour of religion and philosophy. He is also famous for his work in literature. This is an extraordinarily cursory summary of Pascal's many contributions to intellectual spheres.

Pascal's health worsened significantly when he was about 18. By the age of 24 he was only able to consume liquids. He died at the age of 39 at his sister Gilberte's home in Paris, and an autopsy found extensive damage in his abdomen and brain.

Pascal has been honoured in many ways. The University of Waterloo runs an annual contest for young teenagers that is named for him. Clermont-Ferrand in France is home to Université Blaise Pascal. There is also a school named after him in the Democratic Republic of the Congo. He has been the subject of several videos. The chameleon in the Disney film Tangled is named for Pascal. A unit of atmospheric pressure, and a programming language have also been named after him.

Sources: wikipedia, St. Andrews' math history web site, MIT, University of California at Berkeley, biography.com, and Stanford Encyclopedia of Philosophy.

Subsection B.2.49 Peter Christian Julius Petersen (1839—1910)

Peter Christian Julius Petersen (who went by “Julius”) was an important 19th-century Danish mathematician and educator. He was born in 1839 in Sorø, Denmark. He left school at the age of 15 to work for his uncle as a grocer's apprentice in Kolding, because his family needed the money. About a year later his uncle died and left him some money, so Petersen was able to return to Sorø. He passed the high school exams and enrolled in Copenhagen's College of Technology in 1856. In 1860 he passed the first part of his civil engineering exams. Although he wanted to go on to study math at university, he had run out of money by this time.

From 1859 to 1871 Petersen taught at a private school to support himself. At the start of this time he was still finishing his engineering studies. After saving money from his job for a few years, in 1862 Petersen enrolled in the University of Copenhagen, having passed the entrance exams. He also got married in the same year, and continued to teach while studying, in order to support his family (he and his wife had 3 children over time). Petersen wrote many textbooks over the course of his career, beginning during this period. A number of his textbooks were about geometry; his best-known book was translated from the original Danish into 8 other languages, and was still in print in French as recently as 1990. His style was considered elegant and concise, sometimes to a fault. He wrote textbooks for every level from the age of 14 through undergraduate studies, and his books were used almost universally in Danish schools in the late 19th and early 20th centuries.

Petersen obtained his Master's degree in 1866 and his doctorate in 1871 from the University of Copenhagen. His thesis was entitled “On equations which can be solved by square roots, with application to the solution of problems by ruler and compass”. After completing his doctorate, he began teaching at the Polytechnical School in Copenhagen. He also taught at the Officers School of the Danish Army between 1881 and 1887. He took a post as professor of mathematics at the University of Copenhagen in 1887, and remained there until his retirement in 1909. Petersen died in Copenhagen.

Petersen had broad interests and expertise. He did research in algebra, number theory, geometry, analysis, and mechanics, and published individual articles even more broadly, including such areas as economics, cryptography, and physics. Petersen's work in geometry led him to an interest in graphs, and his 1891 paper “Die Theorie der regulären Graphs” is generally considered to be the first significant paper in graph theory as such. In it he proved (among other things) that every bridgeless cubic graph has a 1-factor. The Petersen graph (shown in Exercise 14.1.18.4 and Example 14.3.13 and discussed in Section 15.3) appears in a paper from 1898. He was sometimes limited by a dogged insistence on maintaining independence in his own thinking, which he cultivated by reading as little as possible of other mathematical work. As a result, he more than once discovered that things he proved were already known.

In addition to his research, Petersen was an important part of developing mathematical research in Denmark. His was one of the first theses to be written in Danish rather than Latin, and he continued to use Danish in much of his writing, to help promote math in Denmark. He was a founding member of the Danish Mathematical Society, which was established in 1873, and was elected to the Royal Danish Academy in 1879. He was one of the three members of the Commission of Education, appointed in 1887 and retaining the position for 13 years; this included the duty of setting problems for standardised exams taken at the end of high school. He was appointed to the Order of the Dannbrog in 1891. He took great pleasure in his work, and said: “When throughout life you have obtained honour and money for enjoying yourself, what more can you ask for!”

Sources: wikipedia, St. Andrews' math history web site, biographical article in Cryptologia, and Julius Petersen Graph Theory Centennial volume.

Subsection B.2.50 David Angus Pike (1968—)

David Angus Pike is a Canadian combinatorist (specialising in graph theory and design theory) who identifies as gay. He was born in 1968 and grew up in Ontario, Canada. He attended the University of Waterloo, where he completed a joint degree in math and computer science. As he was enrolled in the co-operative education programme, Pike held several jobs as a programmer, analyst, or trainer during his undergraduate studies, including stints at the Mutual Life Assurance Company, the Department of National Defence, and McNeil Consumer Products Company. Pike continued to work at McNeil during the summers while completing his Master's degree at Auburn University in Alabama, which he finished in 1994.

Pike remained at Auburn University for his Ph.D. in Discrete Math, which he completed under the supervision of Chris Rodger in 1996. His thesis was entitled “Hamilton Decompositions of Graphs”. During his graduate studies, Pike came out as a gay man.

Pike was appointed to a tenure-track position at East Central University in Oklahoma in 1996. He worked there for 2 years before being offered a position in Canada, at Memorial University of Newfoundland (MUN), in the Department of Mathematics and Statistics. He has worked at MUN since 1998; since 2006 he has held a joint appointment in Computer Science. In June 2018 he was named a University Research Professor.

Pike has more than 50 publications, and has supervised about 20 graduate students and post-docs as of 2021, as well as mentoring many undergraduate students. His research includes the work with Atif Aliyan Abueida (1966—) mentioned in Section 17.1. He received a teaching award from MUN students in 2004, and was awarded the Hall Medal by the Institute of Combinatorics and its Applications (ICA) for his research in 2007.

Pike has provided significant service to the Canadian mathematical community also. He has served the Canadian science granting council (NSERC) for math, as both a member of the grant evaluation group, and the scholarships and fellowships selection committee. Pike has held many roles for the Canadian Math Society, of which he has been named a Fellow. These roles include vice-president, and he was elected president in 2021. He has also been vice-president of the ICA, and has served on both the scientific and equity advisory boards for the Banff International Research Station.

In addition to his mathematical work, Pike is a keen genealogist and has been the president of the Family History Society of Newfoundland and Labrador. He enjoys hiking and curling, and serves on the board of directors for his local LGBTQ+ curling league.

Sources: Memorial University of Newfoundland, Institute of Combinatorics and its Applications, the Math Genealogy Project, and LinkedIn. Updated and confirmed through personal communication.

Subsection B.2.51 Cheryl Elisabeth Praeger (1948—)

Cheryl Elisabeth Praeger is an Australian mathematician specialising in group theory, who has led the way for women in math in Australia. She was born in 1948 in Toowoomba, Australia. Her family moved several times during her childhood, whenever her father (who worked in a bank for most of her childhood) was transferred. A career guidance counsellor tried to discourage her from pursuing mathematics because she was a girl, but she persisted. Her parents had not been able to participate in higher education themselves, but encouraged Praeger to do so. She began her university studies at the University of Queensland, where she earned her bachelor's degree in 1969. Praeger then moved to England to study at Oxford University, where she received her Master's in 1972 and completed her D.Phil. in 1973 (though it was not awarded until 1974) under the supervision of Peter Neumann. Her thesis was entitled “On the Sylow Subgroups of Primitive Permutation Groups”.

After completing her doctorate, Praeger was offered a three-year research fellowship at Australian National University, during which time she also worked for one semester at the University of Virginia in the USA. Praeger next moved to Western Australia in 1976, where she accepted a short-term position. This turned into a permanent position that she held until her retirement in 2017. She was the first female professor of math in Western Australia, and the second in all of Australia. A photograph used in a newspaper story about the appointment shows her on a bicycle, pulling a modified trailer that holds her two young children (both preschoolers). Praeger remains active as a professor emeritus, still conducting research, supervision, and outreach.

Praeger has had a noteworthy career as a researcher, teacher, and mentor. Her research is primarily in abstract algebra (more specifically, permutation groups), but her interests are broad and she has studied actions of permutation groups on many types of combinatorial structures, including codes, designs, and graphs. She has also worked on computational group theory. It would be impossible in a brief sketch to indicate all of the areas to which Prager has contributed, as she has authored more than 400 publications with more than 180 co-authors. She served her university as department head, and as a dean and deputy dean at various times. Praeger has supervised more than 30 Ph.D. students as of 2021, in addition to many post-docs, undergraduate and Master's students, and young researchers. Even beyond her direct mentoring Praeger has served as an inspiration to many, particularly to young female mathematicians. The Praeger-Xu graphs that bear her name are described in Section 12.5.

Praeger's involvement in mathematics promotion and education has been enormous. She has been a member of the Curriculum Development Council of the (Australian government's) Commonwealth Schools Commission, the (Australian) Prime Minister's Science Council, and the Australian Federation of University Women. She became the first female president of the Australian Math Society in 1992. She chaired the committee that chose and trained Australia's team for the International Mathematical Olympiad (a high school competition) for about 20 years. She spent 8 years (2007—2014) on the executive of the International Mathematical Union; 4 years (2013—2016) as a vice-president of the International Commission on Mathematical Instruction; 4 years (2014—2018) as Foreign Secretary for the Australian Academy of Science, and also served the Association of Academies and Societies of Sciences in Asia on its executive committee and as chair of its Women in Science and Engineering Committee, for 6 years. Praeger also gives a lot of lectures on popular applications of mathematics at public events and in schools on topics such as the relationships between weaving and mathematics. She gives the advice: “There's nothing so wonderful as working at something you're passionate about. Go for it, grasp all the opportunities you can.”

Along with this extraordinary record have come many honours. Praeger was elected a Fellow of the Australian Academy of Science in 1996, and named Western Australian Scientist of the Year in 2009. She was also made a Fellow of the American Math Society in 2012, and an honorary member of the London Math Society in 2014. Praeger was inducted into the Western Australian Science Hall of Fame in 2015. She has been awarded 6 honorary degrees, at universities in Europe and Asia in addition to Australia. Her many medals include the Centenary Medal (Australia); Moyal Medal (Macquarie University); Euler Medal (Institute of Combinatorics and its Applications); Thomas Ranken Lyle Medal (Australian Academy of Science); George Szekeres Medal (Australian Math Society); and the Ruby Payne-Scott Medal (Australian Academy of Science). She was made a Member of the Order of Australia in 1999, and then appointed a Companion of the Order in 2021. This is the highest civil honour available from the Australian government, and it was awarded for “eminent service to mathematics, and to tertiary education, as a leading academic and researcher, to international organisations, and as a champion of women in STEM careers”.

Praeger was married in 1975 and raised two sons while pursuing her career. She and her husband figured out how to make things work, which wasn't always easy. She spent 6 weeks visiting Cambridge by herself for an important research project when her children were still very young, which was extremely unusual for a mother at that time. Her husband has a Ph.D. in statistics and runs his own consulting firm. Praeger also holds an Associate in Music, Australia (AMusA) in piano performance. An accident to one of her fingers when she was 11 made it impossible for her to pursue music as a career, but this is one of her keener interests along with hiking, sailing, and cycling.

Sources: wikipedia, St. Andrews' math history web site, University of Western Australia, STEM Women, Agnes Scott College, the Math Genealogy Project, Australian Government Honours, Australian Broadcasting, International Science Council, and Australian Academy of Science. Updated and confirmed through personal communication.

Subsection B.2.52 Richard Rado (1906—1989)

Richard Rado was a German-British mathematician who was involved in laying the foundations of Ramsey theory for infinite cardinals. He was born in Berlin, Germany, to Jewish parents. He was educated at the Universities of Berlin and Göttingen, completing his Ph.D. in Berlin in 1933 under the supervision of Issai Schur (1875—1941). His thesis was entitled “Studien Zur Kombinatorik” (“Studies of Combinatorics”). Rado also got married in 1933.

With Hitler coming into power in 1933, new laws made it impossible for Rado to work at a university in Germany. He obtained a scholarship to pursue further studies in Cambridge, and he and his wife moved to England. In 1935 Rado received a second Ph.D. from the University of Cambridge. His thesis was “Linear Transformations of Bounded Sequences”, and his advisor was G.H. Hardy. While in Cambridge Rado met Pál Erdős (1913—1996) with whom he co-authored 18 papers over time.

Rado remained in Cambridge in a temporary position for one year after completing his Ph.D. He moved to the University of Sheffield in 1936, and then to King's College, London in 1947. His only child was born while he and his wife lived in Sheffield. Rado's last move was to take a chair at the University of Reading in 1954; he remained there until his retirement in 1971. Rado spent a year as a visiting professor at the University of Waterloo in Canada immediately after his retirement, in 1971—1972, and held a similar appointment at the University of Calgary from 1973—1974. He and his wife continued to live in Reading, dying in a nursing home in nearby Henley-on-Thames in 1989; the health of both was severely affected by a car accident in 1983.

Rado's most important mathematical work was in finite and transfinite combinatorics, including the early development of Ramsey theory for infinite cardinals (with Erdős), an area that decisively influenced subsequent developments in set theory. The Erdős-Ko-Rado Theorem became a cornerstone of the theory of extremal set systems. A number of objects and results bear his name, including the Rado graph, (see Section 11.5) which had been discovered earlier and independently (in different forms) by Ackermann, Erdős, and Alfréd Rényi (1921—1970). He also contributed to analysis, number theory, algebra, geometry, and measure theory.

Rado is only credited by the Math Genealogy Project with supervising 4 doctoral students, but his mentorship was broader and more influential than this would indicate. He served as secretary and then vice-president of the London Math Society (LMS) from 1953 to 1956, and was awarded the Senior Berwick Prize by the LMS in 1972. He founded the British Combinatorial Committee in 1977 and served as its chair from its inception until 1983. He was elected a Fellow of the Royal Society in 1978, and received an honorary doctorate from the Free University of Berlin in 1981, and from the University of Waterloo in 1986. At his retirement, Rado said, “There are almost as many types of mathematician as there are types of human being. Among them are technicians, there are artists, there are poets, there are dreamers…”.

Rado was also a very talented pianist, but chose to pursue mathematics on the grounds that he believed that he could continue music as a hobby, while he couldn't do that with math. His wife was a professional-quality singer who also played the piano (they met when he was seeking someone to play piano duets with), and they enjoyed performing together in public and private recitals throughout their lives.

Sources: wikipedia, St. Andrews' math history web site, University of Reading, the Math Genealogy Project, Times obituary, and Royal Society.

Subsection B.2.53 Frank Plumpton Ramsey (1903—1930)

Frank Plumpton Ramsey was a British mathematician, philosopher, and economist. The work he produced during his short life had a lasting influence on all of these fields. Within Mathematics, his influence is strongly felt in mathematical logic, combinatorics, and set theory. Ramsey was born in Cambridge in 1903. His father, also a mathematician, was president of Magdalene College there. After attending boarding school in Winchester, Ramsey returned to the University of Cambridge, where he received his bachelor's degree in mathematics in 1923.

Ramsey travelled to Vienna in 1924, seeking psychoanalysis for depression. He returned to England and became a fellow of King's College, Cambridge in October of that year. He was married a year later, and had two children. In 1926 he became a lecturer at King's College. Ramsey had been diagnosed with jaundice and became increasingly ill late in 1929. He underwent an operation in London early in 1930 to remove a suspected blockage. The operation revealed no blockage, but showed that his liver and kidneys were in very bad condition. Ramsey died in the hospital, shortly thereafter. The exact cause of his unexpected death is uncertain.

Ramsey's mathematical research was in the area of logic. In math, he is best known for the theorem that forms the basis of the subject now known as Ramsey theory (introduced in Section 1.3 and discussed in more detail in Section 14.2). To him it was a lemma being used for another purpose. Ironically, he was working toward a problem that is now known to be undecidable, and a more direct proof has since been found for the special case that he was able to prove using this lemma. So it has been said that “Ramsey's enduring fame in mathematics … rests on a theorem he didn't need, proved in the course of trying to do something we now know can't be done!” This whole paper was only 8 pages long.

Although he was employed as a mathematician, only 3 of Ramsey's publications appeared in mathematical journals. Ramsey also published important work in philosophy and economics; he studied with John Maynard Keynes at Cambridge and continued to work with him closely through his brief career. Ramsey was regarded as a genius; many of his important ideas were not fully appreciated for decades.

Sources: wikipedia, St. Andrews' math history web site, Stanford Encyclopedia of Philosophy, the New Yorker, Oxford University Press, and Times Literary Supplement.

Subsection B.2.54 Dwijendra Kumar Ray-Chaudhuri (1933—)

Dwijendra Kumar Ray-Chaudhuri (known as “Dijen”) is an Indian-American combinatorist who has contributed to very important developments in both coding theory and design theory. Ray-Chaudhuri studied at the prestigious Rajabazar Science College of the University of Calcutta in India, where he received his M.Sc. in 1956. He went on to earn a Ph.D. in 1959 from the University of North Carolina at Chapel Hill. His advisor was Raj Chandra Bose (1901—1987), and his thesis was entitled “On the Application of the Geometry of Quadrics to the Construction of Partially Balanced Incomplete Block Designs and Error Correcting Binary Codes.”

For about seven years after completing his dissertation, Ray-Chaudhuri served as a research associate or consultant at a number of universities and private businesses, including Bell Labs, IBM Research, Cornell Medical Center, and Sloan Kettering Institute. In 1966, he was hired at Ohio State University in Columbus, Ohio, where he spent the remainder of his career, and remains a professor emeritus. Ray-Chaudhuri served two terms as chair of his department, and was a visiting professor at a number of universities in Europe during stays there, as well as at the Tata Institute in Mumbai.

Ray-Chaudhuri is most famous for his work in error-correcting codes and in designs. With Bose, he discovered “BCH” codes independently, at about the same time as their discovery by Alexis Hocquenghem in 1959. These codes are still used in applications such as compact disc players, DVDs, and solid-state drives, and the “C” in their name is for Ray-Chaudhuri. With his Ph.D. student Richard Michael Wilson (1945—) in 1968, he determined exactly when Kirkman Triple Systems exist (see Theorem 18.1.11). During his career, Ray-Chaudhuri published over 80 papers, and supervised at least 33 doctoral students. He devoted significant effort and attention to developing research in combinatorics at Ohio State University, hiring and mentoring young researchers in addition to his own students.

Ray-Chaudhuri was a founding Fellow of the Institute of Combinatorics and its Applications (ICA), and was awarded the ICA's Euler Medal in 1999 for career achievements in combinatorics. He was an invited speaker at the International Congress of Mathematicians in 1970, and became a Fellow of the American Math Society in 2012. Ray-Chaudhuri is married with three children.

Sources: wikipedia, Ohio State University, wikipedia, and the Math Genealogy Project, as well as MathSciNet for a publication list.

Subsection B.2.55 Alfréd Rényi (1921—1970)

Alfréd Rényi (called “Buba” by his friends, including Pál Erdős (1913—1996)) was a Hungarian mathematician of Jewish descent, whose research centred around probability. In graph theory, he is particularly known for introducing the use of probabilistic methods in the study of graphs. Rényi was born in Budapest, Hungary. He became attracted to astronomy as a child, which led him to learn physics, and his interest in mathematics stemmed from this. Due to laws limiting Jewish students at Hungarian universities he was not able to attend university immediately upon finishing high school, so he spent some months working at a shipyard. Winning a prize in a math contest led to his admission to the science university in Budapest in 1940. (This university has been named Loránd Eötvös University since 1950, and will be referred to as Eötvös University in the rest of this biography.) He graduated with a bachelor's degree in 1944, and, as a Jewish male of military age, was sent to forced labour in a “labour batallion”. While these batallions saw scenes of atrocities against their enlisted Jewish men throughout World War II, with the Nazi takeover of the Hungarian government on 16 October 1944, many of the labour batallions turned into death marches. Rényi managed to escape before this turn of events.

Rényi continued to live in hiding in Budapest for 6 months using false documents. During this period he rescued his parents, who were being confined in the Budapest ghetto, by the expedient of pretending to be a soldier and marching them out, wearing a uniform he somehow managed to obtain. This was an extraordinarily risky act that required quick thinking and bravery. While all of this was going on, he also managed to enrol at the University of Szeged, where he completed his doctorate in 1945 with Frigyes (Frederic) Riesz as his advisor. Riesz, one of the creators of functional analysis, had narrowly missed being transported to the Auschwitz gas chambers a year earlier. Rényi's thesis was entitled “Egy Stieltjes-féle inegrálról” (“On the Stieltjes transform”). He married in 1946; his wife was also a mathematician, Katalin Schulhof (she went by Kató Rényi after marriage). They had one child.

Rényi worked for a time as a statistician in Budapest. He then spent most of a year in Leningrad, USSR (now St. Petersburg, Russia), doing a postdoc with Yuri Linnik (ending in 1947). During this period he developed his own version of Linnik's number-theoretic “large sieve” method and used it to prove an approximate version of the famous Goldbach Conjecture. He briefly held an appointment at Eötvös University, until he was appointed Professor Extraordinary at the University of Debrecen in 1949. He held this role only until he became the founding director of the Mathematics Research Institute of the Hungarian Academy of Sciences in 1950. He continued to serve as its director until his death; it has since been renamed in his honour. In 1952 he became a professor in the Department of Probability and Statistics at Eötvös University. Rényi died of cancer in 1970 at the age of 48. His wife Kató had also died tragically young, about 6 months previously. In the summer of 1969, shortly after Kató's passing and unaware of his own deadly disease, Rényi delivered a plenary lecture on his recent joint work with his wife on “a theory of trees” at an international conference on combinatorics in Balatonfüred, Hungary.

Rényi's research was generally in the areas of number theory, graph theory, analysis, and probability. Among his more than 200 publications, he wrote 32 joint papers with Erdős. These included their seminal 1960 paper on “The evolution of random graphs” that created the Erdős-Rényi model of random graph described in Section 11.5 and with it, an entirely new and highly influential field of study. Their joint work also included the result showing that almost all graphs are asymmetric, as mentioned in Section 12.5. Rényi also wrote a highly influential textbook in probability theory. He was a charismatic teacher. Through teaching, advising, and mentoring generations of mathematicians, he created a school of probabilists in Hungary, and this area remains one of the strongest suits of mathematics in that country. During his career he held visiting appointments at Stanford, Michigan State, the University of Michigan, Cambridge University, the University of Erlangen, and the University of North Carolina. Among the honours he received were the Kossuth Prize from the Hungarian government (twice).

Rényi used to say “A mathematician is a device for turning coffee into theorems,” which Erdős loved to quote. Pál Turán (1910—1976) built on this joke by adding that weak coffee is only good for a lemma. Rényi was popular as a raconteur, played the piano, and enjoyed rowing, swimming, and skiing. He wrote a series of witty essays on mathematics in the form of imaginary dialogues or exchanges of letters between historical mathematicians.

Sources: wikipedia, St. Andrews' math history web site, Cambridge University obituary, the Math Genealogy Project, and Project Euclid obituary. Additional information and clarification by personal communication from Laci Babai.

Subsection B.2.56 George Neil Robertson (1938—)

George Neil Robertson (who goes by “Neil”) is a Canadian-American mathematician, famous for having settled, in monumental series of works with coauthors, some of the most formidable open problems of graph theory. He was born in Killarney, Manitoba (in Canada). For several years during World War II the family moved through Manitoba, Alberta, and British Columbia, as Robertson's father served in Canada's “Veteran's Guard”, but they returned to Killarney in 1944 and from 1945 on the family lived on a nearby farm. In 1956 Robertson enrolled at Brandon College (at that time a branch campus of the University of Manitoba), graduating in 1959 with a B.Sc.

Robertson's interest in mathematics solidified during his studies at Brandon College, and he went on to graduate school at the University of Manitoba. He taught first-year calculus to support himself during his Master's, which he completed in 1962. His Master's thesis was on systems of distinct representatives, and its quality earned him the attention of top advisors during his Ph.D. He spent one year in the Ph.D. program at Syracuse University in New York (in the United States) where he worked with Herbert Ryser. He then moved back to Canada to study at the University of Waterloo, where he was advised by William Tutte. Robertson completed his Ph.D. in 1969 with a thesis entitled “Graphs Minimal under Girth, Valency and Connectivity Constraints”. For several summers during his doctoral studies he worked in Operations Research at an air force base near Montreal. Robertson also married during this period; he and his wife have 2 children. While still a student he discovered the Robertson graph, which has the fewest vertices among all graphs in which every vertex has 4 neighbours and the smallest cycle has length 5.

Although the degree was not conferred until 1969, Robertson's doctoral work was completed in 1968 and he spent a year at McGill University, working as a postdoc with William Brown. Immediately after completing his postdoc, Robertson was hired at Ohio State University, where he remained throughout his career. While working there, he also acted as a consultant for Bell from 1984 to 1996, and held a number of visiting positions, including at Princeton University and Victoria University of Wellington, New Zealand. He was awarded the title of Distinguished Professor of Mathematics and Physical Sciences at Ohio State in 2006, where he remains a professor emeritus since his retirement in 2008. He maintained a workload of one-third time for 6 years after retirement, not all of it at Ohio State. During his career, Robertson supervised at least 23 Ph.D. students, and published more than 80 papers. Robin Thomas (1962—2020) worked with him as a postdoc.

Robertson's research is in graph theory. His most significant work has been in the areas of structural and topological graph theory. In a ground-breaking series of 22 papers dating from 1983 to 2004, he and Paul Seymour (1950—) proved that any family of graphs that is closed under minors can be characterised by a finite set of forbidden minors; this had been conjectured by Klaus Wagner (1910—2000), and is discussed in Section 15.2. Robertson was also an author (with Daniel Sanders, Seymour, and Thomas) of a new proof of the Four-Colour Theorem (which still required computers but was significantly simpler than the original proof), and (with Maria Chudnovsky (1977—), Seymour, and Thomas) of the proof of the Strong Perfect Graph Theorem. Robertson's work has been key to increasing the respect accorded to research in graph theory by mathematicians in other disciplines.

Robertson won the American Math Society's Fulkerson Prize on 3 occasions: in 1994, 2006, and 2009. He won the Ohio State University Distinguished Scholar Award in 1997, and in 2002 the Waterloo Alumni Achievement Award. In 2004 he also won the Pólya Prize from the Society for Industrial and Applied Math. Robertson was named a Fellow of the American Math Society in 2013, and was elected an Honorary Fellow of the Institute of Combinatorics and its Applications in 2018.

Sources: wikipedia, wikipedia, University of Waterloo, Institute of Combinatorics and its Applications,and the Math Genealogy Project, as well as MathSciNet for a publication list. Updated and confirmed through personal communication.

Subsection B.2.57 Gordon F. Royle (1962—)

Gordon F. Royle is an Australian mathematician and computer scientist, working primarily in the area of computational combinatorics. He was born in Australia, and received his Ph.D. from the University of Western Australia in 1987, under the joint supervision of Cheryl Elisabeth Praeger (1948—) and Brendan McKay. His thesis was entitled “Constructive Enumeration of Graphs”.

Royle completed a postdoc at the University of Waterloo in Canada, where he worked with Chris Godsil. He also met his wife in Waterloo. They have 2 children. Royle and his wife returned to Perth, Australia, where he became a professor at the University of Western Australia, and where he still works.

Royle's research is primarily in computational combinatorics. He has expertise in algebra, computers and algorithms, and many branches of combinatorics. He has more than 60 publications. He is probably best known for the graduate textbook on algebraic graph theory that he co-authored with Godsil, and his work on the mathematics of Sudoku, including the web site mentioned in Section 16.1. Royle has supervised at least 2 doctoral students.

Sources: wikipedia, University of Western Australia, and the Math Genealogy Project. Updated and confirmed through personal communication.

Subsection B.2.58 Al-Samaw’al ben Yahyā al-Maghribī (c.1130—c.1180)

Al-Samaw'al ben Yahyā al-Maghribī was a 12th-century mathematician and doctor from the area that is now Iraq, whose writings include both mathematics and religion. He was born in Baghdad, capital of the Abbasid Caliphate (now capital of Iraq), to a Jewish family; his father was a Rabbi. Baghdad was the centre of scholarship in the Islamic world. Al-Samaw'al began to study mathematics alongside medicine as a youth. After surpassing his teachers in math, al-Samaw'al read all of the books he could find, and began to work out his own improvements to them.

When he was 19, al-Samaw'al wrote al-Bahir fi'l-jabr (the “Brilliant in Algebra”). In addition to including original material, it is valuable because it contains and builds on work by al-Karaji which has not survived. Al-Bahir fi'l-jabr includes 4 parts. In the first part he defines and uses the basic arithmetic operations of addition, subtraction, multiplication, and division, on polynomials. He also describes methods for finding the roots of polynomials. The second part is largely about solving quadratic equations. It also includes the binomial theorem, though al-Samaw'al attributes this to al-Karaji. He also uses an inductive type of argument, though it is not clear or well-developed and it would be a stretch to say that he fully understood the principle of induction. The third part is about calculating with irrational numbers, and does not contain much original material. The fourth and final part includes the problem described in Example 3.2.9, in which (to be more precise) he asks for the values of 10 unknowns, given the 210 equations that show the sums of any 6 of the variables.

After writing this treatise, al-Samaw'al travelled extensively through countries in the region of Iraq. He practised medicine during his travels. In 1163, during his travels, he converted to Islam but kept this a secret for 4 years because he was concerned about his father's reaction. In fact, when his father received the letter in which al-Samaw'al told him of his conversion, he immediately set out to see his son. He died along the way.

Including his first book, al-Samaw'al is said to have written about 85 works; most of these have not survived. Al-Bahir fi'l-jabr contains his only surviving mathematical writing of value. His surviving works also include a book explaining the problems he saw with Judaism and Christianity.

Sources: wikipedia, St. Andrews' math history web site, Encyclopedia.com, and Columbia University.

Subsection B.2.59 Issai Schur (1875—1941)

Issai Schur was a Russian-German mathematician of Jewish descent, best known for his work in algebra, but whose important discoveries have impacted a number of fields including Ramsey theory. He was born in Mogilev, which at the time was part of the Russian Empire (it is now in Belarus). When he was 13, Schur moved to Libau (now Liepāja, Latvia) to live with his sister. The local Jewish community spoke German, and Schur attended a German-language school.

In 1894 Schur moved to Germany to attend the University of Berlin. He completed all of his studies there, finishing in 1901. His Ph.D. thesis, “Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen” (“On a class of matrices which can be assigned to a given matrix”), was written under the joint supervision of Ferdinand Frobenius and Lazarus Fuchs.

Schur worked at the University of Berlin from 1903 to 1911, and then moved to the University of Bonn, where he worked until 1916. In 1916, Schur returned to the University of Berlin. He was promoted to professor in 1919. In 1922, Schur was elected to the Prussian Academy of Sciences. After the Nazis rose to power, anti-Jewish laws passed in Germany in 1933 required that Schur be suspended and barred from the university. Due to an outcry and a technicality (he was respected and popular in the community, and had held an appointment prior to World War I, which made him qualify for an exemption under the law) Schur was given a temporary dispensation to lecture in 1933—1935, and was in fact the last Jewish professor to lose his position. Schur was invited to move to various universities in other countries during this period, but he declined these offers. He continued to think of himself as a German and was baffled by the Nazis' inability to see anything but his Jewish heritage. However, he was increasingly isolated and disrespected, and was ultimately fired in 1935. By the end of 1938 he was pressured into resigning from the Prussian Academy of Sciences.

In 1938 Schur was summoned to an interview with the Gestapo. In his increasing isolation and depression, he had told his wife that he would commit suicide if summoned by the Gestapo, so she managed to send him out of town and go in his place. She was asked why they had not left Germany. A Reich Flight Tax amounting to a quarter of their assets had to be paid if they were to be allowed to leave. They were unable to raise enough money, because a substantial portion of their assets were tied up in a mortgage on a house in Lithuania that at that time could neither be sold nor used as collateral. The tax was ultimately paid; the source of the money is not known, but the Schurs left Germany in 1939. After a brief stay with their daughter in Bern, Switzerland, the Schurs moved to Palestine. Schur had to sell his collection of books to the Institute for Advanced Study in Princeton in order to cover their living expenses in Tel Aviv. Schur died there of a heart attack on his 66th birthday.

Schur's research was mostly in the field of algebra; more specifically, group theory and representation theory. Among his almost 90 papers he also published important results in number theory, analysis, and Ramsey theory. There are numerous structures and results across these fields that bear his name, including the theorem whose statement is given in Exercise 14.2.10. Schur was extremely popular as a lecturer and mentor, and built a school of researchers who worked with him and learned from him. These included the doctoral students he supervised (there were at least 22 of these, and some sources credit him with 35), one of whom was Richard Rado (1906—1989).

Sources: wikipedia, St. Andrews' math history web site, the Math Genealogy Project, Encyclopedia.com, and London Math Society.

Subsection B.2.60 Paul Seymour (1950—)

Paul Seymour is a British mathematician and a towering figure in contemporary combinatorics. He was born in Plymouth, England. He attended Exeter College at the University of Oxford throughout his university education, earning his bachelor's degree in 1971 and his Master's of Science in 1972. Working under the supervision of Aubrey Ingleton, Seymour completed his doctorate in 1975 with a thesis entitled “Matroids, Hypergraphs and the Max-Flow Min-Cut Theorem”. Seymour was also awarded an M.A. from Oxford in 1975.

Between 1974 and 1980 Seymour held research positions: first for 2 years at the University College of Swansea; then for 4 years at Merton College, Oxford University. During his time at Oxford, Seymour also spent a year at the University of Waterloo, Canada, as a visiting researcher. This is where he met George Neil Robertson (1938—). In 1980, he moved to the United States, and took a position at Ohio State University, partly to collaborate with Robertson. Seymour continued to work there until 1983. In 1984 he was hired as a senior scientist at Bellcore (Bell Communications Research) in New Jersey, where he continued to work until 1996. He maintained his academic connections throughout this period, with adjunct positions first at Rutgers University and then at the University of Waterloo, followed by a visiting position at Princeton University. In 1996 he accepted a position as a professor at Princeton, where he has remained ever since. In 2016, he was appointed to the Albert Baldwin Dod Professorship there. Seymour married in 1979 and has two children.

Seymour's influence on combinatorics has been immense. He began his research career working on matroids, where he made some significant contributions before moving into graph theory. He works primarily in the areas of structural and topological graph theory. His research has been instrumental in significantly increasing the respect with which other mathematicians view graph theory. In a ground-breaking series of 22 papers dating from 1983 to 2004, he and Robertson proved that any family of graphs that is closed under minors can be characterised by a finite set of forbidden minors; this had been conjectured by Klaus Wagner (1910—2000), and is discussed in Section 15.2. Seymour was also an author of a new proof of the Four-Colour Theorem (this still required computers but was significantly simpler than the original proof), and of the proof of the Strong Perfect Graph Theorem. Seymour has over 250 publications. He has supervised 11 doctoral students in addition to undergraduate and Master's students as of 2021. He is joint editor-in-chief of the Journal of Graph Theory. Seymour also holds two patents.

Seymour has an impressive list of honours, as befits his illustrious career. He won the American Math Society's Fulkerson Prize on 4 occasions (on his own in 1979, and jointly in 1994, 2006, and 2009). He also won the Pólya Prize from the Society for Industrial and Applied Math in 1983, and again (jointly with Robertson) in 2004. Seymour was awarded a Sloan Fellowship in 1983, and the Ostrowski Prize in 2004. He holds honorary doctorates from the University of Waterloo (2008) and the Technical University of Denmark (2013), and was given the Commemorative Medal of Comenius University (Slovakia) in 2019.

Sources: wikipedia, Princeton University, the Math Genealogy Project, and University of Waterloo. Updated and confirmed through personal communication.

Subsection B.2.61 Sharadchandra Shankar Shrikhande (1917—2020)

Sharadchandra Shankar Shrikhande was an Indian mathematician who was nicknamed one of “Euler's Spoilers” for his work in disproving a conjecture of Leonhard Euler (1707—1783). Shrikhande was born in Sagar, India in 1917. He completed his bachelor's degree at the Government College of Science in Nagpur, after which he spent a year as a Research Fellow at the Indian Statistical Institute in Calcutta, where he met Raj Chandra Bose (1901—1987). Shrikhande briefly taught at the college in Jabalpur and then the College of Science in Nagpur, but regularly returned to Calcutta to visit Bose.

Shrikhande moved to the United States in 1947 for his doctorate, which he completed in 1950. He worked on this under the supervision of Bose at the University of North Carolina in Chapel Hill (Bose moved there from India in 1949). Shrikhande's thesis was entitled “Construction of Partially Balanced Designs and Related Problems”.

Shrikhande moved back to India after completing his doctorate, and held a position at the Science College in Nagpur from about 1953 to 1958. He spent some time back in the United States from 1959 to 1960; it was on this visit that he collaborated with Bose and Ernest Tilden Parker (1926—1991) to disprove Euler's 1782 conjecture that there are no pairs of orthogonal Latin squares of order \(\displaystyle n\) when \(\displaystyle n=4k+2\) (see Section 16.2); this earned them the nickname of “Euler's Spoilers”. Shrikhande took a position as a professor at Banaras Hindu University on his return to India in 1960. In 1963 he became the founding head of the math department at the University of Mumbai and the founding director of the Center of Advanced Study in Mathematics of Mumbai; he retired from these positions in 1978. Shrikhande then took on the directorship of Mehta Research Institute (which has since been renamed after Harischchandra) in Allahabad. He stayed there until about 1988, when his wife died. After this reports vary, but it seems that he lived with his sons in the United States and New Delhi for a couple of decades. In 2009 he moved to an Ashram in Vijayawada run by one of his grandchildren, where he lived for his final years of life. He died there in 2020 at the age of 102.

In addition to Shrikhande's status as one of “Euler's Spoilers”, he also has a graph named after him. He published more than 75 research papers, and was a mentor to many, although there is no formal record of most of these relationships. The techniques that he and his co-authors developed for disproving Euler's conjecture played a significant role in the proof by Richard Michael Wilson (1945—) of Wilson's Theorem. He was a Fellow of the Indian National Science Academy and the Indian Academy of Science. He was also a Fellow of the Institute of Mathematical Statistics in the United States, and was named an Honorary Fellow of the Institute for Combinatorics and its Applications in 1991.

Shrikhande was married with three children. One of his sons was a professor of mathematics at Central Michigan University until his own retirement in 2015.

Sources: wikipedia, Institute of Mathematical Statistics, the Math Genealogy Project, Resonance, Institute for Combinatorics and its Applications,and News 18 (India).

Subsection B.2.62 Vera Sós (1930—)

Vera Sós [pronounced “Shosh”] is a Hungarian number theorist and combinatorist, who has been influential through both her impressive research results and her mentorship. She was born to Jewish parents in Budapest, Hungary. Anti-semitic laws meant that her father lost his job as a teacher during her childhood. After surviving the Holocaust, Sós was able to resume her studies in the Jewish Girls' High School in 1945. Her math teacher was renowned graph theorist Tibor Gallai, a close friend of Pál Erdős (1913—1996). Her classmates included another future math professor, Edit Gyarmati (1930—2014), attesting to the extraordinary qualities of Gallai as a teacher. Sós subsequently studied at the science university in Budapest, which was renamed Loránd Eötvös University while she was there in 1950. Sós studied math and physics, graduating in 1952, and began to teach at the university in 1950 while still a student. Although she was surrounded by men through most of her career, Sós never felt that her gender was an issue in her work.

Sós completed her Ph.D. in 1957, with Fourier analyst Lipót (Leopold) Fejér as her advisor. The title of her thesis (written in Hungarian) translates as “A geometric treatment of continued fractions and its application to the theory of diophantine approximation”; the thesis includes her famous “three-gaps-theorem” in diophantine approximation. She worked at Eötvös University in the Department of Analysis until 1987. At that time, she moved to work at the Alfréd Rényi Institute of Mathematics. She continued to give lectures at the university until about 2006.

While studying at university, Sós met Pál Turán (1910—1976), who was a professor. They married in 1952, and had two children. Sós's name often appears (for example in publications) as Vera T. Sós; the “T” is for Turán. She met Alfréd Rényi (1921—1970) and Erdős while still in high school, as her teacher (Gallai) fostered her abilities not only by assigning her beautiful, challenging problems, but also by introducing her to prominent mathematicians. Recognising Sós's talent, Rényi soon became her mentor, inspiring and challenging her in weekly sessions while she was still in high school and starting a research collaboration with her during university. Sós began working more closely with Erdős in the mid-1960s, and became one of his most frequent collaborators; the two published 35 joint papers. The period of Sós's graduate education was a turbulent time in Hungary, as the Hungarian revolution against Soviet rule took place (and was crushed) in 1956. Sós had a 3-year-old son at this time. Schools and universities closed, and Sós avoided leaving the house to remain safe during fighting on the streets. She said in an interview that this gave her more time to work on mathematics, and downplayed the challenges it presented in comparison with those of mathematicians like her husband who found ways to work on mathematical research while in forced labour camps during World War II.

Sós's research is primarily in the areas of number theory and combinatorics, and her more than 100 publications include results that now bear her name in both of these fields. She initiated several new areas of research, including Ramsey-Turán theory and the extremal theory of structured intersections. In the 2000s she joined a group of researchers led by László Lovász in exploring graph limit theory; their joint paper appeared in 2012, when she was 82, in the Annals of Math. Sós's mentorship and influence have been profound. In 1965, she jointly founded a weekly combinatorics seminar at the Mathematical Institute of the Hungarian Academy of Sciences that has been running ever since. It serves as a forum for new results in combinatorics, and often features the research of young mathematicians including undergraduate students. Starting in the 1960s, she was also a key organiser of a long series of international conferences on combinatorics held in Hungary; these allowed mathematicians from the Soviet bloc (including Hungary) to interact with others worldwide, at a time when such opportunities were very limited. One of her joint works with her husband is mentioned in Section 14.2.

Sós won the Tibor Szele Prize for research mentorship from the János Bolyai Math Society in 1974, and the Academy Award of the Hungarian Academy of Sciences in 1983. She was elected to the Hungarian Academy of Sciences as a corresponding member in 1985, and as an ordinary member in 1990. She won Hungary's Benedikt Otto Prize in 1994, and was elected to the Austrian Academy of Sciences in 1995. Sós also won Hungary's Széchenyi Prize in 1997, the Cross of Merit in 2002, and the “My Country” award in 2007. She became a member of the Academia Europaea (a pan-European Academy of scholarly inquiry) in 2013. Sós received an honorary degree from the Hebrew University of Jerusalem in 2018.

Sources: wikipedia, St. Andrews' math history web site, the Math Genealogy Project, Gil Kalai's blog, and Simons Foundation. Additional information and clarification by personal communication from Laci Babai.

Subsection B.2.63 Jakob Steiner (1796—1863)

Jakob Steiner was a 19th-century Swiss-German mathematician whose research focus was geometry, but who had a profound impact on the early development of design theory. He was born in the village of Utzenstorf in Switzerland. He grew up on a farm, helping his parents, and did not learn to read and write until he was 14. He defied his parents by leaving home to go to school when he was 18, in 1814. After studying at a school in Yverdom for about 4 years, he moved to Heidelberg in 1818, where he continued his studies at the university, while earning his living by tutoring.

In 1821 Steiner moved to Berlin, Germany, where he continued to tutor. In order to be allowed to teach he attempted to pass qualifying exams. Due to his performance in some of the humanities exams Steiner was only granted a restricted license. Nonetheless, he was appointed to teach mathematics at a local school. He was soon dismissed as he refused to use the methods followed by the director of the school or to use the textbook the director had written. Steiner went back to tutoring while studying at the University of Berlin from 1822 to 1824. In 1825 he found another position as a teacher, but again came into conflict with the director of the school. Steiner was carrying out research and publishing during this period, and despite his disagreements with the director he was promoted to the level of senior teacher in 1829.

In 1832 Steiner published his first book. He was awarded an honorary degree from Königsberg University in 1833 in recognition of the research he had published. He was elected to the Prussian Academy of Sciences in 1834, and also accepted a new extraordinary chair in geometry at the University of Berlin. He retained this position for the rest of his life. During a stay in Paris in 1855 he was elected to the Académie des Sciences. Steiner was also elected to the Royal Society but died before the vote was ratified, so never became a member.

Steiner's research was almost entirely in geometry. He made many fundamental discoveries in this field, and a number of theorems and structures are named for him. His work in combinatorics (on Steiner systems) was published in 1853. Although Reverend Thomas Penyngton Kirkman (1806—1895) preceded him in finding Steiner triple systems (see Section 18.1), Steiner was unaware of Kirkman's work. The fundamental design theory structure of Steiner systems (see Section 18.2) also bear his name. Steiner published extensively, including more than 60 papers in Crelle's journal alone (one of the first pure math journals that was not the proceedings of an academy; it was established by a friend of Steiner's). He also produced a number of books.

Steiner apparently had an abrasive personality. He never married; he had many challenging relationships with colleagues and students, and was described as being crude and blunt. He gave nicknames to everyone, and these were never flattering. During his last decade, ill health kept him in Switzerland for most of each year, and he only visited Berlin to give his lectures each winter. He eventually became completely confined to his bed and was unable to continue teaching. He died in Switzerland. He bequeathed one third of his estate to the Berlin Academy for the establishment of a prize named after him.

Sources: wikipedia, St. Andrews' math history web site, University of Evansville, Your Dictionary, The Mathematical Intelligencer, and encyclopedia.com.

Subsection B.2.64 Sushruta (c.800BCE—c.700BCE)

Sushruta was a renowned Indian doctor from the 8th century BCE whose influence on medicine (and surgery in particular) in India may have been as profound as that of Hippocrates on western medicine, and for similar reasons. The work attributed to Sushruta in Section 4.1 (mentioned again in passing in Section 4.2) appears in a book called the Sushruta Samhita, meaning the “Compendium of Sushruta”. Scholars have estimated the date of this work at anywhere from 1000BCE to 500CE. It is likely that the core of the book dates from the early end of this range, and that successive people (perhaps even doctors named Sushruta) added to it over the years. We cannot be certain of the original date of any given piece of the work. The book lists its author as Sushruta, and locates him in Varanasi, India.

There was a very eminent Indian doctor named Sushruta who lived sometime between approximately 800BCE and approximately 700BCE. He is referred to as the “father of Indian surgery” and the “father of plastic surgery” as the topics of surgery and plastic surgery are also covered in the Sushruta Samhita. The original part of the manuscript is most likely attributable to this doctor.

The Sushruta Samhita is a very important work in the history of medicine. The surviving text includes 184 chapters describing illnesses, injuries, surgeries, medicines, and treatments. Sushruta taught others, referred to as “Saushrutas”, who were required to undertake 6 years of study and to take an oath (not the Hippocratic Oath, but with similar intent) to do no harm, before being allowed to practice hands-on surgery under supervision.

Sources: wikipedia, Journal of Indian Philosophy, and wikipedia.

Subsection B.2.65 Stan Swiercz (1955—)

Stan Swiercz is a Canadian computer programmer and software engineer who contributed to the computer-assisted proof that there is no finite projective plane of order 10. He studied electrical engineering at McGill University in Montreal, Canada, but decided that he preferred software to hardware. In 1978, Swiercz was hired by the Department of Computer Science at Concordia University in Montreal as a programmer. He has worked there ever since.

Swiercz is the Manager of Software Applications for what is now the Gina Cody School of Engineering and Computer Science at Concordia University. He is responsible for installing and maintaining required software. It was in this capacity that he worked with Clement Wing Hong Lam (1949—) and Larry Henry Thiel (1945—) on writing, optimising, and running the code to prove the nonexistence of a finite projective plane of order 10, as mentioned in Section 16.2. At the time, Thiel was Swiercz's boss, and Swiercz worked on this under his supervision.

Swiercz is married with two children.

Sources: Concordia University, and Mathematical Association of America. Updated and confirmed through personal communication.

Subsection B.2.66 György Szekeres (1911—2005)

György Szekeres was a Hungarian-Australian mathematician and educator of Jewish descent, who played a key role in establishing the foundations of Ramsey theory. He was born in Budapest, Hungary. He later adopted the anglicised version of his given name, “George”, which is what he is most commonly known by. Laws in Hungary at the time when Szekeres finished high school limited the enrolment of Jewish students in university. Nonetheless, Szekeres secured a place at the Technological University in Budapest. He had long been interested in math, but studied chemical engineering at university, to learn skills that would be useful in his family's leather business.

During university, Szekeres met regularly with a group of students including Tibor Gallai (a notable teacher and mathematician mentioned in the biographical sketch of Sós), Eszter Klein (1910—2005), Pál Turán (1910—1976), and Pál Erdős (1913—1996), who gathered to talk about math. Even though they did not all attend the same university in Budapest, they knew of each other from high school, when they had all worked on solving problems for the famed “Hungarian High School Mathematical Monthly”. This was a national publication that inspired generations of Hungarian mathematicians in their love of mathematics and problem-solving. During one of this group's meetings in about 1933, Klein proposed the Happy Ending Problem (see Section 14.2). Erdős and Szekeres worked on this problem and were able to obtain bounds on the solution; the Erdős-Szekeres Theorem appeared in this same paper. Klein and Szekeres subsequently married in 1937.

After graduating from university in 1933, Szekeres spent about 6 years working as an analytical chemist in a leather factory in Simontornya (his family's business collapsed while he was in university). He and Klein were only able to spend weekends together during this period. Because of their Jewish heritage, Szekeres and Klein were not safe in Hungary, which was an ally of Nazi Germany. In 1939 Szekeres and his wife (as well as one of his brothers) fled to Shanghai, China, which did not require much in the way of documentation for immigrants. Their son was born there in 1940. Life was not easy in China during World War II. Szekeres found work in a leather factory, but it closed in 1940. There were times when they had to flee for their lives from Japanese bombing. Food was sometimes hard to come by, and Szekeres traded a bicycle for a bag of rice on one occasion. Later Szekeres did obtain work as a clerk in an American air force base.

Szekeres' formal qualifications were his bachelor's degree in chemistry, and he had taken virtually no mathematics at university (in an interview he reported having taken none; other sources claim that he took one calculus course). Nonetheless, he had a number of noteworthy publications in math research, beginning in 1935, and he had friends in Australia who held him and his work in high regard. In 1948, Szekeres was hired by the University of Adelaide as a lecturer in math, and the family moved to Australia. Szekeres and Klein had a daughter born there in 1954. Szekeres accepted a professorship at the University of New South Wales in Sydney in 1963, and remained there for the rest of his career. He retired in 1975 and became an emeritus professor, remaining active in research for many years.

Szekeres was a talented musician who played both violin and viola. He started learning violin at the age of 6, and was a member of the Ku-ring-gai Philharmonic Orchestra in Sydney as well as the North Sydney Symphony Orchestra. In addition to maintaining his research in retirement, music and hiking were his principal interests. When Szekeres lost his driver's license the somewhat remote family home near Sydney became too hard to live in, and he and Klein returned to Adelaide, where their children lived. They died within an hour of each other in 2005, in hospital beds side-by-side, after almost 70 years of marriage.

While in Australia, Szekeres was active in mathematical outreach as well as in research. He devised problems for math competitions, and with his wife organised weekly enrichment and problem-solving sessions for high school students that grew and spread, and are still running. He also started a journal for high school students interested in math. Combinatorics was the focus of his research, which included a variety of areas of mathematics. In particular, he was very interested in general relativity, and is credited for developing the mathematical theory that forms the basis for our understanding of black holes. One of the central ideas in this work is the “Kruskal-Szekeres coordinate system” (discovered independently by these two researchers) that is an important element in Carl Sagan's science fiction novel Contact. Szekeres' other best-known contribution to math was in the early development of Ramsey theory, which grew out of his work with Erdős on the Happy Ending Problem. Szekeres had an early interest in the use of computers for mathematical research; he taught himself programming in about 1960, and used computers regularly in his research from that time forward. He produced about 90 papers in all, and supervised at least 16 doctoral students.

Szekeres was elected to the Australian Academy of Science' in 1963, and was awarded its Thomas Ranken Lyle Medal in 1968. He was also elected to the Hungarian Academy of Science. He was a founding member of the Australian Math Society in 1956, and served as its president from 1972 to 1974. The University of New South Wales granted him an honorary doctorate in 1976, shortly after his retirement. In 2001, Szekeres received the Australian Centenary Medal “for service to Australian society and science”. In 2002, he was appointed a Member of the Order of Australia with a similar citation. The Australian Math Society established a medal in 2001 that is named after him.

Sources: wikipedia, St. Andrews' math history web site, Australian Academy of Science, Journal of the Australian Math Society, the Math Genealogy Project, and obituary in the Sydney Morning Herald. Additional information and clarification by personal communication from Laci Babai.

Subsection B.2.67 Peter Guthrie Tait (1831—1901)

Peter Guthrie Tait was a 19th-century Scottish mathematician and physicist. He is probably best known in physics; his most significant work in mathematics was his research on the Four-Colour Conjecture. He was born in Dalkeith, Scotland. His father died when Tait was 6 years old, and the family moved to Edinburgh where they lived with one of his uncles. This uncle was enthusiastic about science, and likely sparked Tait's own interest.

In 1847 Tait enrolled at the University of Edinburgh. He stayed there for only one year before moving to Peterhouse College at Cambridge University. He completed his bachelor's degree in 1852. He remained at Cambridge as a fellow and lecturer until 1854, when he moved to Ireland to become a professor at Queen's College in Belfast.

In 1860 Tait returned to Scotland, becoming a professor of natural philosophy at the University of Edinburgh, a chair that he held almost until he died. It was only after his return to Scotland that Tait began to publish research papers. Some of his early publications were based on things he had studied in Ireland, where he met Sir William Rowan Hamilton (1805—1865). He had also published a book while still in Ireland. Tait knew and worked closely with Lord Kelvin; much of Tait's research was closer to physics than to pure mathematics. He also did research in knot theory and topology, and of course in combinatorics. It was in 1880 that Tait published his proof that the Four-Colour Conjecture is equivalent to the nonexistence of a planar snark (discussed in Section 15.3), which at the time he thought was a proof of the Four-Colour Conjecture. He wrote 16 textbooks, one of which was a joint work with Lord Kelvin that became a standard reference. He also wrote 133 research papers and 232 other publications.

Tait was married in 1857, and had 7 children. He died in Edinburgh in 1901. Tait was a keen golfer and also studied the mathematics and physics of golf. He had some significant disputes with other researchers, which seem to have been based on clinging to his own preferences, whether scientific (holding quaternions as superior to vectors) or patriotic (pro-British). Lord Kelvin quoted Tait as having said that “nothing but science is worth living for,”. Kelvin added that Tait seemed not to live by this maxim as he loved to read and had a terrific memory, which allowed him to be ready with apposite quotations for many situations.

Tait was elected a Fellow of the Royal Society of Edinburgh shortly after moving back to Scotland, and served as its general secretary from 1879 until his death in 1901. He won both the Gunning Victoria Jubilee Prize and the Keith Prize (twice) from the Royal Society of Edinburgh, and in 1866 received the Royal Medal of the Royal Society of London. He was awarded honorary degrees by the University of Glasgow and the University of Ireland. Tait was elected an honorary member of the scientific academies of Denmark, Holland, Ireland, and Sweden. A road in Edinburgh and a chair in the department of physics at the University of Edinburgh have been named for him.

Sources: wikipedia, St. Andrews' math history web site, Clerk Maxwell Foundation, and encyclopedia.com.

Subsection B.2.68 Larry Henry Thiel (1945—)

Larry Henry Thiel is an American-Canadian computer programmer and analyst who contributed to the computer-assisted proof that there is no finite projective plane of order 10. He was born in Trenton, New Jersey (in the United States). He received a bachelor's degree in mathematics from Michigan State University in 1967. He immigrated to Canada in 1968 to work at the University of Alberta as a programmer/analyst.

Thiel moved to Montreal in 1973 to work as a programmer/analyst in the brand new department of Computer Science at Sir George Williams University (which in 1974 merged with Loyola College to form Concordia University). Over time, he was promoted and ended up managing the technical staff and computing labs for the department. In this role, Thiel was often called on to write and/or optimise code being used by researchers. He retired in 2000.

Thiel is a co-author on at least 15 papers to which he was asked to contribute with his expertise in programming. Most notably, he was involved in proving the non-existence of a projective plane of order 10, with Clement Wing Hong Lam (1949—) and Stan Swiercz (1955—), as mentioned in Section 16.2. In his expository paper about this result, Lam says that Thiel “has a reputation of making any computer program run faster.”

Thiel has 4 children, including a son who now teaches computer science at Concordia, and with whom he sometimes collaborates.

Sources: Mathematical Association of America, Concordia University, Theorem of the Day, MathSciNet, and Concordia University. Updated and confirmed through personal communication.

Subsection B.2.69 Robin Thomas (1962—2020)

Robin Thomas was a Czech-American mathematician, renowned for his work in proving (with co-authors) tremendous results in graph theory including the Strong Perfect Graph Theorem. He was born in Prague, Czechoslovakia, where he completed his education with a doctorate from Charles University in 1985, under the supervision of Jaroslav Nešetřil. In 1987, while Czechoslovakia was still under communist control, Thomas emigrated to the United States for a postdoc at Ohio State University in Columbus, where he began to work with George Neil Robertson (1938—). In 1989 Thomas began working at Georgia Institute of Technology in Atlanta. He became a Regents' Professor there in 2010, and remained there until his death.

Thomas was diagnosed with Amyotrophic Lateral Sclerosis (ALS, also known as Lou Gehrig's disease) in about 2008. He was confined to a wheelchair for most of the last decade of his life. He died in 2020 at the age of 57. Thomas was married with 3 children. His wife is a professor at Georgia Tech whose research is in stochastic systems and computer simulation. He advised students to “Follow your passion, value your education, and work hard. Don't give up in the face of hardship, and have fun.”

Thomas' research interests were broad. He published in algebra, geometry, topology, and theoretical computer science as well as in combinatorics. His best-known achievements, though, lie in his outstanding work in structural graph theory, particularly his result on the Hadwiger Conjecture with Robertson and Paul Seymour (1950—), and their work on the Strong Perfect Graph Theorem, joined by Maria Chudnovsky (1977—). During his lifetime Thomas published more than 100 research papers, and supervised at least 16 doctoral students (several others were in progress when he died). His influence as a mentor was far broader, though; Thomas served as the head of the Algorithms, Combinatorics and Optimization Ph.D. program and kept a watchful eye on the progress of all of the students in that program. He maintained close ties with Charles University in what had become the Czech Republic, and frequently invited promising researchers to visit or complete a postdoc with him at Georgia Tech.

Thomas won the Fulkerson Prize twice: in 1994, and in 2009. He gave an invited talk at the International Congress of Mathematicians in 2006. In 2011 Thomas received the Karel Janeček Foundation Neuron Prize for Lifetime Achievement in Mathematics. He was named a fellow of the American Math Society in 2012, and of the Society for Industrial and Applied Mathematics in 2018. In 2016 he received the Class of 1934 Distinguished Professor Award at Georgia Tech, and gave the commencement address to graduate students. After his death, a graduate fellowship was established at Georgia Tech in his name.

Sources: wikipedia, Georgia Institute of Technology, Charles University, Charles University, and the Math Genealogy Project. Updated and confirmed through personal communication with Sigrun Andradöttir.

Subsection B.2.70 Pál Turán (1910—1976)

Pál Turán (who often used the anglicised version of his given name, “Paul”) was a Hungarian mathematician of Jewish descent. His research focus was in analytic number theory and although he is best known for his results in that field, in combinatorics he was responsible for establishing the research area of extremal graph theory. Turán was born in Budapest, Hungary. Until 1919, his surname was Rosenfeld. Anti-Jewish laws in Hungary limited the number of students of Jewish descent who could enter university, but Turán was able to enrol in the science university in Budapest (now Loránd Eötvös University) in 1928, thanks to placing highly in a national competition. While at university, Turán was part of an active group of young mathematical problem-solvers that included Tibor Gallai (who would go on to become a remarkable teacher and an eminent graph theorist, see the biographical sketch of Vera Sós (1930—)), Pál Erdős (1913—1996), Eszter Klein (1910—2005), and György Szekeres (1911—2005). Turán's collaborations with Erdős continued through the rest of his life.

Turán graduated with a teaching degree in 1933, and received a Ph.D. in 1935 with Fourier analyst Lipót (Leopold) Fejér as his advisor. His thesis was entitled “Az egész számok prímosztóinak számáról” (“On the number of prime divisors of integers”). Anti-semitic laws also prevented Turán from obtaining a position after his graduation, either at a university or as a school teacher. He survived by tutoring, finally obtaining a teaching job in 1938 at the high school of the Rabbinical Seminary. Turán married in 1939 and had a son from this first marriage.

From 1940 to 1944 (during the war) Turán spent several long periods in the forced labour batallions to which men of Jewish descent were conscripted. The situation was horrific; men in the labour batallions were subject to daily abuse. Turán's primary interest throughout his career was in analytic number theory. This was not a subject that he could easily work on without writing down large formulas as he went. During his time in labour camps, he discovered that he could perform graph theory research in his head. Many of the problems that Turán came up with in graph theory, as well as the solutions for some of them, came out of this terrible time. He said, “I got my best ideas while pulling wires, because then I could be alone and nobody noticed that I was thinking.” The full force of the Holocaust hit Hungary during the last year of the war. Turán survived, but his three siblings (a sister and two brothers) were murdered.

Remarkable stories came out of Turán's time in the labour camps. Some of these are told beautifully in his own words in the final source listed below, in an effort to illustrate “the enchantment and help [graph theory] gave me in the most difficult times of my life”. I summarise them here. Turán's initial task at labour camp involved carrying railway ties. The officer overseeing this work happened to hear one of Turán's companions call him by name; this officer was an engineer with an interest in math. In an amazing coincidence, as a civilian the officer worked as a proofreader in a publishing house of the Hungarian Academy of Sciences, and had been involved in the production of some of Turán's papers. This officer reassigned Turán to a lighter duty: directing crews to the proper piles of logs. This lighter work freed Turán's mind and he was able to consider (and ultimately solve) a problem that had occurred to him earlier in the year, as he mulled over the mathematical ramifications of some items in a letter he had received from his friend Szekeres, then a refugee in Shanghai.

In 1944 Turán was in a labour camp outside Budapest, working at a brick factory. Small wheeled trucks carried the bricks from the kilns along tracks to the storage yards; every kiln was connected by rail to every yard. Where the rails crossed, the trucks often jumped the tracks and spilled their loads, which caused problems for the workers. In this context, Turán began to wonder: in a complete bipartite graph (as this system represented), what is the minimum number of edge-crossings? Turán considered this and solved some related extremal problems, but not the general case of this one. Even later in 1944, while expecting at any day to be sent to the gas chambers, Turán kept up his spirits by trying to find bounds on a problem very similar to the Ramsey numbers: given \(\displaystyle n\text{,}\) what is the largest value \(\displaystyle M(n)\) such that any 2-edge-colouring of \(\displaystyle K_n\) (using red and blue, say) must contain either a red or a blue \(\displaystyle K_{M(n)}\text{?}\) One of his results in Ramsey theory is mentioned in Section 14.2.

After the war, in 1945 Turán was given a position at the science university in Budapest. He was promoted to the Chair of Algebra and Number Theory there in 1949. In 1952 he remarried, to Vera Sós (1930—); they had two children, one of whom also became a mathematician. In 1955 Turán was named head of the complex function theory department in the Mathematical Institute of the Hungarian Academy of Sciences.

Turán died at the age of 66. He had been ill from leukemia for about 6 years, but Sós concealed the diagnosis from him as she felt that the knowledge would only depress and limit him. Turán once referred to his research work as “building my pyramid”: developing his claim to immortality.

Turán's most significant research is likely his work in analytic number theory. He developed the power sum method, which has seen many uses over the years. He worked on many problems related to the Riemann Hypothesis; Erdős jokingly refers to Turán as a “heretic” for not believing in the Riemann Hypothesis. In graph theory he was the founder of what is now known as extremal graph theory: finding largest or smallest graphs that have certain properties, or bounding various characteristics of such graphs. This is now a major area of research, and it came out of a result Turán proved in his mind while performing forced labour. Turán produced more than 200 publications, and mentored generations of Hungarian mathematicians.

Turán was elected to the Hungarian Academy of Sciences in 1948, initially as an associate member, and then as a regular member in 1952. He was awarded Hungary's Kossuth Prize in 1948 and again in 1952. He served the János Bolyai Math Society in a variety of ways, including by acting as its president from 1963 to 1966. In 1975 he was awarded the Szele Prize for research mentorship by the János Bolyai Math Society. He was elected a fellow of the American Math Society, the Austrian Math Society, and the Polish Math Society.

Sources: wikipedia, St. Andrews' math history web site, Acta Arithmetica, Yivo Encyclopedia of Jews in Eastern Europe, the Math Genealogy Project, encyclopedia.com, Simons Foundation, and Journal of Graph Theory. Additional information and clarification by personal communication from Laci Babai.

Subsection B.2.71 John Cameron Urschel (1991—)

John Cameron Urschel is a Black Canadian-American who was a professional American football player, and is a mathematician. He was born in Winnipeg, Manitoba (in Canada). His parents divorced when he was very young, and he moved with his mother to Buffalo, New York (in the United States), which is where he grew up. When Urschel was a child, his first-grade teacher contacted his mother to say that he might need to repeat first grade, as he was not fitting in or demonstrating understanding. His mother, suspecting a possible racial element to this assessment, challenged the school to test him; Urschel performed extremely well, and the school suggested that he pass immediately into third grade instead.

In middle school, Urschel began to play sports, primarily as a way of fitting in, but later also as something to do with his father, who had played football at the University of Alberta. His father, who had moved to Buffalo to live nearby, also took graduate courses in math and engineering at the University of Buffalo to keep his mind active, and talked about them with Urschel. In 2004 Urschel's father gave him a math book in which he had inscribed: “To live a happy life, one has to be able to see the beauty that is around us. That sounds easy, but it is surprisingly difficult to do. It requires mental training. Studying mathematics is an ideal form of mental training. Mathematics strips away the dirt of the world to leave the beauty and purity of mathematical reasoning.”

Urschel studied mathematics at Pennsylvania State University, where he also played football as an offensive lineman. He earned his bachelor's degree and his Master's at Penn State, graduating in 2014. Upon graduation, Urschel was drafted by the Baltimore Ravens of the National Football League. He played three seasons for the Ravens before announcing his retirement in 2017 at the age of 26.

Even before his retirement, Urschel enrolled in the Ph.D. program in math at the Massachusetts Institute of Technology (MIT). He first applied during his rookie year, in 2015. His application was very atypical, but MIT decided to admit him to begin his studies in 2016. This was the major factor in Urschel's decision to retire from football. He quickly found that trying to complete a doctorate while playing professional football was too demanding, as the MIT program required that he maintain full-time student status. There were other factors involved in his decision also: his fiancée was pregnant, and new evidence about long-term effects of concussions kept emerging. Urschel completed his doctorate in 2021, with a thesis entitled “Topics in Applied Linear Algebra”, under the supervision of Michel Goemans. Urschel published a number of results before the end of his Ph.D., including his work with Jake Wellens (1992—) on the difficulty of testing whether or not a graph is \(\displaystyle k\)-planar that is mentioned in Section 15.1.

Urschel is married, with one child. His wife is a writer who assisted in the writing of his memoir, Mind and Matter: A Life in Math and Football. Urschel has been actively involved in outreach, particularly as a mentor and role model. He gives many public talks and interviews, in attempts to increase the representation of African American youth in mathematics. His status as a professional football player draws in an audience that might not otherwise be receptive to his messages about mathematics. As Urschel points out, “It's very hard to dream of being in a career if you can't relate to anyone who's actually in that field.” Of his own motivation he says, “Mathematics speaks to this side of me where I'm really curious and want to know why.” Urschel also enjoys playing chess.

Subsection B.2.72 Varāhamihira (499—587)

Varāhamihira was a 6th-century Indian mathematician, astronomer, and astrologer. He is also referred to as Varāha or Mihira. He was born in India; the exact place of his birth is unknown. One of his books says that he was educated at Kapitthaka, but it is not certain what this is now called, or where it is. What is known is that he worked at Ujjain; it was already an important mathematical centre, and he increased its prominence. There is a story that he used astrology to predict the death of the prince, and was given the emblem of the boar (varāha) in recognition of his wisdom, which led to his being known as Varāhamihira instead of Mihira.

One of Varāhamihira's most notable works is the Brhatsambitā (the “Great Compilation”), which covers topics including architecture, astronomy, weather, mathematics, perfumes, and more. The book has 106 chapters. It is in this work that he discussed the problem described in Example 3.2.6. He wrote that at least some of his work was a summary of previous results in works that have not survived. The mathematical content of the Brhatsambitā includes versions of the arithmetic triangle, formulas for computing binomial coefficients, and trigonometric identities.

Varāhamihira also wrote some other books on astronomy and astrology; his most famous work is the Pancasiddhantika (“Five Astronomical Canons”), which describes five books (by previous sages) that are no longer in existence. He knew Greek, and was aware of Greek scientific understanding as well as Indian and Babylonian knowledge.

Sources: wikipedia, St. Andrews' math history web site, the Free Press, and Sanskriti magazine.

Subsection B.2.73 Vadim Georgievich Vizing (1937—2017)

Vadim Georgievich Vizing was a pioneering graph theorist in the Soviet Union, best known for Vizing's Theorem. This is the decisive result on the chromatic index of graphs that now appears in many textbooks, including the one by Bondy and Murty referenced in the biographical sketch of John Adrian Bondy (1944—). He was born in Kiev, USSR (now Kyiv, Ukraine). As his mother was partly German, in 1947 (after World War II) the family was forced by the Stalin government to move to Siberia. Beginning in 1954, Vizing studied math at Tomsk State University in Siberia, graduating in 1959. He then moved to Moscow to work on his Ph.D. at the Steklov Institute of Mathematics. He was working in the area of function approximation, which he did not enjoy.

After being refused permission to switch topics, Vizing left Moscow in 1962 without completing his graduate work. He returned to Novosibirsk in Siberia, and worked at the Siberian branch of the Soviet Academy of Sciences in nearby Akademgorodok (one of the premier research centres in the Soviet Union) from 1962 to 1968, studying graph theory. He earned his doctorate there in 1966. A.A. Zykov acted as a mentor to him.

It was during his doctoral studies that Vizing published the result for which he is best known: that the edges of a graph can be coloured with at most \(\displaystyle \Delta+1\) colours, where \(\displaystyle \Delta\) is the maximum number of neighbours of any vertex. This is described in Section 14.1. The paper was written in Russian; the theorem is now considered essential content in graph theory courses. Vizing also introduced the much-studied idea of list colouring, in which the permitted colours for a vertex have to come from a predetermined list (each vertex has its own list). He posed the conjecture that the vertices and edges of a graph can be coloured with at most \(\displaystyle \Delta+2\) colours so that no pair of adjacent or incident objects has the same colour. This problem remains open to this day.

Vizing wanted to move back to Ukraine where it would be less cold. He was not given permission to return to Kiev. He worked in a number of small towns, and in 1974 was offered a position at the Academy for Food Technology in Odessa, where he remained for the rest of his career. In 1976 Vizing switched his research interests to scheduling problems, and only returned to the study of graph theory in 1995. He had retired by 2000, while continuing to pursue research. Vizing published more than 50 papers.

Vizing was unable to travel much outside of the Soviet Union during his career, as the Soviet government did not allow him to accept the invitations he received prior to the Perestroika reforms of the late 1980s. Vizing's pension was the equivalent of just $70USD/month; remarkably, he said that the additional $45USD/month he received from a European Union research grant enabled him to travel and meet colleagues. (Despite the effects of inflation, these numbers are as unimaginably low as they seem.) A world-class mathematician, the highest recognition Vizing received was the “Great Silver Medal of the Institute of Mathematics of the Siberian Department of the Russian Academy of Sciences”.

Sources: wikipedia, European Math Society, the Math Genealogy Project, and MathSciNet. Additional information and clarification by personal communication from Laci Babai.

Subsection B.2.74 Klaus Wagner (1910—2000)

Klaus Wagner was a German graph theorist best known for his pioneering work on topological graph theory and graph minors. He was born in Cologne, Germany. The family moved (within Cologne) in 1923, and Wagner continued to live in this house for the rest of his life, taking particular interest in maintaining the garden. Wagner enrolled at the University of Cologne in 1930, studying math, physics, and chemistry. By the end of 1934 he had already submitted his doctoral thesis, entitled “Über zwei Sätze der Topologie: Jordanscher Kurvensatz und Vierfarbensatz” (“On two theorems in topology: Jordan's curve theorem, and the four-colour theorem”), under the supervision of Karl Dörge. Dörge had been a student of Issai Schur (1875—1941). Wagner's Ph.D. was awarded in 1935.

In 1937, Wagner published Wagner's Theorem, characterising planar graphs as graphs that have no \(\displaystyle K_{3,3}\) or \(\displaystyle K_5\) minor. Much of his best-known research focussed around similar questions of planarity, minors, and colouring. He also made the conjecture that when proved became the Robertson-Seymour Theorem (a true magnum opus of the careers of George Neil Robertson (1938—) and Paul Seymour (1950—)), showing that every family of graphs that is closed under minors can be characterised by a finite collection of forbidden minors (this result is discussed further in Section 15.2). Wagner did spend a period of his career focussed more on other topics, including topology, calculus, and analysis. He returned to graph theory and particularly topological graph theory in the late 1950s. He organised many conferences on graph theory at Oberwolfach, and wrote an introductory book about graph theory that appeared in 1970.

After completing his doctorate, Wagner worked for a time as a meteorologist for the German air force at the airports in Cologne and Berlin. The Allies took over this work after the war ended, leaving Wagner out of work, so he returned to the University of Cologne. In Germany, one way to become a full professor is to complete a “habilitation”; essentially completing an additional thesis and examination at a more advanced level, typically while working in a junior position. Wagner went through this process, completing his habilitation in 1949.

Wagner was married in 1950, and had one son. He and his wife also adopted the daughter of close friends who had died in a car accident. Wagner taught for many years at the University of Cologne, becoming a professor in 1956. He moved in 1970 to the University of Duisberg, about an hour away from Cologne. He remained there until he retired in 1978, and continued active in mathematics after his retirement. He gave his last lecture at the age of 84.

Over the course of his career, Wagner is credited by the Math Genealogy Project with supervising 25 doctoral students; he wrote more than 60 research papers, and published 7 books. He was made an honorary professor at the University of Cologne in 1971, and received an honorary doctorate from the University of Duisburg in 1997.

Sources: wikipedia, Festcolloquium for Wagner, the Math Genealogy Project, and Memorial article in Results in Mathematics.

Subsection B.2.75 Jake Wellens (1992—)

Jake Wellens is an American combinatorist still in the early stages of his career. He was born in Philadelphia, Pennsylvania (in the United States) in 1992. During high school he enjoyed competing in math competitions. He moved to California for university, attending the California Institute of Technology beginning in 2011 and graduating in 2015.

Wellens moved back east to Massachusetts, enrolling in the Ph.D. program at the Massachusetts Institute of Technology (MIT). He worked under the supervision of Henry Cohn, and graduated in 2020 during the global pandemic. His thesis was entitled “Assorted results in boolean function complexity, uniform sampling and clique partitions of graphs”. It was at MIT that he met John Cameron Urschel (1991—) and they collaborated on several projects, including their work on the difficulty of testing whether or not a graph is \(\displaystyle k\)-planar mentioned in Section 15.1.

Sources: MIT, Main Line Times, and CalTech. Updated and confirmed through personal communication.

Subsection B.2.76 Richard Michael Wilson (1945—)

Richard Michael Wilson is an outstanding American design theorist best known for his impressive proofs establishing the existence of certain kinds of designs. He was born in Gary, Indiana (in the United States). He completed his bachelor's degree at Indiana University, graduating in 1966. He then moved to Columbus, Ohio, where he received a Master's degree from Ohio State University in 1968. He remained at Ohio State University to complete his doctorate in 1969 under the supervision of Dwijendra Kumar Ray-Chaudhuri (1933—). His thesis was entitled “An Existence Theory for Pairwise Balanced Designs”.

After completing his Ph.D., Wilson stayed in Columbus, immediately taking a position at Ohio State University. He worked his way up through the ranks there, becoming a professor in 1974. In 1980 he moved to Pasadena, California, where he accepted a job at the California Institute of Technology (CalTech). Wilson remained there until his retirement in 2014, and is now a professor emeritus at CalTech.

Among Wilson's many enormous contributions to combinatorics, here are just a few that are particularly relevant to the material in this book. During his graduate studies and in collaboration with Ray-Chaudhuri, Wilson determined exactly when Kirkman Triple Systems exist (see Theorem 18.1.11). Subsequently, in 1974 Wilson proved Wilson's Theorem, that BIBDs exist whenever \(\displaystyle v\) is sufficiently large and the natural divisibility conditions are satisfied, as discussed in Section 17.2. Wilson also produced the best known lower bounds for the number of mutually orthogonal Latin squares of order \(\displaystyle n\text{.}\) Wilson published more than 60 papers during his career. He had supervised at least 29 doctoral students as of 2021, including Gary McGuire (1967—). He is also a co-author of a graduate-level introductory textbook in combinatorics.

In 1975, Wilson won the Pólya Prize from the Society for Industrial and Applied Math. He held a research fellowship from 1975 to 1977, and was the Sherman Fairchild Distinguished Scholar at CalTech in the winter of 1976. He has held visiting appointments at the University of London, the University of Illinois at Chicago, and the University of Minnesota, and an adjunct appointment at the University of Waterloo from 1982 to 1987.

In the late 1970s Wilson developed an interest in historical flutes. He collects, studies, and performs on original European flutes from the 18th and 19th centuries, as well as on replicas from these periods or earlier. He jointly wrote a book that has been published, on flute performance in London in about 1830. His interest has broadened to traditional flutes from around the world, which he also collects and performs on, sometimes at poetry-readings (his wife is a poet).

Sources: wikipedia, Wilson's c.v., the Math Genealogy Project, and Designs, Codes, and Cryptography.

Subsection B.2.77 Wesley Stoker Barker Woolhouse (1809—1893)

Wesley Stoker Barker Woolhouse was a 19th century British actuary and editor, whose writings included mathematics, music, and actuarial science. He was born in North Shields, England, in 1809. When he was 13 he won a prize for math offered by The Ladies' Diary that was open to all ages. Woolhouse spent most of his career as an actuary, but also had interests in music, steam engines, and other diverse fields. He was active in publishing through much of his life, including both books and periodicals.

Woolhouse published a number of papers, problems, and solutions to problems over the next years, making his name known in mathematical circles. He was made deputy superintendent of the Nautical Almanac's office from 1830 to 1837, where he published tables and papers on astronomy. After a difference of opinion with his superior, he left the Almanac, and in 1839 he took a position as actuary of the International Loan fund. At one point in his career he was asked by Lord Ashley to calculate the distance walked by factory workers who had to retie threads when they broke, for the purposes of motivating a reform bill limiting the working day to 10 hours. Using probability, Woolhouse estimated the distance at 30 miles per day.

In 1841 the Ladies' Diary and the Gentleman's Diary were succeeded by the Lady's and Gentleman's Diary, and Woolhouse became the editor in 1844. He held this post until 1865; this annual periodical itself only lasted until 1871, so he was the editor for most of its existence. It was in this publication that (in his first year as editor) Woolhouse included the problem (see Section 18.1) that inspired Reverend Thomas Penyngton Kirkman (1806—1895) in his work on designs, and Kirkman's schoolgirl problem also appeared in the Diary in 1850.

During his lifetime, Woolhouse wrote and published books on subjects as diverse as geometry, musical theory, mortality in the Indian Army (this was an actuarial investigation), calculus, and coins and calendars. He co-founded the English Institute of Actuaries in 1848, and played a significant role in the development of actuarial science and the actuarial profession in England. Woolhouse also enjoyed needlework, and amused himself by calculating the exact length of thread he would need for various projects.

Sources: wikipedia, Journal of the Institute of Actuaries, wikipedia, and Monthly Notices of the Royal Astronomical Society.

Subsection B.2.78 Ming-Yao Xu (1941—)

Ming-Yao Xu is a contemporary Chinese mathematician who is responsible for the formation of two significant schools of mathematical research in China. He was born in Tianjin, China, during World War II (part of the period known in China as the War of Resistance Against Japan). During the war years the family traveled through the south of China evading Japanese attacks, settling in Peking in 1945. The Chinese Communist Revolution was still ongoing, and from 1945 to 1949 the Kuomintang (Nationalist Party) controlled Peking; the Chinese Communist Party captured the city in 1949.

His wartime experiences made Xu a strong supporter of the Communist Party. He joined the Communist Youth League in 1956 and was elected as a leader of the League at his school.

Xu enrolled in Peking University in 1959. His university years fell between periods of significant political movements in China, but life was not calm. Xu expressed his own opinions on politics and came under criticism from his classmates twice. Besides these political controversies, chaotic “teaching reform” efforts at the university during the time Xu was there made it difficult to follow prerequisites through anything like a standard curriculum. As a result it took Xu almost 6 years to complete his bachelor's degree; he graduated in 1965. Despite the chaos, Xu studied hard. His teacher Shisun Ding became a close mentor and led him to study group theory. Ding was later president of Peking University (1984—1989).

After completing university, Xu was assigned to work in the Tangshan District. His time in Tangshan coincided very closely with the 10 years of the tumultuous Cultural Revolution. For almost all of this period, Xu was officially assigned to teach at the Tangshan Number 5 Middle School, but the schools worked abnormally during this period and were even closed for the first few years. During these years, Xu was often assigned to manual labour including as a purchaser for the school cafeteria; in the Tangshan Steel Plant; and in the Kailuan Coal Mine. In 1976, Xu escaped the worst of the deadly Tangshan earthquake (7.8 on the Richter scale) as he happened to be visiting Beijing. On his return he found that his bed had been split in two and the floor had collapsed. Toward the end of this time China's restrictions on publishing mathematical research lightened; Xu's first paper (based on his undergraduate thesis) was published in 1976.

At this time there was a shortage of people in higher education in China. Peking University wanted Xu to return to teach, but the Tangshan Education Bureau refused to release him. Eventually Xu got around this by passing the postgraduate entrance examination. Xu returned to Peking University in 1978 at the age of 37 as a graduate student. He worked under the supervision of Xiaofu Duan (also known as Hsio-Fu Tuan) and Efang Wang, and finished his graduate studies in 1980, immediately beginning to teach.

During his 24 years at Peking University, Xu built around himself a school of Chinese researchers working on groups, and on group interactions with graphs and with maps. He also developed expertise in computational group theory. Xu's efforts included writing a new graduate textbook on group theory; organising a seminar; and building relationships with international experts. Chinese researchers had been largely isolated from the international research community for many years, so building international relations was a challenge. Xu organised international conferences, but also travelled extensively in order to build these relationships, which he then used to develop opportunities for his students. He made long visits to Australia in 1985, 1987, and 1989; during the first of these he visited Cheryl Elisabeth Praeger (1948—) and they discovered the Praeger-Xu graphs described in Section 12.5. Xu also visited Canada in 1991 and 1992; Slovenia in 1995; and South Korea in 1999. In 1993 Xu began to supervise his own doctoral students for the first time; in the 11 years between then and his retirement from Peking University, Xu supervised 13 Ph.D. students.

In 2003, just as Xu was preparing to retire from Peking University, he accepted an opportunity to become the first Distinguished Professor at Shanxi Normal University. Here too he developed a school of research. This time the focuses were on \(\displaystyle p\)-groups and computational group theory. Again, one of his first steps was to write a graduate-level textbook. After establishing this school of research, Xu retired again in 2010.

Xu enjoys music, and takes pleasure in reading literature and history. In his leisure time he has independently conducted an extensive study of modern Chinese history, with a particular focus on the political, cultural, and ideological history of China between 1957 and 1976. He has published about 10 scholarly articles in journals that focus on these topics. Xu is married; his daughter Jing Xu is an associate professor of math at Capital Normal University, having completed her doctorate under the supervision of Praeger.

Xu has written more than 90 mathematical research papers, principally in the areas of group theory and group actions on graphs. Several of his papers introduced important concepts that were subsequently studied extensively by other researchers. Groups and group actions on graphs are now major areas of mathematical research in China, and most Chinese researchers in these fields owe something to Xu, whether it is for his textbooks, the international relationships he established, or mentorship (direct or indirect). Xu has won the Science and Technology Progress Award from the Ministry of Education in China.

Sources: personal communications, Amazon, and MathSciNet. Most of the content of this biography comes from material shared by Shaofei Du, Xu's first Ph.D. student, from an article in preparation for the occasion of Xu's 80th birthday.

Subsection B.2.79 Kazimierz Zarankiewicz (1902—1959)

Kazimierz Zarankiewicz was a Polish mathematician, working in topology and other areas of mathematics. In graph theory, his name is best known for a problem he shared with Pál Turán (1910—1976). He was born in Częstochowa, then in the Austro-Hungarian empire (now in Poland). He enrolled at the University of Warsaw in 1919. By then, Warsaw was the capital of the newly-independent Poland. Zarankiewicz graduated with a Ph.D. in topology in 1923, although his thesis was not published until 1927. Topology was a major strength of researchers at the University of Warsaw at that time.

Poland followed the German tradition of the “habilitation”, in which young researchers complete an additional (more advanced) thesis and examination, typically while working at a junior level. Zarankiewicz completed his habilitation in 1929 while working as an assistant at the University of Warsaw, and was promoted. He left in 1930 to visit Vienna and Berlin, and was unable to obtain a position at the University of Warsaw on his return. He taught instead at the Warsaw University of Technology, the Agricultural College, and at the University of Tomsk in the years from 1931 to 1939.

In 1939, Nazi Germany invaded Poland. After the invasion, many academics were killed or sent to concentration camps, and universities were closed. During the war, Zarankiewicz continued to teach at the underground university that had been established, defying Nazi edict. In 1944 he was arrested for this, and sent to a concentration camp. He survived the camp, returned to Warsaw in 1945, and in 1946 took a position at the Warsaw University of Technology, where he was promoted to professor in 1948. Zarankiewicz also visited the United States in 1948, teaching at a number of universities including Harvard.

Zarankiewicz published at least 45 papers. His research was in topology, complex functions, number theory, and graph theory. The problem for which he is best known in graph theory (mentioned in Section 14.2) was originally stated as a problem about matrices, but can be reformulated in terms of bipartite graphs. Zarankiewicz traded problems with Turán: Turán was involved in finding an upper bound for the Zarankiewicz problem, and Zarankiewicz found and proved an upper bound on Turán's problem about the number of crossings in a complete bipartite graph, and conjectured that his upper bound is the correct answer. (This is one of the problems Turán formulated while in a labour batallion, and is mentioned in his biographical sketch.)

Zarankiewicz was also a noteworthy mentor, supervising Poland's Mathematical Olympiad (a competition for high school students) from 1949 to 1957, and remaining on as a member of the board. He also served as an editor for a math journal aimed at high school teachers. From 1948 to 1951 he was president of the Warsaw section of the Polish Mathematical Society, and he was a founder of the Polish Astronautical Society in 1956. Zarankiewicz died at the age of 57 in London, England, where he was attending the Congress of the International Astronautical Federation, of which he was the vice-president. There is a street in Warsaw named for him.

Sources: wikipedia, St. Andrews' math history web site, and encyclopedia.com.

Subsection B.2.80 Zhu Shijie (1249—1314)

Zhu Shijie was a 13th-century Chinese mathematician who wrote, travelled, and taught during the Yuan dynasty. He was born near what is now Beijing, China, in 1249. The Yuan dynasty came into power during his lifetime in 1279, when Kublai Khan united China. Zhu wrote two books that have survived: the Suan hsüeh Ch'i-mong (“Introduction to Computational Studies”) in 1299, and the Siyuan Yuchian (“Jade Mirror of the Four Unknowns”) in 1303. In the preface to the Siyuan Yuchian, Zhu says that he has travelled around China for more than 20 years, teaching mathematics. Widespread travel had only become safe once China was unified.

The Suan hsüeh Ch'i-mong is written as a textbook on elementary mathematics. It discusses basic computations, areas, and volumes. It also includes polynomials. It was lost in China for a long time, but had reached Korea and Japan, where it survived, and was translated back into Chinese in 1839.

The Siyuan Yuchian is Zhu's most important work. It includes algebra in four unknowns, finds cube roots and square roots, and uses matrix reduction to solve systems of linear equations. Most importantly for our purposes, it includes the arithmetic triangle (which Zhu calls “the table of the ancient method of powers”) and binomial coefficients, as discussed in Section 4.3. This book too was lost for a time. A copy was eventually found, but the version had passed through many hands and it is not always clear when or where material and errors may have been introduced.

Sources: wikipedia, St. Andrews' math history web site, and Mathigon.