Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Spring 2012 All talks are at noon on Monday in E575 For more information, or to receive an email announcement of each week's seminar, contact Nathan Ng < ng AT cs DOT uleth DOT ca > or Dave Morris .
 Talks in the series this semester: (Click on any title for more info, including the abstract. Then click on it again to hide the info.)

 Date Speaker Title Jan 16 everyone Open problem session at noon in E575 (University of Lethbridge) Please bring your favourite (math) problems to share with everyone. Jan 23 Habiba Kadiri Zero density estimates for the zeros of the Riemann zeta function at noon in E575 (University of Lethbridge) An important tool in the analytic investigation of the distribution of primes consists in estimates for the density of zeros of the Riemann zeta function in the critical strip $$0< x < 1$$. We denote $$N(a,T)$$ the number of such zeros in the region $$\bigl\{\, a < x < 1 \mid 0< y< T \,\bigr\}$$. It is expected that the zeros close to the 1-line are rare (and thus that $$N(a,T)$$ decreases with $$a$$). Great efforts have been made to establish asymptotic results of the form $$N(a,T) \ll T^{c(a)(1-a)} (\log T)^A,$$ where $$c(a)$$ and $$A$$ are positive. However, there are very few explicit bounds for $$N(a,T)$$. In this talk, we will present the following result together with the ideas for its proof: $$N(a,T) < c_1(a) T + c_2(a) \log T - c_3(a),$$ where each $$c_i$$ is explicit and positive. Jan 30 Brandon Fodden Hilbert's Tenth Problem at noon in E575 (University of Lethbridge) Hilbert's Tenth Problem asks for an algorithm which can determine if an arbitrary Diophantine equation (with integral coefficients) has solutions. In 1970 Yuri Matiyasevich, building on the work of Martin Davis, Julia Robinson and Hilary Putnam, showed that no such algorithm may exist. In this talk, we give an outline of a proof of this, and discuss some applications and related problems. No specialist knowledge is assumed and everyone is welcome. (Click here for PDF slides from the talk.) Feb 6 Nathan Ng At least one-third of the zeros of the Riemann zeta function are on the half line at noon in E575 (University of Lethbridge) In 1942 Selberg proved that a positive proportion of the zeros of the Riemann zeta function lie on the half line. Unfortunately, he never determined a numerical value for this proportion. By a different method Levinson proved in 1974 that the proportion is at least one-third. In this talk I will explain Levinson's proof and I will also briefly discuss Conrey's improvement of Levinson's result to four-tenths. Feb 13 Amir Akbary Maass Forms at noon in E575 (University of Lethbridge) This is an introductory talk on Maass forms for $$\mathrm{SL}(2,\mathbb{Z})$$. These objects were first studied by H. Maass in 1949. They appeared as non-holomorphic analogues of modular forms. Maass called them waveforms. We follow the presentation given in Sections 1.6 and 1.9 of Bump's book (Automorphic Forms and Representations) (no seminar) Reading Week No seminar on Feb 20. Feb 27 Yuri Matiyasevich New conjectures about zeros of Riemann's zeta function at noon in C610 (Steklov Institute, Russia) In http://logic.pdmi.ras.ru/~yumat/personaljournal/artlessmethod/artlessmethod.php the speaker described a surprising method for (approximate) calculation of the zeros of Riemann's zeta function using terms of the divergent Dirichlet series. In the talk this method will be presented together with some heuristic "hints" explaining why the divergence of the series doesn't spoil its use. Several conjectures about the zeros of Riemann's zeta function will be stated including supposed new relationship between them and the prime numbers. Feb 29 Ted Dobson Groups that are transitive on all partitions of a finite set at 11am in E575 (Mississippi State University) Let $$\ell_1,\ldots,\ell_r$$ be positive integers whose sum is $$n$$. Let $$K_1,\ldots,K_r$$ be subsets of the $$n$$-element set $$[n] = \{1,\ldots,n\}$$ such that these sets form a partition $$P$$ of $$[n]$$ and $$\vert K_i\vert = \ell_i$$. We say that $$[\ell_1,\ldots,\ell_r]$$ is the shape of $$P$$. Let $${\cal P}$$ be the set of all partitions of $$[n]$$ with shape $$[\ell_1,\ldots,\ell_r]$$. We determine all subgroups of $$S_n$$ that are transitive on $${\cal P}$$ for every possible shape $$[\ell_1,\ldots,\ell_r]$$, as well as determine all subgroups of $$S_n$$ that are transitive on the set of all ordered partitions of every possible shape. As an application, we determine which Johnson graphs are isomorphic to Cayley graphs. This is joint work with Aleksander Malnič. Mar 5 Ce Bian Two approaches to compute GL(3) automorphic forms at noon in E575 (University of Calgary) In this talk we present methods using the functional equation and Voronoi summation formula to compute the Casimir eigenvalues and Hecke eigenvalues of certain automorphic forms on GL(3). Mar 12 Timothy Trudgian Some mathematics in voting at noon in E575 (University of Lethbridge) Many countries employ different voting systems, each with its own advantages and disadvantages. In Australia the 'preferential' voting system is used: candidates are numbered 1 through to n. One disadvantage is a theoretical occurence of non-monotonicity: essentially candidates can be ranked more highly and yet they, almost paradoxically, perform more poorly. The debate about the likelihood of such a paradox has influenced recent decisions (notably in the UK) about whether a particular voting system should be adopted. I will try to sketch a mathematical approach to this problem. Mar 19 Joy Morris The Probabilistic Method at noon in E575 (University of Lethbridge) Paul Erdös is well-known for the number and breadth of his publications. His most significant contributions to combinatorics involved using the "probabilistic method:" proving that various objects must exist by developing a probability model and proving that the probability of finding such an object is strictly greater than 0. I will introduce the probabilistic method, and work through examples of several theorems that can be proved using this method. Mar 26 Soroosh Yazdani Death of Synthetic Geometry at noon in E575 (University of Lethbridge) Since antiquity people have been intrigued by problems like the "problems of Appolonius:" construct a circle tangent to three given circles. This specific problem can be solved using compass and straight edge. Jacob Steiner in 1848 pointed out that when the circles are replaced by arbitrary conics, these problems become more involved. For one thing it should take 5 conics, rather than 3, to determine the unknown conic. Furthermore, the five conic rarely determine a single conic, which raises the question, known as the Steiner problem, of how many conics are tangent to 5 given conics. Steiner conjectured that there are $$6^5=7776$$ conics, but did not elaborate much on this conjecture. This was proved by Johann Bischoff in 1859, but it was soon realized that this is not the right answer to the question. In this talk I will explain how do we get this answer, why this is not the right answer, how to get the right answer, and why this is an example of a problem responsible for killing synthetic geometry, and for that matter, what do I mean by synthetic geometry. Apr 2 Dave Morris Strictly convex norms on amenable groups at noon in E575 (University of Lethbridge) It is obvious that the usual Euclidean norm is strictly convex, by which we mean that, for all $$x$$ and all nonzero $$y$$, either $$\|x + y\| > \|x\|$$, or $$\|x - y\| > \|x\|$$. We will discuss the existence of such a norm on an abstract (countable) group $$G$$. A sufficient condition is the existence of a faithful action of $$G$$ by orientation-preserving homeomorphisms of the real line. No examples are known to show that this is not a necessary condition, and we will combine some elementary measure theory and dynamics with the theory of orderable groups to show that the condition is indeed necessary if $$G$$ is amenable. This is joint work with Peter Linnell of Virginia Tech. Apr 10 Michael Coons Diophantine Approximation of Mahler Numbers at 12:15 in D631 (University of Waterloo) Let $$F(x) \in Z[[x]]$$ be a Mahler function; that is, there exist positive integers $$k \ge 2$$ and $$d \ge 1$$ and polynomials $$a_0(x), \ldots , a_d(x) \in \mathbb{Z}[x]$$ with $$a_0(x)a_d(x) \neq 0$$ such that $$\sum_{i=0}^d a_i(x)F\bigl(x^{k^i} \bigr) = 0 .$$ Let $$\xi$$ be a real number. The irrationality exponent $$\mu(\xi)$$ of $$\xi$$ is defined as the supremum of the set of real numbers $$\mu$$ such that the inequality $$|\xi − p/q| < q^{-\mu}$$ has infinitely many solutions $$(p, q) \in \mathbb{Z} \times \mathbb{N}$$. Last year (in a talk at the University of Lethbridge), I showed that the sum of the reciprocals of the Fermat numbers (which is a special value of a Mahler function) has irrationality exponent 2 and I conjectured that all reasonable special values of Mahler functions should have finite irrationality exponent. In this talk, I will present some of the history of Mahler functions and Diophantine approximation with a view towards the proof of the above-mentioned conjecture. Apr 16 Paul Buckingham $$p$$-adic $$L$$-functions, and derivatives of Artin $$L$$-functions at $$s = 0$$ at noon in D610 (University of Alberta) $$p$$-adic $$L$$-functions are traditionally obtained from the more classical Artin $$L$$-functions by interpolating their values at negative integers. Iwasawa gave an alternative construction which, surprisingly, involves only values at $$s = 0$$. We will discuss a relatively new object defined in terms of derivatives at $$s = 0$$, instead of simply values, which appears to extend Iwasawa's construction to a potentially new kind of $$p$$-adic $$L$$-function. The talk will assume no prior knowledge of $$p$$-adic numbers.
 Past semesters: Fall 2007 Fall 2008 Fall 2009 Fall 2010 Fall 2011 Spring 2008 Spring 2009 Spring 2010