Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Fall 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 Sept 10 everyone Open problem session at noon in E575 (University of Lethbridge) Please bring your favourite (math) problems. Anyone with a problem to share will be given about 5 minutes to present it. We will also choose most of the speakers for the rest of the semester. Sept 17 Joy Morris Calculating partition numbers at noon in E575 (University of Lethbridge) The partition number $$p(n)$$ is the number of ways that $$n$$ can be partitioned into a sum of smaller positive integers. At the SIAM Discrete Math conference in June, I attended a plenary talk by Ken Ono of Emory on how to calculate partition numbers. This topic incorporates both combinatorics and number theory. Ken Ono was kind enough to give me a copy of his slides so that I could present this topic in our seminar, and I will be using those slides for this talk. Sept 24 Amir Akbary On a Conjecture of Erdös at noon in E575 (University of Lethbridge) Let $$m$$ be an integer bigger than 1 and let $$P(m)$$ denote the largest prime divisor of $$m$$. In 1962, Erdös conjectured that $$\lim_{n\rightarrow \infty} \frac{P(2^n-1)}{n}=\infty.$$ In 2000, Ram Murty and Siman Wong conditionally resolved this conjecture, under the assumption of a celebrated conjecture in number theory. In this talk I will describe their work. Oct 1 Dave Morris Hamiltonian paths in solvable Cayley digraphs at noon in E575 (University of Lethbridge) Cayley graphs are very nice graphs that are constructed from finite groups. If the group is abelian, then it is easy to show that the graph has a hamiltonian cycle. It is conjectured that the nonabelian Cayley graphs also have hamiltonian cycles.       We will discuss a few recent results (both positive and negative) on the related problem where the graph is replaced by a directed graph, and the finite group is assumed to be solvable (which means it is not too far from being abelian). Oct 15 Soroosh Yazdani Local Szpiro Conjecture at noon in E575 (University of Lethbridge) The Szpiro conjecture is one of the big conjectures in number theory and Diophantine equations. It is equivalent to the ABC conjecture, and so it implies many interesting results. In this talk I will mention a conjecture that is motivated by the Szpiro conjecture, which seems much less strong than the Szpiro conjecture, even though it still has many interesting Diophantine applications. We will also present a few cases where we can prove this conjecture. Oct 22 Nathan Ng Additive Divisor Sums at noon in E575 (University of Lethbridge) The divisor function $$d(n)$$ equals the number of divisors of an integer $$n$$. In this talk I will discuss what is known about additive divisor sums of the shape $$D(N,r)=\sum_{n \le N} d(n) d(n+r)$$ where $$r$$ is a fixed positive integer. These sums were introduced by Ingham in 1926, who proved an upper bound for $$D(N,r)$$. This was later refined to an asymptotic formula by Estermann and over the years was further sharpened by a succession of authors, including Heath-Brown, Deshouilliers and Iwaniec, Motohashi, and Meurman. More recent evaluations of $$D(N,r)$$ makes use of the spectral theory of automorphic forms. I will also discuss more general additive divisor sums of the shape $$D_k(N,r) = \sum_{n \le N} d_k(n) d_k(n+r)$$ where $$k$$ is a natural number larger than 2 and where $$d_k(n)$$ equals the number of ordered $$k$$-tuples $$(n_1, \ldots, n_k)$$ such that $$n = n_1 \cdots n_k$$. Oct 29 Majid Shahabi Weil Conjectures at noon in E575 (University of Lethbridge) In 1949, Weil proposed a set of conjectures about the generating functions which are derived from counting the number of points on an algebraic variety over a finite field. Solving Weil's conjectures was one of the central mathematics projects of the twentieth century. These problems were totally solved by a group of people including Dwork, Grothendieck, and Deligne. In this talk, we present a historical background and state the assertions of Weil conjectures. We further explain some sentences about the ideas of the proofs. Nov 5 Farzad Aryan The distribution of $$k$$-tuples of reduced residues at noon in E575 (University of Lethbridge) Let $$q$$ be a natural number, and write $$P = \varphi(q)/q$$, that is $$P$$ is the probability that a randomly chosen integer is relatively prime to $$q$$. Let $$1 = a_1 < a_2 < \cdots < a_{\phi(q)} < q$$ be the reduced residues mod $$q$$ (integers co-prime to $$q$$ in increasing order). A quantity of central interest is $$V_\gamma (q) = \sum_{i=1}^{\phi(q)} (a_{i+1}- a_i )^ \gamma .$$ In 1940, Erdős conjectured that $$V_{\gamma }(q) \ll qP^{1-\gamma }.$$ Let $$\mathcal{D}=\lbrace h_1, h_2 , \cdots, h_s \rbrace$$ be an admissible set. We call $$a+h_1,\ldots, a+h_s$$ an $$s$$-tuple of reduced residues, if each of these numbers is co-prime with $$q$$. Study of $$s$$-tuples of reduced residues is an analogue to the study of $$s$$-tuples of primes. In this talk we prove estimates about the distribution of $$s$$-tuples of reduced residues and finally we prove an extension of Erdős's conjecture for $$s$$-tuples: $$V^{\mathcal{D}}_{\gamma }(q):=\sum_{a_i < q} ( a_{i+1} - a_i )^ \gamma \ll qP^{-s(\gamma-1) },$$ where the sum runs over the integers $$1 = a_1 < a_2 < \cdots < q$$ for which $$a_i+h_1,\ldots, a_i+h_s$$ is an $$s$$-tuple of reduced residues. Nov 23 Chris Godsil Continuous Quantum Walks on Graphs at noon in room B650 (University of Waterloo) If $$A$$ is the adjacency matrix of a graph $$X$$, then the matrix exponential $$U (t) = \exp(itA)$$ determines what physicists term a continuous quantum walk. They ask questions such as: for which graphs are there vertices $$a$$ and $$b$$ and a $$t$$ such that $$| U (t)_{a,b} | = 1$$? The basic problem is to relate the physical properties of the system with properties of the underlying graphs, and to study this we make use of results from the theory of graph spectra, number theory, ergodic theory.... My talk will present some of the progress on this topic. Nov 30 Heinz Bauschke An Invitation to Projection Models at 1pm in room D610 (UBC Okanagan) Feasibility problems, i.e., finding a solution satisfying certain constraints, are common in mathematics and the natural sciences. If the contraints have simple projectors (nearest point mappings), then one popular approach to these problems is to use the projectors in some algorithmic fashion to approximate a solution. In this talk, I will survey three methods (alternating projections, Dykstra, and Douglas-Rachford), and comment on recent advances and remaining challenges. Dec 3 Mark Thom Squarefree Values of Trinomial Discriminants at noon in E575 (UBC) The discriminant of a trinomial of the form $$x^n \pm x^m \pm 1$$ has the form $$\pm n^n \pm (n-m)^{n-m} m^m$$ when $$n$$ and $$m$$ are co-prime. We determine necessary and sufficient conditions for identifying primes whose squares never divide the discriminants arising from coprime pairs $$(n,m)$$. These conditions are adapted into an exhaustive search method, which we use to corroborate a heuristic estimate of the density of all such primes among the odd primes. The same results are used to produce a heuristic estimate of the density of squarefree values of these discriminants. We'll also look at an unlikely seeming family of divisors of the discriminants, arising from an elementary identity on them. This is joint work with David Boyd and Greg Martin.
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