Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Spring 2017 Talks are at noon on Monday in room B660 of University Hall 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 9 Darcy Best Transversals in Latin Arrays with many Symbols at noon in B660 (Monash University, Australia) A transversal of a latin square of order $n$ is a set of $n$ entries picked in such a way that no row, column or symbol is present more than once. As you add more symbols to a latin square, you expect the number of transversals to increase. We show that once the number of symbols reaches a certain threshold, the square is guaranteed to have a transversal. Jan 16 everyone Open problem session at noon in B660 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. Jan 23 Alia Hamieh Non-Vanishing of Central Values of Rankin-Selberg $\boldsymbol{L}$-Functions at noon in B660 (University of Lethbridge) In this talk, we discuss some results on the non-vanishing of the central values of Rankin-Selberg convolutions of families of Hilbert modular forms. Such results are obtained by establishing some asymptotics of certain twisted first and second moments. This is an on-going joint work with Naomi Tanabe. Jan 30 Nathan Ng A subconvexity bound in the $\boldsymbol{t}$-aspect for degree two L-functions at noon in B660 (University of Lethbridge) Let $f$ be a primitive modular form and $L_f(s)$ its associated $L$-function. Anton Good (1982) showed that $L_f(\tfrac{1}{2}+it) \ll |t|^{\frac{1}{3}+\varepsilon}$ in the case $f$ is a modular form on the full modular group. In 1987, Matti Jutila gave a different proof and this was later refined by Martin Huxley. In this talk, we explain how the Jutila-Huxley approach can be generalized to the case of modular forms with arbitrary character $\chi$, weight $k$, and level $N$. This is joint work with Andrew Booker (Bristol) and Micah Milinovich (Mississippi). Feb 6 Joy Morris Oriented Regular Representations at noon in B660 (University of Lethbridge) An oriented graph is a digraph with at most one arc between any pair of vertices. We say that the action of a group on a set of points is regular if it is sharply transitive; that is, there is exactly one group element mapping any point to any other point. An oriented regular representation (ORR) for a group $G$ is an oriented graph whose automorphism group is isomorphic to the regular action of $G$ on the vertices. In 1980, Babai asked which groups admit an ORR. I will discuss this problem, and present joint work with Pablo Spiga in which we showed that every non-solvable group admits an ORR. Feb 13 Habiba Kadiri Explicit results in prime number theory at noon in B660 (University of Lethbridge) The prime number theorem, proven in 1896, is one of the first major theorems in analytic number theory. It provides estimates for prime counting functions. In 1962, Rosser and Schoenfeld gave a method to estimate the error term in the approximation of the prime counting function $\psi(x)$. Since then, progress on the numerical verification of the Riemann Hypothesis and widening the zero-free region of the Riemann zeta function have allowed numerical improvements of these bounds. In this talk, we present various new explicit methods such as introducing some smooth weights and establishing some zero density estimates for the Riemann zeta function. We also present some explicit results for primes in short intervals and for primes in arithmetic progressions. Feb 27 Dave Morris Modern approach to the Traveling Salesman Problem at noon in B660 (University of Lethbridge) The Traveling Salesman Problem asks for the shortest route through a collection of cities. This classical problem is very hard, but, by applying Linear Programming (and other techniques), the optimal route has been found in test cases that have tens of thousands of cities. This talk will present some of the powerful methods that are explained in W.J.Cook's book "In Pursuit of the Traveling Salesman". Mar 6 Hadi Kharaghani An association scheme for twin prime powers at noon in B660 (University of Lethbridge) Twin prime powers are used in the construction of some very interesting combinatorial objects. In an old paper they are used in the construction of regular Hadamard matrices. In a recent work, they are used to show the existence of an infinite class of Hadamard matrices lacking a certain algebraic structure. In this talk I will discuss the use of twin prime powers in order to show the existence of a class of translation commutative association schemes. I will use the twin primes 3 and 5, as an example, to make the talk accessible to everyone. Mar 13 Forrest J. Francis Special Values Of Euler's Function at noon in B660 (University of Lethbridge) In 1909, Landau showed that $\limsup \frac{n}{\phi(n) \log\log{n}} = e^\gamma,$ where $\phi(n)$ is Euler's function. Later, Rosser and Schoenfeld asked whether there were infinitely many $n$ for which ${n}/{\phi(n)} > e^\gamma \log\log{n}$. This question was answered in the affirmative in 1983 by Jean-Louis Nicolas, who showed that there are infinitely many such $n$ both in the case that the Riemann Hypothesis is true, and in the case that the Riemann Hypothesis is false. One can prove a generalization of Landau's theorem where we restrict our attention to integers whose prime divisors all fall in a fixed arithmetic progression. In this talk, I will discuss the methods of Nicolas as they relate to the classical result, and also provide evidence that his methods could be generalized in the same vein to provide answers to similar questions related to the generalization of Landau's theorem. Mar 27 Sahar Siavashi On the Solutions of Certain Congruences at noon in B660 (University of Lethbridge) An odd prime $p$ is called a Wieferich prime (in base $2$), if $$2^{p-1} \equiv 1 \pmod {p^2}.$$ These primes first were considered by A. Wieferich in $1909$, while he was working on a proof of Fermat's last theorem. This notion can be generalized to any integer base $a>1.$ In this talk, we discuss the work that has been done regarding the size of the set of non-Wieferich primes and show that, under certain conjectures, there are infinitely many non-Wieferich primes in certain arithmetic progressions. Also we consider the congruence $$a^{\varphi(m)} \equiv 1 \pmod{m^2},$$ for an integer $m$ with $(a,m)=1,$ where $\varphi$ is Euler's totient function. The solutions of this congruence lead to Wieferich numbers in base $a$. In this talk we present a way to find the largest known Wieferich number for a given base. In another direction, we explain the extensions of these concepts to other number fields such as quadratic fields of class number one. We also consider the solutions of the congruence $$g^m-g^n \equiv 0 \pmod{f^m-f^n},$$ where $m$ and $n$ are two distinct natural numbers and $f$ and $g$ are two relatively prime polynomials with coefficients in the field of complex numbers. We prove this congruence has finitely many solutions. Apr 3 Amir Akbary Elliptic Sequences at noon in B660 (University of Lethbridge) An elliptic sequence is a solution over an arbitrary integral domain of the recursion $$W_{m+n}\, W_{m-n} \, = \, W_{m+1}\, W_{m-1}\, W_{n}^{2} - W_{n+1}\, W_{n-1}\,W_{m}^{2},$$ where $m, n \in \mathbb{Z}$. The theory of integral elliptic sequences was developed by Morgan Ward in 1948. We describe the fundamental classification theorem of Ward for these sequences. Our emphasis will be on the so called "singular" sequences and their relation to the classical Lucas sequences.
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