Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Number Theory Seminar

Number Theory Seminar - M. Ram Murty (Queen's University)

Wednesday, August 7th, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

Speaker: M. Ram Murty (Queen's University)

Title: AN INTRODUCTION TO THE CIRCLE METHOD.

Abstract: In 1916, in their ground-breaking paper deriving an asymptotic formula for the partition function, Hardy and Ramanujan introduced a powerful method in analytic number theory called the circle method. In a series of subsequent papers, Hardy and Littlewood developed the method to study Waring's problem, Goldbach's conjecture and other additive questions in number theory. I will present a gentle introduction to this method and indicate how it can be used to solve Waring's problem following an arrangement due to Linnik.

Number Theory Seminar - Akshaa Vatwani (IIT Gandhinagar)

Wednesday, July 31st, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

Speaker: Akshaa Vatwani (IIT Gandhinagar)

Title: The Voronoi Summation Formula for a generalized divisor function.

Abstract: In 1904, Vorono\" {\i} formulated a conjecture for arithmetical functions, which is a generalization of the Poisson summation formula. He was able to establish this conjecture for the divisor function $d(n)$, giving a relation between Bessel functions and weighted sums of the divisor function. Such formulas have since been generalized to various arithmetical functions. We report on joint work with Atul Dixit, Bibekananda Maji and A. Sankaranarayanan, giving such a formula for the generalized divisor function:
$$
\sigma_{k,z} (n) = \sum_{d^k |n }d^z, \qquad \qquad k\in \mathbb N, \, z\in \mathbb{C}.
$$

Number Theory Seminar - Francesco Cellarosi (Queen's University)

Wednesday, July 24th, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

Speaker: Francesco Cellarosi (Queen's University)

Title: COMPUTING KRONECKER SYMBOLS IN THE UNIVERSAL COVER OF SL_2(R).

Abstract: The Kronecker symbol generalizes Jacobi and Legendre symbols, and is intimately connected to the theory of Dirichlet characters. For co-prime integers $c$ and $d$ such that $c \equiv 0 \bmod 4$ and $d \equiv 1 \bmod 4$, I will explain an algorithm to compute the Kronecker symbol $(c/d)$ using a free group inside the universal cover of SL_2(R). Joint work with J. Griffin.

Number Theory Seminar - Molly Liu (Queen's University)

Wednesday, July 17th, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

Speaker: Molly Liu (Queen's University)

Title: Eigenvalues and diameter of regular graphs.

Abstract: In graph theory, a regular graph is a graph where each vertex has the same number of neighbours. In this talk, we will discuss eigenvalues and diameter of regular graphs.

Number Theory Seminar - Richard Gottesman (Queen's University)

Wednesday, July 10th, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

Speaker: Richard Gottesman (Queen's University)

Title: Vector-Valued Modular Forms and Modular Linear Differential Equations

Abstract: The sequence of denominators of the Fourier coefficients of a modular form on a congruence subgroup is always bounded. It has been conjectured that the converse is also true. We will consider this problem in the context of vector-valued modular forms and explain a strategy for proving such an unbounded denominator result. In this talk, we will also explain how the module structure of vector-valued modular forms leads one to study modular linear differential equations. A key point is the importance of understanding the solutions of the modular linear differential equation at all of the cusps. No previous background on vector-valued modular forms will be assumed.

Number Theory - Robert Osburn (University College Dublin)

Wednesday, June 19th, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

Speaker: Robert Osburn (University College Dublin)

Title: Generalized Kontsevich-Zagier series via knots

Abstract: Over the past two decades, there has been substantial interest in the overlap between quantum knot invariants, q-series and modular forms. In this talk, we discuss one such instance, namely an explicit q-hypergeometric expression for the Nth colored Jones polynomial for double twist knots. As an application, we generalize a duality at roots of unity between the Kontsevich-Zagier series and the generating function for strongly unimodal sequences. This is joint work with Jeremy Lovejoy (Paris 7 and Berkeley).

Number Theory - Ahmet Guloglu (Bilkent University)

Wednesday, June 12th, 2019

Time: 3:00-4:00 p.m.  Place: Jeffery Hall 319

Speaker: Ahmet Guloglu (Bilkent University)

Title: Cubic Characters and some applications

Abstract: I will mainly focus on cubic Hecke characters and related applications such as non-vanishing results for related L-functions, one-level density and Kummer’s conjecture.

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