Department of Mathematics and Statistics

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

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.

Number Theory - Brad Rodgers (Queen's University)

Wednesday, May 29th, 2019

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

Speaker: Brad Rodgers (Queen's University)

Title: The variance of counts of squarefrees in short intervals

Abstract: Consider the number of squarefree integers in a randomly chosen short interval. In this talk we will discuss a method for computing the variance of such a count. The estimate we arrive at improves an old result of R.R. Hall and confirms a conjecture of Keating and Rudnick in a restricted range. This is joint work with Ofir Gorodetsky, Bingrong Huang, and Maksym Radziwill.

Number Theory - Steven Spallone (IISER Pune)

Wednesday, May 22nd, 2019

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

Speaker: Steven Spallone (IISER Pune)

Title: Divisibility of Character Values of Symmetric Groups

Abstract: Fix a permutation $\sigma$ in some symmetric group $S_k$, and consider it as sitting in $S_n$ for $n \geq k$. Also fix a positive integer $d$. The probability that an irreducible character $\chi$ of $S_n$ evaluated at $\sigma$ will be a multiple of $d$ approaches $100\%$ as $n$ approaches infinity. We will sketch a proof, which is joint work with Jyotirmoy Ganguly and Amritanshu Prasad.

Number Theory - Anup Dixit (Queen's University)

Wednesday, May 15th, 2019

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

Speaker: Anup Dixit (Queen's University)

Title: On the distribution of the number of local prime factors of n

Abstract: Let $\omega(n)$ denote the number of distinct prime factors of $n$ and $\omega_y(n)$ denote the number of distinct prime factors of $n$ less than $y$. It was shown by Hardy and Ramanujan that typically $\omega(n)$ is $\log \log n$ up to an error term of $\sqrt{\log \log n}$. This was further generalized in the famous Erd\"{o}s-Kac theorem, which asserts that the probability distribution of $(\omega(n) - \log \log n)/(\sqrt{\log\log n})$ is the standard normal distribution.In this talk, we will prove analogous results for $\omega_y(n)$, which can be thought of as a local Erd\"{o}s-Kac theorem and describe its further implications. This is joint work with Prof. Ram Murty.

Number Theory - Siddhi Pathak (Queen's University)

Wednesday, May 8th, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 110

Speaker: Siddhi Pathak (Queen's University)

Title: Convolution sums of values of the Lerch zeta-function

Abstract: In 1887, Lerch introduced the function, $\Phi(z, \alpha, s) := \sum_{n=0}^{\infty} \frac{ z^n } { {(n + \alpha)}^s },$ for $|z| = 1$, $0 < \alpha \leq 1$ and $\Re(s)>1$. This function is a generalization of the Riemann zeta-function, the Hurwitz zeta-functions as well as the polylogarithms. In this talk, we discuss convolution sum identities of values of the Lerch zeta-function at positive integers. This study is inspired by similar identities for values of the Riemann zeta-function, which were known to Euler, and leads one naturally into the realm of multiple Hurwitz zeta-functions. This is joint work with Prof. M. Ram Murty.

Number Theory - Chantal David (Concordia University)

Tuesday, April 23rd, 2019

Time: 1:30-2:30 p.m.  Place: Jeffery Hall 319

Speaker: Chantal David (Concordia University)

Title: Moments of cubic Dirichlet twists over function fields (Joint work with A. Florea and M. Lalin.)

Abstract: We obtain an asymptotic formula for the mean value of $L$--functions associated to cubic characters over $\F_q[T]$. We solve this problem in the non-Kummer setting when $q \equiv 2 \pmod 3$ and in the Kummer case when $q \equiv 1 \pmod 3$. The proofs rely on evaluating averages of cubic Gauss sums over function fields, which can be done using the theory of metaplectic Eisenstein series. In the non-Kummer setting, we display some explicit cancellation between the main term and the dual term coming from the approximate functional equation of the $L$--functions.

Number Theory - Tariq Osman (Queen's University)

Tuesday, March 26th, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Tariq Osman (Queen's University)

Title: Counting Integer Points on Vinogradov's Quadric.

Abstract: Consider the variety defined by the pair of equations $x_1 + x_2 + x_3 = y_1 + y_2 + y_3$ and $ x_1^2 + x_2^2 + x_3^2 = y_1^2 + y_2^2 + y_3^2$, known as Vinogradov's quadric. Following a brief historical overview and a few motivational remarks, we will derive an asymptotic formula (due to Ragovskya) for the number of integer points on Vinogradov's quadric in a large box.

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