Department of Mathematics and Statistics

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

Number Theory - Siddhi Pathak (Queen's University)

Tuesday, February 12th, 2019

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

Speaker: Siddhi Pathak (Queen's University)

Title: Distribution of special values of $L$-series attached to Erdos functions.

Abstract: Inspired by Dirichlet's theorem that $L(1,\chi) \neq 0$ for a non-principal Dirichlet character $\chi$, S. Chowla initiated the study of non-vanishing of $L(1,f)$ for a rational valued, $q$-periodic arithmetical function $f$. In this context, Erdos conjectured that $L(1,f) \neq 0$ when $f$ takes values in $\{ -1, 1\}$. This conjecture remains open in the case $q \equiv 1 \bmod 4$ or when $q > 2 \phi(q) + 1$. In this talk, we discuss a density theoretic approach towards this conjecture and the distribution of these values.

Number Theory - Ahmet Muhtar Guloglu (Bilkent University, Turkey)

Tuesday, February 5th, 2019

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

Speaker: Ahmet Muhtar Güloğlu (Bilkent University, Turkey)

Title: Cyclicity of elliptic curves modulo primes in arithmetic progressions (joint work with Yildirim Akbal).

Abstract: Let E be an elliptic curve defined over the rational numbers. We investigate the cyclicity of the group of F_p-rational points of the reduction of E modulo primes in a given arithmetic progression as a special case of Serre's Cyclicity Conjecture.

Number Theory - Neha Prabhu (Queen's University)

Tuesday, January 29th, 2019

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

Speaker: Neha Prabhu (Queen's University)

Title: A probabilistic approach to analytic number theory.

Abstract: In 2005, Allan Gut showed that the distribution of a certain random variable, constructed using a zeta-distributed random variable, is compound Poisson. Exploiting this property, he reproved some well-known facts about the Riemann zeta function, and Selberg’s identity, using probability theory. In this talk, I present these results.

Number Theory - Steven Spallone (Indian Institute)

Tuesday, January 22nd, 2019

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

Speaker: Steven Spallone (Indian Institute of Science Education and Research, Pune)

Title: A Chinese Remainder Theorem for Young Diagrams.

Abstract: Given a natural number 't' and a Young Diagram 'Y', there is a notion of a "remainder of Y upon division by t", called the t-core of Y. Let s,t be relatively prime, and consider the map taking a given st-core Y to the pair consisting of its s-core and t-core. The fibres of this map are infinite. More precisely, we have proven that the cardinality of the set of length k members of a given fibre is a quasipolynomial in k, of degree st-s-t. This is joint work with K. Seethalakshmi.

Number Theory - Siddhi Pathak (Queen's University)

Tuesday, January 15th, 2019

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

Speaker: Siddhi Pathak (Queen's University)

Title: Dedekind zeta function at odd positive integers.

Abstract: Let $\zeta(s)$ denote the Riemann zeta-function. Thanks to Euler's evaluation and Lindemann's theorem on transcendence of $\pi$, we understand that $\zeta(2k)$ is transcendental for any positive integer $k$. However, the arithmetic nature of the values of $\zeta(s)$ at odd positive integers remains a mystery. Recently, significant progress was made concerning the irrationality of these values, with perhaps the most remarkable theorem being that infinitely many of $\zeta(2k+1)$ are irrational, which was shown by T. Rivoal in 2000.

Similarly, one can inquire regarding the arithmetic nature of values of the Dedekind zeta-function $\zeta_K(s)$ attached to a number field $K$. When $K$ is totally real, the values $\zeta_K(2k)$ were proven to be algebraic multiples of powers of $\pi$ by Siegel and Klingen, independently. But this question remains unsolved in all other cases. In this talk, we discuss how our current knowledge allows us to deduce certain irrationality results for $\zeta_K(2k+1)$, where $K$ is an imaginary quadratic field. This is joint work with Prof. M. Ram Murty.

Number Theory - Anup Dixit (Queen's University)

Tuesday, January 8th, 2019

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

Speaker: Anup Dixit (Queen's University)

Title: Large values on the 1-line for a family of L-functions.

Abstract: A classical problem in analytic number theory is to understand the values of L-functions in the critical strip. It is well-known that $|\zeta(1+it)|$ takes arbitrarily large values when $t$ runs through the real numbers. In 2006, Granville and Soundarajan conjectured that there exists arbitrarily large $t$ such that $|\zeta(1+it)|$ satisfies a certain lower bound. We discuss recent progress towards this conjecture and also generalize it to a family of L-functions. This is joint work with K. Mahatab.

Number Theory - Sonja Ruzic

Tuesday, November 27th, 2018

Time: 10:00-11:00 a.m.  Place: Jeffery Hall 422

Speaker: Sonja Ruzic

Title: Presentation of the paper "Maxima for Graphs and a New Proof of a Theorem of Turan", by T.S. Motzkin and E.G. Straus.

Abstract: In this talk we consider the following problem: Given a graph G with vertices 1, 2, ..., n, let S be the simplex in $\mathbb{R}^n$ given by the set $x={x_1, x_2, ..., x_n | \sum_{i=1}^{n}x_i=1, x_i \geq 0 \forall I}$. What is $\max_{x \in S} \sum_{(i, j)\in G}x_ix_j?$ Furthermore, a proof of a theorem of Turan, which gives an upper bound to the number of edges of a graph G which contains no complete subgraph of order k, will be presented.

Number Theory - Siqi Li (Queen's University)

Tuesday, November 20th, 2018

Time: 10:00-11:00 a.m.  Place: Jeffery Hall 422

Speaker: Siqi Li (Queen's University)

Title: Realizability of a set of integers as degrees of the vertices of a linear graph.

Abstract: In this talk, we consider the following problem: Let A be a non-decreasing sequence of positive integers of length n. Does there exist a graph G on n vertices v_1 to v_n such that A is the sequence formed by deg(v_1) to deg(v_n)? Furthermore, the realizability of a connected graph, simple graph and the biconnected graph for a given finite sequence of vertices degrees A is considered. The application of the above theorem involves the description of the structure for isomers in an organic chemical compound.

Number Theory - Brad Rodgers (Queen's University)

Tuesday, November 6th, 2018

Time: 10:00-11:00 a.m.  Place: Jeffery Hall 422

Speaker: Brad Rodgers (Queen's University)

Title: Integers in short intervals representable as sums of two squares.

Abstract: Consider the set S of integers that can be represented as a sum of two squares. How are the elements of S distributed? In particular, how many elements fall into a random "short interval". (The definition of short interval will be given in the talk.) For very short intervals elements of S seem to be laid down at random, but I will discuss evidence that this ceases to be the case for longer short intervals. In particular, I will discuss a function field analogue of this problem and a connection to z-measures, an object first investigated in the context of asymptotic representation theory. This is joint work with O. Gorodetsky.

Number Theory - Neha Prabhu (Queen's University)

Tuesday, October 30th, 2018

Time: 10:00-11:00 a.m.  Place: Jeffery Hall 422

Speaker: Neha Prabhu (Queen's University)

Title: Central limit theorems in number theory.

Abstract: In this talk, I will report on joint work in progress with Ram Murty, where we obtain central limit theorems for sums of quadratic characters, and eigenvalues of Hecke operators acting on spaces of holomorphic cusp forms.

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