Department of Mathematics and Statistics

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

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.

Number Theory - Payman Eskandari (University of Toronto)

Tuesday, October 23rd, 2018

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

Speaker: Payman Eskandari (University of Toronto)

Title: On the transcendence degree of the field genarated by quadratic periods of a smooth curve over a number field.

Abstract: Grothendieck's period conjecture predicts the transcendence degree of the field generated by the periods of a smooth projective variety (or more generally, a pure motive) over a number field, in terms of the dimension of its Mumford-Tate group. For mixed motives a similar conjecture was made by Andre. The upper bound part of Grothendieck's conjecture was proved in the case of abelian varieties by Deligne (as a consequence of his "Hodge implies absolute Hodge" theorem for abelian varieties). The lower bound part of Grothendieck's conjecture is known for a CM elliptic curve, thanks to a theorem of G. V. Chudnovsky.

This talk is a report on an aspect of a work in progress with Kumar Murty, in which we use Hodge theoretic methods and Tannakian formalism to study quadratic and higher periods of a punctured curve. We start by some background material and motivation. In the end, we prove the upper bound part of Andre's conjecture for quadratic periods of a punctured elliptic curve, defined over a subfield of $\mathbb{R}$. The argument is quite formal, and in fact, applies to any extension of $H^1\otimes H^1$ by $H^1$.

Number Theory - Siddhi Pathak (Queen's University)

Tuesday, October 16th, 2018

Time: 10:15 a.m.  Place: Jeffery Hall 422

Speaker: Siddhi Pathak (Queen's University)

Title: On the Euler-Kronecker constants.

Abstract: Values of the series $\sum_{n=1}^{\infty} A(n)/B(n)$ where A(X) and B(X) are polynomials with suitable conditions to ensure convergence, have been studied by several authors in the past. The evaluation of these series can be considered as a generalization of Euler's theorem evaluating $\zeta(2k)$. In this talk, we study elliptic analogues of these series.

Number Theory - Anup Dixit (Queen's University)

Tuesday, October 9th, 2018

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

Speaker: Anup Dixit (Queen's University)

Title: On the Euler-Kronecker constants.

Abstract: In 2006, Y. Ihara introduced the Euler-Kronecker constant $\gamma_K$ attached to any number field $K$, which is a generalization of the Euler-Mascheroni constant. This constant surprisingly arises in several seemingly unrelated aspects of analytic number theory. Ihara studied this constant systematically and produced bounds on $gamma_K$ under GRH. In this talk, we prove unconditional bounds for $\gamma_K$ in some cases and discuss its connection to the Brauer-Siegel theorem.

Number Theory - Siddhi Pathak (Queen's University)

Tuesday, October 2nd, 2018

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

Speaker: Siddhi Pathak (Queen's University)

Title: On the values of the Epstein zeta function.

Abstract: Given a positive definite binary quadratic form, $Q(X,Y)$, the Epstein zeta function attached to $Q$ is given by $Z_Q(s) = \sum_{m,n} Q(m,n)^{-s}$, where the sum is over all tuples $(m,n)$ in $\mathbb{Z} \times \mathbb{Z}$, excluding $(0,0)$. This series converges absolutely for $Re(s)>1$. In this talk, we will present a result by J. R. Smart that 'evaluates' $Z_Q(k)$ for positive integers $k > 1$.

Number Theory - Neha Prabhu (Queen's University)

Tuesday, September 25th, 2018

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

Speaker: Neha Prabhu (Queen's University)

Title: The error term in the Sato-Tate theorem of Birch.

Abstract: In 1968, Birch proved a vertical analogue of the Sato-Tate conjecture for elliptic curves showing the asymptotic behaviour of $a_E(p)$, a quantity associated to an elliptic curve $E$ mod p. An error term for this result was obtained by Banks and Shparlinski in 2009 using the results of Katz and Neiderreiter. In this talk, we shall see that this error term can also be obtained in an elementary fashion using ideas in Birch's paper. This is joint work with Ram Murty.

Number Theory - Zhen Zhao (Queen's University)

Tuesday, September 18th, 2018

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

Speaker: Zhen Zhao (Queen's University) 

Title: Asymptotics of the dimensions of irreducible representations of the symmetric group

Abstract: We will discuss the results of the 1985 paper of S. V. Kerov and A. M. Vershik giving the asymptotic rates growth for the dimensions of the largest and 'typical' dimensions of the irreducible representations of the symmetric group of order n.

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