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 - Keshia Yap

Monday, March 16th, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Keshia Yap

Title: Dimension of magic squares over a field.

Abstract: In this talk, we will follow the proof of Charles Small's 1988 paper to compute the dimension of magic squares over fields. A magic square of size $n$ over a field $F$ is an $n \times n$ matrix for which every row, every column, the principal diagonal and the principal backdiagonal all have the same sum. The set of all magic squares is an $F$-vector space. We will prove that for $n \geq 5$, its dimension is $n^2 - 2n$ (for all $F$), and for $n

Number Theory Seminar - David Wehlau

Monday, March 9th, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: David Wehlau

Title: Planes in Finite Fields, Lehmer Numbers and the Multiplicative Order of Elements mod $p$.

Abstract: Let $\mathbb{F}_p$ denote the finite field of order $p$ and $\mathbb{F}$ its algebraic closure. Classifying the $\mathbb{F}$-representations of $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$ leads to a simply stated geometric problem involving $\mathbb{F}_p$-planes in $\mathbb{F}$. Solving this leads in turn to an infinite family of polynomials in $\mathbb{F}[t]$. These polynomials have a number of surprising algebraic and combinatorial properties and satisfy a recursion relation related to that studied by D.H. Lehmer in his thesis. This is joint work with H. E. A. Campbell.

This presentation will be accessible to graduate students and senior undergraduates.

Number Theory Seminar - Didi Zhang

Monday, March 2nd, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Didi Zhang

Title: Induced subgraphs of hypercubes and a proof of the sensitivity conjecture.

Abstract: In this talk, we follow Hao Huang’s 2019 paper and show that every $(2^{n−1} + 1)$-vertex induced subgraph of the $n$-dimensional cube graph has maximum degree at least $\sqrt{n}$. This is the best possible result, and it improves a logarithmic lower bound shown by Chung, Füredi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.

Number Theory Seminar - Ertan Elma (University of Waterloo)

Monday, February 24th, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Ertan Elma (University of Waterloo)

Title: Discrete Mean Values of Dirichlet L-functions.

Abstract: Let $\chi$ be a Dirichlet character modulo a prime number $p \geq 3$ and let $\mathfrak{a}_{\chi}:=(1-\chi(-1))/2$. Define the mean value $$ \mathcal{M}_{p}(s,\chi):=\frac{2}{p-1}\sum_{\substack{\psi \bmod p\\\psi(-1)=-1}}L(1,\psi)L(s,\chi\overline{\psi})$$ for a complex number $s$ such that $s\neq 1$ if $\mathfrak{a} _{\chi}=1.$ Mean values of the form above have been considered by several authors when $\chi$ is the principal character modulo $p$ and $\Re(s)>0$ where one can make use of the series representations of the Dirichlet $L$-functions being considered. In this talk, we will investigate the behaviour of the mean value $\mathcal{M}_{p}(-s,\chi)$ where $\chi$ is a non-principal Dirichlet character modulo $p$ and $\Re(s)>0$. Our main result is an exact formula for $\mathcal{M}_{p}(-s,\chi)$ which, in particular, shows that $$ \mathcal{M}_{p}(-s,\chi)= L(1-s,\chi)+\mathfrak a_\chi 2p^sL(1,\chi)\zeta(-s)+o(1), \quad (p\rightarrow \infty)$$ for fixed $0<\sigma:=\Re(s)<\frac{1}{2}$ and $|\Im s|=o\left(p^{\frac{1-2\sigma}{3+2\sigma}}\right)$.

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

Monday, February 3rd, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

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

Title: On the normal number of prime factors of Fourier coefficients of modular forms.

Abstract: Using recent advances in the theory of Galois representations, we will indicate how to prove that for a prime $p$, the $p$-th Fourier coefficient of any modular form (with integer coefficients) has $\log \log p$ prime factors for almost all primes $p$. This is joint work in progress with Kumar Murty and Sudhir Pujahari. It extends my earlier joint work with Kumar Murty where only eigenforms were considered.

Number Theory Seminar - Seoyoung Kim (Queen's University)

Monday, January 27th, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Seoyoung Kim (Queen's University)

Title: On the density of irreducible polynomials which generate $k$-free polynomials over function fields.

Abstract: Let $M \in \FF_{q}[t]$ be a polynomial, and let $k \geq 2$ be an integer. In this talk, we will compute the asymptotic density of irreducible monic polynomials $P\in\FF_{q}[t]$ for which $P+M$ is not divisible by the $k$-th power of any irreducible polynomial.

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

Monday, January 20th, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Francesco Cellarosi (Queen's University)

Title: Smooth arithmetical sums over k-free integers.

Abstract: We use partial zeta functions to analyse the asymptotic behaviour of certain smooth arithmetical sums over smooth $k$-free integers. This is joint work with Ram Murty.

Number Theory Seminar - Anup Dixit (Queen's University)

Monday, January 13th, 2020

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Anup Dixit (Queen's University)

Title: On Picard-type theorems involving $L$-functions.

Abstract: The little Picard's theorem states that any non-constant entire function takes all complex values or all complex values except one point. In a similar flavour, suppose $f$ is an entire function such that for complex values $a$ and $b$, the set of zeros of $f$ is same as the set where $f'$ takes values $a$ and $b$, then it is possible to show that $f$ is a constant function. Such results are called Picard-type theorems. In this talk, we will discuss similar questions for $L$-functions, where it is possible to prove much stronger results.

Number Theory Seminar - Seoyoung Kim (Queen's University)

Thursday, November 28th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Seoyoung Kim (Queen's University)

Title: Artin's primitive root conjecture for function fields without Riemann Hypothesis.

Abstract: Artin's primitive root conjecture for function fields is known by Bilharz in his thesis in 1937, which was conditional on the proof of the Riemann hypothesis for global function fields, which was proved by Weil in 1948. In this talk, we suggest a simple proof of Artin's primitive root conjecture for function fields unconditional on the Riemann hypothesis for global function fields by using the technique from the proof of the prime number theorem by Hadamard and de la Vall/'ee Poussin. This is joint work with M. Ram Murty.

Number Theory Seminar - Richard Leyland

Thursday, November 21st, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Richard Leyland

Title: Isogenies of Elliptic Curves with Complex Multiplication.

Abstract: In my thesis work I seek to answer Mazur's Question which asks if there exists any isomorphisms of mod $N$ Galois representations attached to elliptic curves that are not induced by isogenies. The first step in answering this question is determining which isogenies of elliptic curves are defined over a field $F$. In this talk, I will show how to construct isogenies between CM elliptic curves by using ideals of the endomorphism rings. In particular, we will see that if the field of definition $F$ does not contain the CM field, then we can reduce the problem to finding cyclic isogenies.

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