at noon
online

(University of Lethbridge)

Please bring your favourite (math) problems. Anyone with a problem to share will be given about 5 minutes to present it. We will also choose most of the speakers for the rest of the semester.

at noon
online

Let $\chi$ be a nonprincipal Dirichlet character modulo a prime number $p\geqslant 3$ and $L(s,\chi)$ denote the Dirichlet $L$-function associated with the character $\chi$. Define \begin{align*} \mathcal{M}(-s,\chi):=\frac{2}{p-1}\sum_{\substack{\psi \ (\mathrm{mod}\ p) \\\psi(-1)=-1}}L(1,\psi)L(-s,\chi\overline{\psi}), \quad \left(0<\Re(s)<\frac{1}{2}\right) \end{align*} and \begin{align*} \Delta(s,\chi):=\sum_{n=2}^{\infty}\frac{\chi(n)\Delta(n)}{n^{s}}, \quad (\Re(s)>2) \end{align*} where $\Delta(n)$ is the error term in the Prime Number Theorem given by \begin{align*} \Delta(n)=\sum_{m\leqslant n}\Lambda(m)-\frac{\Lambda(n)}{2}-n \end{align*} and $\Lambda(.)$ stands for the von Mangoldt function.

In this talk, we will obtain some identities for $\mathcal{M}(-s,\chi)$ and $\Delta(s,\chi)$. These identities show that $\mathcal{M}(-s,\chi)$ is related to $L(1-s,\chi)$ for $0<\Re(s)<\frac{1}{2}$, and $\Delta(s,\chi)$ is related to the zeros of the Dirichlet $L$-functions $L(s,\chi\psi)$ where $\psi(-1)=-1$ and $\frac{1}{2}<\Re(s)<1$, if such zeros exist.

at noon
online

Given a smooth plane curve $C$ defined over an arbitrary field $k$, we say that $C$ is transverse-free if it has no transverse lines defined over $k$. If $k$ is an infinite field, then Bertini's theorem guarantees the existence of a transverse line defined over $k$, and so the transverse-free condition is interesting only in the case when $k$ is a finite field $F_q$. After fixing a finite field $F_q$, we can ask the following question: For each degree $d$, what is the fraction of degree $d$ transverse-free curves among all the degree $d$ curves? In this talk, we will investigate an asymptotic answer to the question as $d$ tends to infinity. This is joint work with Brian Freidin.

at noon
online

Let $E$ be an elliptic curve over $\mathbb Q$, and let $a_p$ be the Frobenius trace for each prime $p$. In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies the convergence of the Nagao-Mestre sum $$ \lim_{x\to\infty} \frac{1}{\log x} \sum_{p < x} \frac{a_p \log p}{p}=-r + \frac{1}{2} , $$ where $r$ is the order of the zero of the $L$-function of $E$ at $s=1$, which is predicted to be the Mordell-Weil rank of $E(\mathbb Q)$. We show that if the above limit exists, then the limit equals $-r + \frac{1}{2}$, and study the connections to the Riemann hypothesis for $E$. We also relate this to Nagao's conjecture for elliptic curves. Furthermore, we discuss a generalization of the above results for the Selberg classes and hence (conjecturally) for the $L$-function of abelian varieties and graphs.

at noon
online

Paley graphs connect many branches of mathematics, notably combinatorics and number theory. Inspired by the Erdős-Ko-Rado (EKR) theorem for Paley graphs of square order (first proved by Blokhuis) and recent development of the study of EKR-type results, we study the EKR property, the EKR-module property, and the strict-EKR property of pseudo-Paley graphs from unions of cyclotomic classes using a mixture of ideas from number theory, finite geometry, and combinatorics. In particular, we extend the EKR theorem for Paley graphs to several families of pseudo-Paley graphs and solve a problem proposed by Godsil and Meagher.

The talk is based on joint work with Shamil Asgarli, Sergey Goryainov, and Huiqiu Lin.

at noon
in Room TBA

Given a finite transitive permutation group $G\leq \operatorname{Sym}(\Omega)$ and a set of permutations $\mathcal{F}\subset G$, we say that $\mathcal{F}$ is intersecting if any two elements of $\mathcal{F}$ agree on an element of $\Omega$. The intersection density of an intersecting set $\mathcal{F}$ of $G$ is $$\rho(\mathcal{F}):= |\mathcal{F}|\left(\frac{|G|}{|\Omega|}\right)^{-1}$$ and the intersection density of $G$ is $$\rho(G) =\max \left\{ \rho(\mathcal{F}) \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\}.$$

It was conjectured by Meagher, Razafimahatratra and Spiga in [On triangles in derangement graphs, J. Combin. Theory Ser. A, 180:105390, 2021] that if $G\leq \operatorname{Sym}(\Omega)$ is transitive of degree $pq$, where $p$ and $q$ are odd primes, then $\rho(G)=1$. I will talk about some recent progress on this conjecture for primitive groups.