 Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Spring 2022 Until further notice, all talks will be given online For more information, contact Bobby Miraftab , Nathan Ng , or Raghu Pantangi . Talks are usually at noon on Monday. All times are Mountain Time.
 Click on any title for more info, including the abstract. Then click on it again to hide the info. When available, click on the mathtube icon to access a videorecording of the talk.

 Date Speaker Title Jan 24 everyone Organizational meeting and problem session 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. Feb 7 Ertan Elma Some identities concerning Dirichlet $L$-functions at noon online (University of Waterloo and University of Lethbridge) 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. No seminars while faculty was on strike February 10 to March 21 Mar 28 Shamil Asgarli On the proportion of transverse-free curves at noon online (University of British Columbia) 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. Apr 25 Seoyoung Kim From the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture at noon online (Queen's University) 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. May 2 Chi Hoi Yip Erdős-Ko-Rado properties of pseudo-Paley graphs of square order at noon online (University of British Columbia) 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. May 30 Andriaherimanana Sarobidy Razafimahatratra EKR problems for permutation groups at noon in Room TBA (University of Regina) 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.
 Past semesters: Fall F2007 F2008 F2009 F2010 F2011 F2012 F2013 F2014 F2015 F2016 F2017 F2018 F2019 F2020 F2021 Spring S2008 S2009 S2010 S2012 S2013 S2014 S2015 S2016 S2017 S2018 S2019 S2020 S2021