
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}{p1}\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 $L(1s,\chi)$ 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 transversefree 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 transversefree 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 transversefree
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 $d$ transversefree 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 SwinnertonDyer 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 SwinnertonDyer formulated a conjecture which implies the convergence of the NagaoMestre 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 MordellWeil 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 
ErdősKoRado properties of pseudoPaley 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 ErdősKoRado (EKR) theorem for Paley graphs of square order (first proved by Blokhuis) and recent development of the study of EKRtype results, we study the EKR property, the EKRmodule property, and the strictEKR property of pseudoPaley 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 pseudoPaley 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.

