
Date 
Speaker 
Title 


Jan 10 
everyone 
Organizational meeting and problem session 

at 1:45pm
in M1060

(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.



Jan 24 
Hadi Kharaghani 
Projective Planes and Hadamard Matrices 

at 1:45pm
in M1060

(University of Lethbridge)

It is conjectured that there is no projective plane of order 12. Balanced splittable Hadamard matrices were introduced in 2018. In 2023, it was shown that a projective plane of order 12 is equivalent to a balanced multisplittable Hadamard matrix of order 144. There will be an attempt to show the equivalence in a way that may require little background.



Jan 31 
Ertan Elma 
A discrete mean value of the Riemann zeta function and its derivatives 

at 1:45pm
in M1060

(University of Lethbridge)

In this talk, we will discuss an estimate for a discrete mean value of the Riemann zeta function and its derivatives multiplied by Dirichlet polynomials. Assuming the Riemann Hypothesis, we obtain a lower bound for the 2$k$th moment of all the derivatives of the Riemann zeta function evaluated at its nontrivial zeros. This is based on a joint work with Kübra Benli and Nathan Ng.



Feb 7 
Samprit Ghosh 
Moments of higher derivatives related to Dirichlet $L$functions 

at 1:45pm
in M1060

(University of Calgary)

The distribution of values of Dirichlet $L$functions $L(s, \chi)$ for variable $\chi$ has been studied extensively and has a vast literature. Moments of higher derivatives has been studied as well, by Soundarajan, Sono, HeathBrown etc. However, the study of the same for the logarithmic derivative $L'(s, \chi)/ L(s, \chi)$ is much more recent and was initiated by Ihara, Murty etc. In this talk we will discuss higher derivatives of the logarithmic derivative and present some new results related to their distribution and moments at $s=1$.



Feb 14 
Abbas Maarefparvar 
Hilbert Class Fields and Embedding Problems 

at 1:45pm
in M1060

(University of Lethbridge)

The class number one problem is one of the central subjects in algebraic number theory that turns back to the time of Gauss. This problem has led to the classical embedding problem which asks whether or not any number field $K$ can be embedded in a finite extension $L$ with class number one. Although Golod and Shafarevich gave a counterexample for the classical embedding problem, yet one may ask about the embedding in 'Polya fields', a special generalization of class number one number fields. The latter is the 'new embedding problem' investigated by Leriche in 2014.
In this talk, I briefly review some wellknown results in the literature on the embedding problems. Then, I will present the 'relativized' version of the new embedding problem studied in a joint work with Ali Rajaei.



Thursday
Feb 29 
Félix Baril Boudreau 
The Distribution of Logarithmic Derivatives of Quadratic $L$functions in Positive Characteristic 

at 1:45pm
in M1040

(University of Lethbridge)

To each squarefree monic polynomial $D$ in a fixed polynomial ring $\mathbb{F}_q[t]$, we can associate a real quadratic character $\chi_D$, and then a Dirichlet $L$function $L(s,\chi_D)$. We compute the limiting distribution of the family of values $L'(1,\chi_D)/L(1,\chi_D)$ as $D$ runs through the squarefree monic polynomials of $\mathbb{F}_q[t]$ and establish that this distribution has a smooth density function. Time permitting, we discuss connections of this result with EulerKronecker constants and ideal class groups of quadratic extensions. This is joint work with Amir Akbary.



Mar 6 
Andrew Fiori 
Tight approximation of sums over zeros of $L$functions 

at 1:45pm
in M1060

(University of Lethbridge)

In various contexts explicit formulas relate sums over primes (eg: numbers or ideals) to sums over zeros of some corresponding $L$function(s). The aim of this talk is to explain how we tightly approximate these sums over zeros in the context where one has zero free regions and zero density results for the corresponding $L$function(s) and how we use this to get essentially best possible bounds for the error term in the prime number theorem.
This talk discusses joint work with Habiba Kadiri and Joshua Swidinsky as well as ongoing work with Mikko Jaskari and Nizar Bou Ezz.



Mar 13 
Sho Suda 
On extremal orthogonal arrays 

at 1:45pm
in M1060

(National Defense Academy of Japan)

An orthogonal array with parameters $(N,n,q,t)$ ($OA(N,n,q,t)$ for short) is an $N\times n$ matrix with entries from the alphabet $\{1,2,...,q\}$ such that in any of its $t$ columns, all possible row vectors of length $t$ occur equally often.
Rao showed the following lower bound on $N$ for $OA(N,n,q,2e)$:
\[
N\geq \sum_{k=0}^e \binom{n}{k}(q1)^k,
\]
and an orthogonal array is said to be complete or tight if it achieves equality in this bound.
It is known by Delsarte (1973) that for complete orthogonal arrays $OA(N,n,q,2e)$, the number of Hamming distances between distinct two rows is $e$.
One of the classical problems is to classify complete orthogonal arrays.
We call an orthogonal array $OA(N,n,q,2e1)$ extremal if the number of Hamming distances between distinct two rows is $e$.
In this talk, we review the classification problem of complete orthogonal arrays with our contribution to the case $t=4$ and show how to extend it to extremal orthogonal arrays.
Moreover, we give a result for extremal orthogonal arrays which is a counterpart of a result in block designs by Ionin and Shrikhande in 1993.



Mar 20 
Sarah Dijols 
Parabolically induced representations of $p$adic $G_2$ distinguished by $\mathrm{SO}_4$ 

at 1:45pm
in M1060

(University of British Columbia)

I will explain how the Geometric Lemma allows us to classify parabolically induced representations of the $p$adic group $G_2$ distinguished by $\mathrm{SO}_4$. In particular, I will describe a new approach, in progress, where we use the structure of the $p$adic octonions and their quaternionic subalgebras to describe the double coset space $P \backslash G_2/\mathrm{SO}_4$, where $P$ stands for the maximal parabolic subgroups of $G_2$.

