
Date 
Speaker 
Title 


Jan 13 
everyone 
Open problem session 

at noon
in B543

(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 20

Joy Morris

Regular Representations of Groups


at noon
in C620

(University of Lethbridge)

A natural way to understand groups visually is by examining objects on which
the group has a natural permutation action. In fact, this is often the way we
first show groups to undergraduate students: introducing the cyclic and dihedral
groups as the groups of symmetries of polygons, logos, or designs. For example,
the dihedral group $D_8$ of order $8$ is the group of symmetries of a square.
However, there are some challenges with this particular example of visualisation,
as many people struggle to understand how reflections and rotations interact as
symmetries of a square.
Every group $G$ admits a natural permutation action on the set of elements of $G$ (in fact,
two): acting by right (or left) multiplication. (The action by rightmultiplication is
given by $\{\tau_g: g \in G\}$, where $\tau_g(h)=hg$ for every $h \in G$.) This action is
called the right (or left) regular representation of $G$. It is straightforward
to observe that this action is regular (that is, for any two elements of the underlying set,
there is precisely one group element that maps one to the other). If it is possible to find
an object that can be labelled with the elements of $G$ in such a way that the symmetries
of the object are precisely the rightregular representation of $G$, then we call this
object a regular representation of $G$.
A Cayley (di)graph Cay$(G,S)$ on the group $G$ (with connection set $S \subset G$) is defined
to have the set $G$ as its vertices, with an arc from $g$ to $sg$ for every $s \in S$. It is
straightforward to see that the rightregular representation of $G$ is a subset of the
automorphism group of this (di)graph. However, it is often not at all obvious whether or not
Cay$(G,S)$ admits additional automorphisms. For example, Cay$(\mathbb Z_4, \{1,3\})$ is a
square, and therefore has $D_8$ rather than $\mathbb Z_4$ as its full automorphism group,
so is not a regular representation of $\mathbb Z_4$. Nonetheless, since a regular representation
that is a (di)graph must always be a Cayley (di)graph, we study these to determine when regular
representations of groups are possible.
I will present results about which groups admit graphs, digraphs, and oriented graphs as
regular representations, and how common it is for an arbitrary Cayley digraph to be a regular
representation.



Jan 27

Habiba Kadiri

Explicit results about primes in Chebotarev's density theorem


at noon
in W561

(University of Lethbridge)

Let $L/K$ be a Galois extension of number fields with Galois group $G$,
and let $C \subset G$ be a conjugacy class. Attached to each unramified
prime ideal $\mathfrak{p}$ in $\mathcal{O}_K$ is the Artin symbol
$\sigma_{\mathfrak{p}}$, a conjugacy class in $G$. In 1922 Chebotarev
established what is referred to his density theorem (CDT). It asserts
that the number $\pi_C(x)$of such primes with $\sigma_{\mathfrak{p}}=C$
and norm $N_{\mathfrak{p}} \le x$ is asymptotically
$\frac{C}{G}Li(x)$ as $x\to \infty$, where $Li(x)$ is the usual
logarithmic integral. As such, CDT is a generalisation of both the prime
number theorem and Dirichlet's theorem on primes in arithmetic
progressions. In light of Linnik's result on the least prime in an
arithmetic progression, one may ask for a bound for the least prime
ideal whose Artin symbol equals $C$. In 1977 Lagarias and Odlyzko proved
explicit versions of CDT and in 1979 Lagarias, Montgomery and Odlyzko
gave bounds for the least prime ideal in the CDT. Since 2012 several
unconditional explicit results of these theorems have appeared with
contributions by Zaman, Zaman and Thorner, Ahn and Kwon, and Winckler.
I will present several recent results we have proven with Das, Ng, and
Wong.



Feb 3

Selcuk Aygin

On the eta quotients whose derivatives are also eta quotients


at noon
in C620

(University of Lethbridge)

In classical qseries studies there are examples of eta quotients whose
derivatives are also eta quotients. The most famous examples can be found in
works of S. Ramanujan and N. Fine. In 2019, in a joint work with P. C. Toh, we
have given 203 pairs of such eta quotients, which we believe to be the complete
list (see "When is the derivative of an eta quotient another eta quotient?", J.
Math. Anal. Appl. 480 (2019) 123366). Recently, D. Choi, B. Kim and S. Lim have
given a complete list of such eta quotients with squarefree levels (see "Pairs
of etaquotients with dual weights and their applications", Adv. Math. 355
(2019) 106779). Their findings support the idea that our list is complete.
In this talk we introduce a beautiful interplay between eta quotients, their
derivatives and Eisenstein series. Then we share our work in progress (joint
with A. Akbary) in proving the completeness of our list beyond squarefree
levels.



Feb 10

Amir Akbary

Reciprocity Laws


at noon
in W561

(University of Lethbridge)

The Artin reciprocity law provides a solution to Hilbert's ninth
problem (9. Proof of the Most General Law of Reciprocity in any Number Field).
In this talk we provide an exposition of this theorem with emphasis on its
relation with the classical law of quadratic reciprocity and describe its
motivating role in the far reaching Langlands Reciprocity Conjecture.



Feb 24

PengJie Wong

Cyclicity of CM Elliptic Curves Modulo $p$


at noon
in W561

(University of Lethbridge)

Let $E$ be a CM elliptic curve defined over $\Bbb{Q}$. In light of the LangTrotter conjecture,
there is a question asking for an asymptotic formula for the number of primes $p\le x$ for which the
reduction modulo $p$ of $E$ is cyclic. This has been studied by Akbary, Cojocaru, Gupta, M.R. Murty,
V.K. Murty, and Serre. In this talk, we will discuss their work and some variants of the question.



Mar 2

Nathan Ng

Moments of the Riemann zeta function and mean values of long Dirichlet polynomials


at noon
in W561

(University of Lethbridge)

The $2k$th moments $I_k(T)$ of the Riemann zeta function have been studied extensively. In the late
90's, KeatingSnaith gave a conjecture for the size of
$I_k(T)$. At the same time ConreyGonek connected $I_k(T)$ to mean values of long Dirichlet
polynomials with divisor coefficients. Recently this has been further developed by ConreyKeating in
a series of 5 articles. I will discuss my work relating $I_3(T)$ to smooth shifted ternary additive
divisor sums and also recent joint work with Alia Hamieh on mean values of long Dirichlet
polynomials with higher divisor coefficients.



Mar 9

Nathan Ng

The size of prime number error terms


at noon
in W561

(University of Lethbridge)

Montgomery made a conjecture for the size of the error term
in the prime number theorem. Similarly, Gonek made a conjecture for
the size of the error term for the summatory function of the Mobius function.
I will discuss these conjectures and the connection to large
deviations of infinite sums of identically distributed random variables.
This is joint work with Amir Akbary and Majid Shahabi.



May 20

Daniel Fiorilli

On the distribution of the error term in Chebotarev's density theorem and applications


WED
at noon
(8pm in France)
online

(University of Ottawa and University of Paris  Saclay)

We will discuss both extreme and generic values of the error term in Chebotarev's density theorem. This will allow us to deduce applications on a conjecture of K. Murty on the least unramified prime ideal in a given Frobenius set as well as on asymptotic properties of Chebyshev's bias. This is joint work with Florent Jouve, and builds on ideas of RubinsteinSarnak, Ng, K. Murty, R. Murty, and others.



May 25

Olivier Ramaré

Explicit Average Orders: News & Problems


at noon
online

(Institut de Mathématiques de Marseille)

We mix some of the novelties that have occurred recently in the field
of explicit multiplicative number theory, together with some
questions that have not been answered yet and with several new results.



June 8

Forrest Francis

Explicit Improvements to the Burgess Bound


at 4pm
online

(UNSW Canberra, Australia)

The standard estimate for the size of a (Dirichlet) character sum is the wellknown Pólya–Vinogradov inequality. However, this inequality only uses the modulus of the character and does not consider the length of the sum. For a bound that does consider the length of the sum, we have a family of estimates known as the Burgess bound. Owing to the size of their effective ranges, the Burgess bound is sometimes considered a "better" estimate.
Despite this, in an explicit setting, some improvements to the Pólya–Vinogradov inequality end up yielding improvements to the Burgess bound. In this talk, we will look at two such results. One involves the leading constant for the Burgess bound for characters of prime modulus, while the other makes improvements to the effective range of the Burgess bound for odd quadratic characters. The second topic is joint work with Matteo Bordignon.



June 15

Ethan Lee

Goldbach's Conjecture and Squarefree Numbers


at 4pm
online

(UNSW Canberra, Australia)

We introduce the Goldbach conjecture and some heuristic evidence to support it. Goldbach's conjecture is hard, so we consider some relaxations of Goldbach's conjecture, which has lead to theorems by Helfgott and Dudek. Then, I give an account of the results and the main method used in joint work with F. Francis (submitted), which builds upon Dudek's result. Finally, we observe some conjectures, which would build upon the aforementioned results from myself and F. Francis.



June 22

Asif Zaman

An effective Chebotarev density theorem for fibres


at noon
online

(University of Toronto)

Let $K/k$ be a Galois extension of number fields. The Chebotarev density theorem asserts that, as $x$ tends to infinity, the primes of $k$ with norm at most $x$ equidistribute according to their splitting behaviour in $K$. How large must $x$ be to actually observe this equidistribution? Pierce, TurnageButterbaugh, and Wood have made remarkable progress on this central question for certain families of number fields. Their results rely on two key ingredients: (1) the Artin conjecture is known for every field in the family and (2) there are few common intermediate field extensions within the family. Dependence on (1) was removed in recent joint work with Thorner.
I will discuss forthcoming work with Lemke Oliver and Thorner where we prove an unconditional result that allows both (1) and (2) to be bypassed in many cases of interest. For example, we prove that almost all degree $n$ $S_n$extensions have GRHquality bounds on the $\ell$torsion subgroups of their class groups.



June 29

Shabnam Akhtari

How to count (cubic and quartic) binary forms


at noon
online

(University of Oregon)

In order to study interesting arithmetic, algebraic, analytic and geometric properties of binary forms, we often need a way to order them. I will discuss some natural ways to order binary forms F(x , y) with integer coefficients, especially those of degree 3 and 4. I will show some of my recent works as examples of the importance of understanding the invariant theory of integral binary forms in order to count these important arithmetic objects.

