at 2pm
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.

at 2pm
in M1060

Many results in algebraic graph theory can be viewed as upper bounds on the size of the automorphism group of graphs satisfying various hypotheses. These kinds of results have many applications. For example, Tutte's classical theorem on 3-valent arc-transitive graphs led to many other important results about these graphs, including enumeration, both of small order and in the asymptotical sense. This naturally leads to trying to understand barriers to this type of results, namely graphs with large automorphism groups. We will discuss this, especially in the context of vertex-transitive graphs of fixed valency. We will highlight the apparent dichotomy between graphs with automorphism group of polynomial (with respect to the order of the graph) size, versus ones with exponential size.

at 2pm
in M1060

In 1979, Erdős conjectured that for $k \ge 9$, $2^k$ is not the sum of distinct powers of $3$. That is, the set of powers of two (which is $2$-automatic) and the $3$-automatic set consisting of numbers whose ternary expansions omit $2$ has finite intersection. In the theory of automata, a theorem of Cobham (1969) says that if $k$ and $\ell$ are two multiplicatively independent natural numbers then a subset of the natural numbers that is both $k$- and $\ell$-automatic is eventually periodic. A multidimensional extension was later given by Semenov (1977). Motivated by Erdős' conjecture and in light of Cobham's theorem, we give a quantitative version of the Cobham-Semenov theorem for sparse automatic sets, showing that the intersection of a sparse $k$-automatic subset of $\mathbb{N}^d$ and a sparse $\ell$-automatic subset of $\mathbb{N}^d$ is finite. Moreover, we give effectively computable upper bounds on the size of the intersection in terms of data from the automata that accept these sets.

at 2pm
in M1060

(University of Lethbridge)

In 1992, Erdős, Granville, Pomerance, and Spiro conjectured that if $\mathcal{A}$ is a set of integers with asymptotic density zero then the preimage of $\mathcal{A}$ under $s(n)$, sum-of-proper-divisors function, also has asymptotic density zero. In this talk, we will discuss the verification of this conjecture when $\mathcal{A}$ is taken to be the set of integers with missing digits (also known as ellipsephic integers) by giving a quantitative estimate on the size of the set $s^{-1}(\mathcal{A})$. This is joint work with Giulia Cesana, Cécile Dartyge, Charlotte Dombrowsky and Lola Thompson.

at 2pm
in M1060

Given an abelian variety $A$ defined over a number field, a conjecture attributed to Serre states that the set of primes at which $A$ admits ordinary reduction is of positive density. This conjecture had been proved for elliptic curves (Serre, 1977), abelian surfaces (Katz 1982, Sawin 2016) and certain higher dimensional abelian varieties (Pink 1983, Fite 2021, etc).

In this talk, we will discuss ideas behind these results and recent progress for abelian varieties with non-trivial endomorphisms, including the case where $A$ has almost complex multiplication by an abelian CM field, based on joint work with Cantoral-Farfan, Mantovan, Pries, and Tang.

Apart from ordinary reduction, we will also discuss the set of primes at which an abelian variety admits basic reduction, generalizing a result of Elkies on the infinitude of supersingular primes for elliptic curves. This is joint work with Mantovan, Pries, and Tang.

at 2pm

We establish a Weyl-type estimate for exponential sums over irreducible elements in function fields. As an application, we generalize an equidistribution theorem of Rhin. Our estimate works for polynomials with degree higher than the characteristic of the field, a barrier to the traditional Weyl differencing method. In this talk, we briefly introduce Lê-Liu-Wooley's original argument for ordinary Weyl sums (taken over all elements), and how we generalize it to estimate bilinear exponential sums with general coefficients. This is joint work with Jérémy Campagne (Waterloo), Thái Hoàng Lê (Mississippi) and Yu-Ru Liu (Waterloo).

at 1:30pm
in M1040

Let $\mathbb{Z}$ be the ring of integers, and let $\mathbb{F}_p[t]$ be the ring of polynomials in one variable defined over the finite field $\mathbb{F}_p$ of $p$ elements. Since the characteristic of $\mathbb{Z}$ is 0, while that of $\mathbb{F}_p[t]$ is the positive prime number $p$, it is a striking theme in arithmetic that these two rings faithfully resemble each other. The study of the similarity and difference between $\mathbb{Z}$ and $\mathbb{F}_p[t]$ lies in the field that relates number fields to function fields. In this talk, we will investigate some Diophantine problems in the settings of $\mathbb{Z}$ and $\mathbb{F}_p[t]$, including Fermat's Last Theorem and Waring's problem.

at 2pm

(University of Lethbridge)

Graphical and Digraphical Regular Representations (GRRs and DRRs) are a concrete way to visualise the regular action of a group, using graphs. More precisely, a GRR or DRR on the group $G$ is a (di)graph whose automorphism group is isomorphic to the regular action of $G$ on itself by right-multiplication.

For a (di)graph to be a DRR or GRR on $G$, it must be a Cayley (di)graph on $G$. Whenever the group $G$ admits an automorphism that fixes the connection set of the Cayley (di)graph setwise, this induces a nontrivial graph automorphism that fixes the identity vertex, which means that the (di)graph is not a DRR or GRR. Checking whether or not there is any group automorphism that fixes a particular connection set can be done very quickly and easily compared with checking whether or not any nontrivial graph automorphism fixes some vertex, so it would be nice to know if there are circumstances under which the simpler test is enough to guarantee whether or not the Cayley graph is a GRR or DRR. I will present a number of results on this question.

This is based on joint work with Dave Morris and with Gabriel Verret.

at 2pm
in M1060

(University of Lethbridge)

Historically, the notion of Pólya fields dates back to some works of George Pólya and Alexander Ostrowski, in 1919, on entire functions with integer values at integers; a number field $K$ with ring of integers $\mathcal{O}_K$ is called a Pólya field whenever the $\mathcal{O}_K$-module $\{f \in K[X] \, : \, f(\mathcal{O}_K) \subseteq \mathcal{O}_K \}$ admits an $\mathcal{O}_K$-basis with exactly one member from each degree. Pólya fields can be thought of as a generalization of number fields with class number one, and their classification of a specific degree has become recently an active research subject in algebraic number theory. In this talk, I will present some criteria for $K$ to be a Pólya field. Then I will give some results concerning Pólya fields of degrees $2$, $3$, and $6$.

at 2pm
in M1060

(University of Lethbridge)

This talk will begin with a study on explicit bounds for $\psi(x)$ starting with the work of Rosser in 1941. It will also cover various improvements over the years including the works of Rosser and Schoenfeld, Dusart, Faber-Kadiri, Platt-Trudgian, Büthe, and Fiori-Kadiri-Swidinsky. In the second part of this talk, I will provide an overview of my master's thesis which is a survey on 'Estimating $\pi(x)$ and Related Functions under Partial RH Assumptions' by Jan Büthe. This article provides the best known bounds for $\psi(x)$ for small values of $x$ in the interval $[e^{50},e^{3000}]$. A distinctive feature of this paper is the use of Logan's function and its Fourier Transform. I will be presenting the main theorem in Büthe's paper regarding estimates for $\psi(x)$ with other necessary results required to understand the proof.

at 2pm
in M1060

The size function $h^0$ for a number field is analogous to the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. Van der Geer and Schoof conjectured that $h^0$ attains its maximum at the trivial class of Arakelov divisors if that field is Galois over $\mathbb{Q}$ or over an imaginary quadratic field. This conjecture was proved for all number fields with the unit group of rank $0$ and $1$, and also for cyclic cubic fields which have unit group of rank two. In this talk, we will discuss the main idea to prove that the conjecture also holds for totally imaginary cyclic sextic fields, another class of number fields with unit group of rank two. This is joint work with Peng Tian and Amy Feaver.