Events

Alex Cowan, Harvard University

"A twisted additive divisor problem"

What correlation is there between the number of divisors of N and the number of divisors of N + 1? This is known as the classical additive divisor problem. This talk will be about a generalized form of this question: I’ll give asymptotics for a shifted convolution of sum-of-divisors functions with nonzero powers and twisted by Dirichlet characters. The spectral methods of automorphic forms used to prove the main result are quite general, and I’ll present a conceptual overview. One step of the proof uses a less well-known technique called “automorphic regularization” for obtaining the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable.

MC 5417

Joey Lakerdas-Gayle, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory V"

We will discuss 1-generics following Antonio Montalbán's monograph.

MC 5479

Xiao Zhong, Department of Pure Mathematics, University of Waterloo

"Harmonic Functions on the Berkovich Projective Line"

We introduce the harmonic functions and explore their properties. As a byproduct, we prove the uniqueness of the equilibrium distribution. The materials in this presentation cover the first half of the chapter 7 in Baker-Rumely's monograph on "Potential Theory and Dynamics on the Berkovich Projective Line".

MC 5417

Timothy Ponepal, Wilfrid Laurier University

"The flow of the horizontal lift of a vector field"

Let $E$ be a vector bundle over a manifold $M$, and let $\nabla$ be a connection on $E$. Given a vector field $X$ on $M$, the connection determines its horizontal lift $X^h$, which is a vector field on the total space of $E$. We will show that the flow of $X^h$ is related to parallel transport with respect to $\nabla$. If time permits, we will show that in the special case when $E$ is a rank 3 oriented real vector bundle with fibre metric, the flow of $X^h$ preserves the cross product on the fibres.

MC 5403

Yash Totani, Department of Pure Mathematics, University of Waterloo

"Mellin Transforms"

Mellin transforms, a powerful mathematical tool that often stands in the shadow of its more popular counterpart, the Laplace transform, have found remarkable applications across various disciplines. In this talk, we explore the analytic properties of Mellin transforms and as an example, provide a rich solution to the following equation due to Ramanujan

$$\sum_{n=1}^\infty\frac{n^{13}}{e^{2\pi n}-1}=\frac{1}{24}. $$

Given extra time, we will delve into their connection with Dirichlet series.

MC 5501

Gian Cordana Sanjaya, Department of Pure Mathematics, University of Waterloo

"The Gluing Construction of Schemes"

We work on some concrete examples of morphisms of schemes. We will then continue with the gluing construction.

MC 5417

Kaleb D. Ruscitti, Department of Pure Mathematics, University of Waterloo

"Understanding the local behaviour of a toric degeneration of the moduli of holomorphic bundles"

There is a toric degeneration of the moduli space of holomorphic semi-stable rank 2 bundles on a Riemann surface, induced by a degeneration of the Riemann surface along 2g-2 loops. Biswas and Hurtubise gave an explicit local description of this degeneration in terms of the connection matrices that define the holomorphic structure on the bundles.

In this talk, I will discuss my ongoing project to understand how the functions on the moduli space behave under this degeneration. I will begin by reviewing the relationship between sections of bundles on toric varieties and lattice points in their moment polytopes. Then I will try to use this theory to work out explicitly what happens to functions in the case of the toric degeneration for the aforementioned moduli space.

MC 5417

Nicolas Chavarria, Department of Pure Mathematics, University of Waterloo

"Continuous Stable Regularity"

We discuss joint work with G. Conant and A. Pillay regarding a version of the Malliaris-Shelah stable regularity lemma realized in the context of continuous logic, which allows us to speak about the structure of stable functions of the form $f:V\times W\to [0,1]$, where we think of $V$ and $W$ as the parts of a "weighted'' bipartite graph. In the process, we will also mention some results about the structure of local Keisler measures in this setting.

MC 5479

Carlos Valero, McGill University

"The Calderón problem for U(N)-connections coupled to spinors"

The Calderón problem refers to the question of whether one can determine the Riemannian metric on a manifold with boundary from its "Dirichlet-to-Neumann (DN) map", which maps a function on the boundary to the normal derivative of its harmonic extension. In this talk, we define the analogue of the DN map for the spinor Laplacian twisted by a unitary connection and show that it is a pseudodifferential operator of order 1, whose symbol determines the Taylor series of the metric and connection at the boundary. We go on to show that if all the data are real-analytic, then the spinor DN map determines the connection modulo gauge.

MC 5417

Myrto Mavraki, University of Toronto Mississauga

"Dynamics, number theory, and unlikely intersections"

Fruitful interactions between arithmetic geometry and dynamical systems have emerged in recent years. In this talk I will illustrate how insights from complex dynamics can be employed to study problems from arithmetic geometry. And conversely how arithmetic geometry can be used in the study of dynamical systems. The motivating questions are inspired by a recurring phenomenon in arithmetic geometry known as 'unlikely intersections' and conjectures of Pink and Zilber therein. More specifically, I will discuss work toward understanding the distribution of preperiodic points in subvarieties of families of rational maps. 

MC 5479