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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

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

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

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

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

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

Andy Zucker, Department of Pure Mathematics, University of Waterloo

"Ultracoproducts and weak containment for flows of topological groups"

We develop the theory of ultracoproducts and weak containment for flows of arbitrary topological groups. This provides a nice complement to corresponding theories for p.m.p. actions and unitary representations of locally compact groups. For the class of locally Roelcke precompact groups, the theory is especially rich, allowing us to define for certain families of G-flows a suitable compact space of weak types. When G is locally compact, all G-flows belong to one such family, yielding a single compact space describing all weak types of G-flows.

his seminar will be held both online and in person:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Ákos Nagy, BEIT Canada

"On the hyperbolic Bloch transform"

Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, I will introduce the noncommutative Bloch transform for Fuchsian groups which I will call the hyperbolic Bloch transform (HBT). The HBT transforms wave functions on the hyperbolic plane to sections of irreducible, flat, Hermitian vector bundles over the orbit space and transforms the hyperbolic Laplacian into the covariant Laplacian. I will prove that the HBT is injective and “asymptotically unitary”. If time permits, I will talk about potential applications to hyperbolic band theory. This is a joint work with Steve Rayan (arXiv:2208.02749).

MC 5417

Rachael Alvir, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory IV"

We will continue looking at degree spectra, and see an application of forcing.

MC 5479

Jacques van Wyk, Department of Pure Mathematics, University of Waterloo

"Essentials of Schemes"

We continue reading Eisenbud and Harris, actually starting with morphisms of schemes this time.

MC 5417