Events

Dylan McGinley, McMaster University

"Cohomogeneity One Ricci Solitons"

Here we study the Ricci Soliton equation in the steady case, utilising symmetry to reduce the problem to an ordinary differential equation. I will focus in particular on the case where the principal orbits of the group action are line bundles over Fano Kahler manifolds.

MC 5403

Wanchun Rosie Shen, Harvard University

"Du Bois singularities, rational singularities, and beyond"

We survey some extensions of the classical notions of Du Bois and rational singularities, known as the k-Du Bois and k-rational singularities. By now, these notions are well-understood for local complete intersections (lci). We explain the difficulties beyond the lci case, and propose new definitions in general to make further progress in the theory. This is joint work (in progress) with Matthew Satriano, Sridhar Venkatesh and Anh Duc Vo.

MC 5417

Shirly Geffen, Universität Münster

"Dynamical comparison and nonamenable groups"

We pull back paradoxical dynamical systems (e.g. hyperbolic  groups acting on their Gromov boundary), to paradoxical decompositions of the acting group itself. This allows to show that whenever such groups admit a minimal amenable topologically free action on a compact Hausdorff space, the attached crossed product C*-algebra is classified by K-theoretic data.

This is joint work with Eusebio Gardella, Julian Kranz, and Petr Naryshkin.

MC 5501

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