Events

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

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

Alina Stancu, Concordia University

"On the fundamental gap of convex domains in hyperbolic space"

The difference between the first two eigenvalues of the Dirichlet Laplacian on convex domains of R^n and, respectively S^n, satisfies the same strictly positive lower bound depending on the diameter of the domain. In work with collaborators, we have found that the gap of the hyperbolic space on convex sets behaves strikingly different even if a stronger notion of convexity is employed. This is very interesting as many other features of first two eigenvalues behave in the same way on all three spaces of constant sectional curvature. 

MC 5501

Andrés Chirre, Pontifical Catholic University of Peru

"Remarks on a formula of Ramanujan"

In this talk, we will discuss a well-known formula of Ramanujan and its relationship with the partial sums of the Möbius function. Under some conjectures, we analyze a finer structure of the involved terms. It is a joint work with Steven M. Gonek.

Zoom link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09