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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

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

Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo

"An exercise in Riemannian geometry (or how to make a Riemannian geometric omelet without breaking any eggs)"

 I will describe a particular class of Riemannian metrics on the total space of a vector bundle, depending only on one natural coordinate $r$, and which are thus of cohomogeneity one. Such metrics arise frequently in the study of special holonomy, By carefully thinking before diving in, one can extract many useful formulas for such metrics without needing to explicitly compute all of the Christoffel symbols and the curvature. For example, these include the rough Laplacian of a function or of a vector field which are invariant under the symmetry group. If time permits, I will explain why I care about such formulas, as they are ingredients in the study of cohomogeneity one solitons for the isometric flow of $\mathrm{G}_2$-structures.

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