Events

Alex Waldron, University of Wisconsin-Madison

"Parabolic gap theorems for the Yang-Mills functional"

Given a principal bundle over a compact Riemannian 4-manifold or special-holonomy manifold, it is natural to ask whether a uniform gap exists between the instanton energy and that of any non-minimal Yang-Mills connection. This question is quite open in general, although positive results exist in the literature. We'll review several of these gap theorems and strengthen them to statements of the following type: the space of all connections below a certain energy deformation retracts (under Yang-Mills flow) onto the space of instantons. As applications, we recover a theorem of Taubes on path-connectedness of instanton moduli spaces on the 4-sphere, and obtain a method to construct instantons on quaternion-Kähler manifolds with positive scalar curvature.

MC 5417

Katarzyna Wyczesany, Carnegie Mellon University

"Dualities on sets and how they appear in optimal transport"

In this talk, we will discuss order reversing quasi involutions, which are dualities on their image, and their properties. We prove that any order reversing quasi-involution is of a special form, which arose from the consideration of optimal transport problem with respect to costs that attain infinite values. We will discuss how this unified point of view on order reversing quasi-involutions helps to deepen the understanding of the underlying structures and principles. We will provide many examples and ways to construct new order reversing quasi-involutions from given ones. This talk is based on joint work with Shiri Artstein-Avidan and Shay Sadovsky.

This seminar will be held both online and in person:

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

Tianyi Zheng, UC San Diego

"Random walks on self-similar groups and conformal dimension"

Conformal dimension was introduced in the late 1980s by P. Pansu; it is a natural invariant in the study of the geometry of hyperbolic spaces and their boundaries. In this talk we will discuss how conformal geometry can be used to study random walks on iterated monodromy groups, in particular, random walk entropy bounds when the limit set has Ahlfors-regular conformal dimension strictly less than 2. Based on joint work with N. Matte Bon and V. Nekrashevych.

MC 5501

Jakub Krásenský, Czech Technical University in Prague

"Criterion sets for quadratic forms over number fields"

By the celebrated 15 theorem of Conway and Schneeberger, a classical positive definite quadratic form over Z is universal if it represents each element of {1,2,3,5,6,7,10,14,15}. Moreover, this is the minimal set with this property. In 2005, B.M. Kim, M.-H. Kim and B.-K. Oh showed that such a finite criterion set exists in a much general setting, but the uniqueness of the criterion set is lost. Since then, the question of uniqueness for particular situations has been studied by several authors.

We will discuss the analogous questions for totally positive definite quadratic forms over totally real number fields. Here again, the existence of criterion sets for universality is known, and Lee determined the set for Q(sqrt5). We will show the uniqueness and a strong connection with indecomposable integers. A part of our uniqueness result is (to our best knowledge) new even over Z. This is joint work with G. Romeo and V. Kala.

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

Rachael Alvir, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory X"

We will conclude our series of talks on computable structure theory.

MC 5479

Gian Sanjaya, Department of Pure Mathematics, University of Waterloo

"Arithmetic Schemes"

We now look at examples of arithmetic schemes.

MC 5417

Franklin Tall, University of Toronto

"An undecidable extension of Morley’s theorem on the number of countable models"

We show that Morley’s theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of equivalence relations obtained by countable intersections of projective sets in several models of set theory. Our methods include random and Cohen forcing, large cardinals, and Inner Model Theory.

MC 5479

Laindon Burnett, Department of Pure Mathematics, University of Waterloo

"Sturmian Sequence Decidability Does Not Generalize"

In 2022, it was shown that much like Presburger arithmetic itself, Presburger arithmetic along with a Sturmian sequence is a decidable theory. We give an overview of Konieczny's 2024 proof that this does not extend to generalized polynomials, themselves a generalization of Sturmian sequences.

MC 5479

Corey Jones, North Carolina State University

"Constructing actions of fusion categories on C*-algebras"

A fusion category is an algebraic object that simultaneously generalizes finite groups and their representation categories. Fusion categories can ``act" on C*-algebras by bimodules, extending the familiar concept of a group acting by automorphisms to a non-invertible setting. Building actions of specific fusion categories on specific C*-algebras is hard. In this talk, we will discuss a general method that allows for the construction of actions of fusion categories on interesting C*-algebras with minimal algebraic input. As an application, we construct actions of exotic fusion categories on noncommutative tori. Based on joint work with David Evans.

This seminar will be held both online and in person:

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

AJ Fong, Department of Pure Mathematics, University of Waterloo

"The mathematics of juggling (and perhaps a geometric application)"

Before videos could be easily transmitted over the internet, mathematical notation for juggling patterns was used by jugglers to share instructions and new patterns with each other. After introducing these, I will show that a mild generalisation of this gives a natural partial order on juggling patterns. If time permits, I will describe a relatively recent result that demonstrated that juggling patterns can be used to index a natural stratification of Grassmannians, which naturally arises in Poisson geometry, total positivity and cluster algebras.

MC 5417