Events

Rahim Moosa, University of Waterloo

Curve excluding fields I

Recently, Johnson and Ye have proved an attractive and somewhat surprising result: Suppose C is an algebraic curve of genus at least two having no rational points. Then the class of fields over which C has no rational points, has a model companion. This model companion, they call it CXF, turns out to answer several old questions.

I will start presenting the results of the paper.

MC 5403

Daniel Gromada, Czech Technical University

A brief introduction to quantum symmetries

In this talk, I would like to explain the concept of a quantum symmetry. We will focus on symmetries of simple combinatorial objects like finite sets and graphs. This can be approached either from the viewpoint of quantum groups or via diagrammatic categories. I will try to explain how drawing simple string diagrams can reveal interesting findings about quantum symmetries of certain objects.

MC 5501

Krishnarjun Krishnamoorthy, BIMSA

Moments of non-normal number fields

Let K be a number field and a_K(m) be the number of integral ideals in K of norm equal to m. We asymptotically evaluate the sum \sum_{m\leqslant X} a_K^l(m) as X grows to infinity. We also consider the continuous moments of the associated Dedekind zeta function and prove lower bounds of the expected order of magnitude.

Join on Zoom

Catherine St-Pierre, University of Waterloo

Sheppard-Todd-Chevalley theorem (and beyond)

Sheppard-Todd-Chevalley's theorem is one of the most significant results in invariant theory. It provides necessary and sufficient conditions for the fixed subring k[x_1, \dots , x_n]^G under a finite subgroup G of GL_n(k) to be a polynomial ring. We will review the theorem and its applications and summarise some generalisations.

MC 5479

Leigh Foster, University of Waterloo

Introduction to planar tilings

From tangrams to tessellations to brick pavers, we have many real life examples of tilings of a planar region. In this learning seminar, we will get a gentle introduction to the math behind these ideas, and over the course of the term will be able to answer the questions: Given a set of tiles, can we determine if a given region is tilable? If so, do we have more than one way of laying out the tiles? How can we know when a tiling does not exist? To answer these questions, we'll use techniques including counting and coloring arguments, height functions from a more topological point of view, and a combinatorial group-theoretic approach, among others. No previous knowledge is needed, and no outside work is required! Come and listen and ask questions - everyone is welcome, and interruptions are expected.

For our first meeting, we will discuss the basics of tilings. What does it mean to tile a region, and what are some ways that these questions arise in mathematics?

MC 5403

Faisal Romshoo, University of Waterloo

Topological calibrations and their moduli spaces

We discuss an approach to deformation problems of geometric structures laid out in https://arxiv.org/abs/math/0112197 by Ryushi Goto. In particular, we will explore the cohomological conditions under which the moduli space of the geometric structures become smooth manifolds of finite dimension. As an application, we will prove the unobstructedness of G2 structures and if time permits, of Spin(7)-structures as well.

MC 5479

Daniel Gromada, Czech Technical University

Quantum association schemes

This talk is based on a preprint arXiv:2404.06157. We start by briefly explaining what a quantum group is and how quantum graphs are defined. Then, we recall what association schemes are and we apply the quantization procedure here. As a result, this allows to define distance regular and strongly regular quantum graphs. In addition, we observe that the duality for translation association schemes extends to the quantum setting.

MC 5417 or Join on Zoom

Rahim Moosa, University of Waterloo

Curve excluding fields II

Recently, Johnson and Ye have proved an attractive and somewhat surprising result: Suppose C is an algebraic curve of genus at least two having no rational points. Then the class of fields over which C has no rational points, has a model companion. This model companion, they call it CXF, turns out to answer several old questions.

I will start presenting the results of the paper.

MC 5403

Luis Fernandez (City University of New York)

The Dirac operator in the Clifford bundle and Kaehler identities for almost complex manifolds

We use the Dirac operator in the Clifford bundle of an almost complex manifold to obtain a different formulation of the Kaehler identities which, when viewed in the exterior bundle, give the known generalization of these identities for complex manifolds found by Demailly, thus obtaining a generalization of the Kaeher identities for almost complex manifolds. This result was also proved by de la Ossa, Karigiannis, and Svanes.

In the process we will define a number of operators in the Clifford bundle, together with relations between them, that should give an alternative way to study almost complex manifolds.

All the work presented is joint with Sam Hosmer.

MC 5417

Michael Chapman, NYU (Courant Institute)

Subgroup Tests and the Aldous-Lyons conjecture

The Aldous-Lyons conjecture from probability theory states that every (unimodular) infinite graph can be (Benjamini-Schramm) approximated by finite graphs. This conjecture is an analogue of other influential conjectures in mathematics concerning how well certain infinite objects can be approximated by finite ones; examples include Connes' embedding problem (CEP) in functional analysis and the soficity problem of Gromov-Weiss in group theory. These became major open problems in their respective fields, as many other long standing open problems, that seem unrelated to any approximation property, were shown to be true for the class of finitely-approximated objects. For example, Gottschalk's conjecture and Kaplansky's direct finiteness conjecture are known to be true for sofic groups, but are still wide open for general groups.

In 2019, Ji, Natarajan, Vidick, Wright and Yuen resolved CEP in the negative. Quite remarkably, their result is deduced from complexity theory, and specifically from undecidability in certain quantum interactive proof systems. Inspired by their work, we suggest a novel interactive proof system which is related to the Aldous-Lyons conjecture in the following way: If the Aldous-Lyons conjecture was true, then every language in this interactive proof system is decidable. A key concept we introduce for this purpose is that of a Subgroup Test, which is our analogue of a Non-local Game. By providing a reduction from the Halting Problem to this new proof system, we refute the Aldous-Lyons conjecture.

This talk is based on joint work with Lewis Bowen, Alex Lubotzky, and Thomas Vidick.

MC 5501

2:30pm - 3:30pm