Events

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

Jesse Huang, University of Waterloo

Organizational Meeting

This is an organizational meeting for the mirror symmetry learning seminar. We will skim through the reading list and topics to cover and assign talks. All are welcome!

MC 2017

Akash Singha Roy, University of Georgia

Residue-class distribution and mean values of multiplicative functions

The distribution of values of arithmetic functions in residue classes has been a problem of great interest in elementary and analytic number theory. The analogous question commonly studied for multiplicative functions is the distribution of their values in coprime residue classes. In work studying this problem for large classes of multiplicative functions, Narkiewicz obtained criteria deciding when a family of such functions is jointly uniformly distributed among the coprime residue classes to a fixed modulus. In the first part of this talk, we shall extend Narkiewicz's criteria to moduli that are allowed to vary in a wide range. Our results are essentially the best possible analogues of the Siegel-Walfisz theorem in this setting. One of the primary themes behind our arguments is the quantitative detection of a certain "mixing" (or ergodicity) phenomenon in multiplicative groups via methods belonging to the "anatomy of integers", but we also rely heavily on more classical analytic arguments, tools from arithmetic and algebraic geometry, and from linear algebra over rings.

In the second part of this talk, we shall gain a finer understanding of these distributions, such as the second-order behavior. This shall rely on extending some of the most powerful known estimates on mean values of multiplicative functions (precisely, the Landau-Selberg-Delange method) to a result that is much more uniform in certain important parameters. We will see several applications of this extended result in other interesting settings as well.

This talk is partially based on joint work with Prof. Paul Pollack.

Join on Zoom

Jesse Huang, University of Waterloo

Mirror symmetry for complex tori

We discuss various forms of mirror symmetry using the example of a complex torus and its compactifications.

MC 5479