Events

Aristomenis Papadopoulos, University of Maryland

Zarankiewicz's Problem and Model Theory

"A shower thought that anyone interested in graph theory must have had at some point in their lives is the following: 'How ""sparse"" must a given graph be, if I know that it has no ""dense"" subgraphs?'. This curiosity definitely crossed the mind of Polish mathematician K. Zarankiewicz, who asked a version of this question formally in 1951. In the years that followed, many central figures in the development of extremal combinatorics contemplated this problem, giving various kinds of answers. Some of these will be surveyed in the first part of my talk.

So far so good, but this is a logic seminar and the title says the words ""Model Theory""… In the second part of my talk, I will discuss how the celebrated Szemerédi-Trotter theorem gave a starting point to the study of Zarankiewicz's problem in ""geometric"" contexts, and how the language of model theory has been able to capture exactly what these contexts are. I will then ramble about improvements to the classical answers to Zarankiewicz's problem, when we restrict our attention to semilinear/semibounded o-minimal structures, Presburger arithmetic, and various kinds of Hrushovski constructions.

The new results that will appear in the talk were obtained jointly with Pantelis Eleftheriou."

MC 5479

Sourabh Das, University of Waterloo

Tools in Analytic Number Theory

Analytic number theory provides several classical tools to explore the distribution of arithmetic functions, including the rearrangement of sums, generating series, and counting arguments. In this talk, we will use these techniques to investigate the distribution of h-free and h-full numbers, which play an important role in understanding the structure of integers. If time allows, we will extend our discussion to prove results about the distribution of the prime-counting ω-function over these subsets of integers. Designed as an introductory talk, this presentation will be accessible to anyone with a basic background in number theory.

MC 5403

Spiro Karigiannis, University of Waterloo

Infinitesimal deformations of G-structures

I will introduce the setting of G-structures on an oriented Riemannian n-manifold, where G is a closed Lie subgroup of SO(n). These can be understood in terms of global sections of the SO(n) bundle which is the quotient of the SO(n)-prinicipal bundle of oriented orthonormal frames by the free action of G. We will define the intrinsic torsion of a G-structure, and explain how to describe infinitesimal deformations of G-structures. If time permits, we will discuss a Dirichlet energy type of functional on the space of G-structures, whose critical points are called harmonic G-structures. This condition includes the torsion-free G-structures but is more general. These ideas were developed recently by Fowdar, Loubeau, Moreno, Sa Earp building on earlier work in the G2 and Spin(7) cases by myself from 2006-2007.

MC 5479

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.

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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