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Spencer Unger, University of Toronto

Proofs of countable Ramsey theorems

We discuss the various proofs of Ramsey theorems involving colorings of countable sets with additional structure.  To illustrate a typical argument which proves an infinite Ramsey statement from a finite one, we sketch Baumgartner's proof of Hindman's theorem and report on some ongoing related projects.

MC 5479

Meenakshi McNamara, Perimeter Institute for Theoretical Physics

Exact quantum chromatic numbers of Hadamard graphs and products

Quantum chromatic numbers are defined in terms of non-local games on graphs. This talk gives a proof of the exact quantum chromatic number of Hadamard graphs using a conjugacy class graphs approach. This further allows us to consider graph products, and we compute the exact quantum chromatic number of the categorical product of Hadamard graphs. This work makes use of several results for the quantum chromatic numbers of quantum graphs, an operator algebraic generalizations of graphs. In particular, we also discuss results on products of quantum graphs from joint work with Rolando de Santiago.

MC 5417

Erik Seguin, University of Waterloo

Selected Topics on Fourier Algebras of Locally Compact Hausdorff Groups

We discuss some selected topics on Fourier algebras of locally compact Hausdorff groups.

MC 5403

Mark Rudelson, University of Michigan

When a system of real quadratic equations has a solution

The existence and the number of solutions of a system of polynomial equations in n variables over an algebraically closed field is a classical topic in algebraic geometry. Much less is known about the existence of solutions of a system of polynomial equations over reals. Any such problem can be reduced to a system of quadratic equations by introducing auxiliary variables. Due to the generality of the problem, a computationally efficient algorithm for determining whether a real solution of a system of quadratic equations exists is believed to be impossible. We will discuss a simple and efficient sufficient condition for the existence of a solution. While the problem and the condition are of algebraic nature, the proof relies on Fourier analysis and concentration of measure.

Joint work with Alexander Barvinok.

MC 5501

Andy Zucker, University of Waterloo

Minimal dynamics of topological groups: A set-theoretic perspective

This talk explores the minimal actions of topological groups on compact spaces. By a classical result of Ellis, every topological group admits a largest such action called the universal minimal flow. Here, we take a set-theoretic perspective and ask how the universal minimal flow can change when considering different models of set theory. In particular, we will take the opportunity to give a gentle introduction to set-theoretic forcing. Our main result is a characterization of those topological groups for which the universal minimal flow is absolute. Joint work with Gianluca Basso.

MC 5501

Liam Orovec, University of Waterloo

Greedy beta-expansions for families of Salem numbers

We give criteria for finding the greedy beta-expansion for 1 under families of Salem numbers that approach a given Pisot number. We show these expansions are related to the greedy expansion under the Pisot base. This expands the work of Hare and Tweedle to include more Pisot numbers and more families of Salem numbers.

MC 5403

Miho Mukohara, University of Tokyo

On a Galois correspondence for minimal actions of compact groups on C*-algebras

Inclusions arising from compact quantum group actions on factors have been studied by Izumi-Longo-Popa and Tomatsu. For a minimal action of a compact group on a factor, there is an isomorphism from the lattice of closed subgroups onto that of intermediate subfactors between the factor and the fixed point subfactor. The correspondence between intermediate subfactors and subgroups is called a Galois correspondence. As a duality result, a Galois correspondence for discrete group actions is also known. Analogues for actions on C*-algebras were also studied by Izumi, Cameron-Smith, and others. In this talk, I will discuss a Galois correspondence for compact group actions on C*-algebras. A crucial result for our main theorem is the proper outerness of finite index endomorphisms of purely infinite simple C*-algebras. This was shown by Izumi recently. If time permits, I will also explain an extension of our main result to actions of compact quantum groups of Kac type and a relationship between our main result and the C*-discrete inclusion introduced by Hernández Palomares and Nelson.

MC 5417 or Join on Zoom

Noah Slavitch, University of Waterloo

Measurable Cardinals and Non-Constructible Sets

In this talk we will explain how the existence of a measurable cardinal implies that V≠L, that is, that there exist nonconstructible sets.

MC 5479

Robert Cornea, University of Waterloo

Stable Pairs on P2 via Spectral Correspondence

In this talk we will consider stable wild Vafa-Witten-Higgs bundles (or stable pairs for short) (E, ϕ) on P^2 where E is a rank two holomorphic vector bundle and ϕ : E -> E(d) is a holomorphic bundle map with d > 0. There is a way to construct stable pairs on called the spectral correspondence. This states that given a stable pair (E,ϕ) on P^2, there exists a surface Y and a 2:1 covering map pi: Y -> P^2 such that E is the push forward of a line bundle on Y and ϕ comes from the multiplication of a section on Y. So studying stable pairs (E,ϕ) on P^2 boils down to finding 2:1 covering maps Y -> P^2 and line bundles on Y. The study of constructing rank two vector bundles on P^2 via 2:1 coverings was studied by Schwarzenberger in 1960. We will demonstrate examples of stable pairs when d=1 and explain the cases briefly for d=2 and 3.

MC 5479

Emily Quesada-Herrera, University of Lethbridge

Fourier optimization and the least quadratic non-residue

We will explore how a Fourier optimization framework may be used to study two classical problems in number theory involving Dirichlet characters: The problem of estimating the least character non-residue; and the problem of estimating the least prime in an arithmetic progression. In particular, we show how this Fourier framework leads to subtle, but conceptually interesting, improvements on the best current asymptotic bounds under the Generalized Riemann Hypothesis, given by Lamzouri, Li, and Soundararajan. Based on joint work with Emanuel Carneiro, Micah Milinovich, and Antonio Ramos.

MC 5479