Events

Amanda Petcu, University of Waterloo

Stability and Lyapunov Functions

When working with a nonlinear system of differential equations, finding explicit, closed-form solutions can be difficult. A tool in such situations is to determine the stability of the equilibrium points of the system. This analysis allows us to predict the long-term behavior of the system by examining its trajectories and how they behave near an equilibrium point: specifically, do they remain bounded in some compact set, converge to the point, or escape to infinity? In this talk, we will discuss Lyapunov's Direct Method, a technique that allows us to determine the stability of an equilibrium point without explicitly solving the differential equations.

MC 5403

Josse van Dobben de Bruyn, Charles University

Asymmetric graphs with quantum symmetry

Quantum isomorphisms of graphs form a bridge between noncommutative geometry (NCG) and quantum information theory (QIT), as they connect quantum automorphism groups of graphs with nonlocal games. This makes it possible to use techniques from QIT in NCG and vice versa. In this talk, I will present a striking application of this connection, where we use ideas from QIT to prove a surprising result in NCG. Using a construction similar to the Mermin–Peres magic square from QIT, we construct graphs with trivial automorphism group and non-trivial quantum automorphism group, which shows that even graphs with no symmetry at all can have hidden quantum symmetries. These are the first known examples of any kind of commutative spaces in NCG with this property.

This talk is based on joint work with David E. Roberson (Technical University of Denmark) and Simon Schmidt (Ruhr University Bochum).

QNC 1507 or Join on Zoom

Gian Cordana Sanjaya, University of Waterloo

Density of Special Classes of Polynomials with Squarefree Discriminant

In this talk, we compute the density of monic polynomials of fixed degree over Z_p, which has squarefree discriminant, provided some restriction on the coefficients of the polynomial. This is a natural extension to a previous result by Yamamura, who solved the case where the coefficients have no restrictions at all.

MC 5479

Java Villano, University of Toronto

Relativizing computable categoricity

A computable structure A is said to be computably categorical if for all computable copies B of A, there exists a computable isomorphism between A and B. We can relativize this notion to any Turing degree d by asking that for any d-computable copy B of A, there is a d-computable isomorphism between A and B. In this talk, we will discuss results about this relativization. In particular, we will discuss how for directed graphs, categoricity relative to a degree need not be monotonic in the c.e.~ degrees, and how for other structures besides directed graphs, categorical behavior relative to a degree stabilizes on certain Turing cones.

MC 5403

Liam Orovec, University of Waterloo

Greedy and Lazy Parry Numbers

We say a real number \beta is a Parry number provided the greedy \beta-expansion for 1 is eventually periodic or finite. We show conditions for when \beta is a Parry number and provide a family of real numbers which are always Parry numbers, the PV-numbers. The related Salem numbers, constructed from PV-numbers are then considered. We split this into four cases, the first of which was shown by Hare and Tweedle, we give criteria for finding Salem numbers which are Parry numbers. Time permitting we will explore the case where we look at lazy expansions instead of greedy expansions, we call these numbers lazy Parry numbers.

MC 5417

Elan Roth, University of Waterloo

A Continuation of Random Binary Sequences 2.0

Inspired by probability theory, we can define a new notion of randomness using betting strategies. We'll discuss some properties of this notion of randomness and, you guessed it, prove its equivalence to ML-randomness and 1-randomness.

MC 5403

Joel Kamnitzer, McGill University

The top-heavy conjecture and the topology of (real) matroid Schubert varieties

Suppose we are giving a spanning set S in a vector space V and we consider all subspaces of V spanned by subsets of S. The top-heavy conjecture states that the number of dimension k subspaces is less than or equal to the number for codimension k subspaces. This elementary statement was first conjectured by Dowling and Wilson in 1975 and resisted any proof for 40 years. Finally though, it was resolved by Huh and Wang in 2017, and partially led to Huh's 2022 Fields Medal. I will outline the details of the proof, which relies on the study of the topology of a beautiful space called a matroid Schubert variety. Finally, I will discuss our own contribution to this subject, which is the study of the topology of the real locus of this space (which unfortunately does not lead to the proof of any famous conjecture).

MC 5501

Facundo Camano, University of Waterloo

Dimensional Reduction of S^1-Invariant Instantons on the Multi-Taub-NUT

In this talk I will discuss the dimensional reduction of S^1-invariant instantons on the multi-Taub-NUT space to singular monopoloes on \mathbb{R}^3. I will first introduce the multi-Taub-NUT space, followed up by a discussion on S^1-equivariant principal bundles. Next, I will go over the natural decomposition of S^1-invariant connections into horizontal and vertical pieces, and then show how the self-duality equation reduces to the Bogomolny equation under said decomposition. I will then show how the smoothness of the instanton over the NUT points determines the asymptotic conditions for the singular monopole. Finally, I will go over the reverse construction: starting with a singular monopole on \mathbb{R}^3 and building up to an S^1-invariant instanton on the multi-Taub-NUT space.

MC 5403

Brent Nelson, Michigan State University

Closable derivations are anticoarse, of course

The anticoarse space of an inclusion $N\subset M$ of tracial von Neumann algebras is an $N$-subbimodule of $L^2(M)$ whose size is sensitive to several structural properties of the inclusion. It has become a staple of so-called microstates techniques in free probability, where it allows one to parlay finite dimensional approximations into algebraic properties. On the other hand, non-microstates techniques, which exploit the regularity of certain derivations on a von Neumann algebra, have not made use of the anticoarse space, until now. In this talk, I will discuss how deformations given by closable derivations provide a natural connection to anticoarse spaces and consequently yield new applications of free probability. This is based on joint work with Yoonkyeong Lee.

QNC 1507 or Join on Zoom

Jérémy Champagne, University of Waterloo

Small fractional parts of polynomials (aka 11J54)

In the eary 1900's, Hardy and Littlewood asked the following question: given a real number α and integer k>1, what is the smallest distance obtained between αn^k and the nearest integer as n runs over the set {1,...,N}? More specifically, does there exist an exponent theta_k>0 such that the smallest distance is at most N^-theta_k for sufficiently large N? This question was answered positively by Vinogradov a couple decades later, but the question of finding the largest possible theta_k with this property is still open.

In this talk, I will discuss some historical results around this problem and present some typical methods used in the literature.

MC 5479