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Statistics Seminar

Monday, October 20th, 2021

Time: 11:00 a.m.  Place: Jeffery Hall 225 & Online via Zoom (contact Brian Ling for Zoom link)

Speaker: Simon Barthelmé (CNRS, Gipsa-lab)

Title: Kernel matrices in the flat limit

Abstract: Kernel matrices are ubiquitous in statistics and machine learning. Within Bayesian statistics they occur most often as covariance matrices of Gaussian processes, in non-parametric or semi-parametric models. Most of the theoretical work on kernel methods has focused on a large-$n$ asymptotics, characterising the behaviour of kernel matrices as the amount of data increases. Fixed-sample analysis is much more difficult outside of simple cases, such as locations on a regular grid.

In this talk I will describe a fixed-sample analysis that was first studied in the context of approximation theory by Fornberg \& Driscoll (2002), called the "flat limit". In flat-limit asymptotics, the goal is to characterise kernel methods as the length-scale of the kernel function tends to infinity, so that kernels appear flat over the range of the data. Even though flat kernel matrices may seem trivial, because their rank goes to one, detailed analysis reveals very interesting structure. We have been able to show that the eigenvectors and eigenvalues in that regime are tightly related to orthogonal polynomials or splines, depending on the smoothness of the kernel.

With the results on the spectrum of kernel matrices in hand, one may study a wide range of kernel methods. In this talk I'll describe an application to Gaussian Process regression. Results show that Gaussian process regression tends in the flat limit to (multivariate) polynomial regression, or (polyharmonic) spline regression, depending on the kernel. Since these methods are simpler, the results may have practical implications for GP regression.

Algebra & Geometry Seminar

Monday, October 25th, 2021

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Eric Hanson (LaCIM, Montreal)

Title: tau-perpendicular wide subcategories.

Abstract: Let $\Lambda$ be a finite-dimensional algebra. Jasso's $\tau$-tilting reduction, later extended by Buan-Marsh, allows one to relate the $\tau$-tilting theory of mod-$\Lambda$ to that of a certain type of subcategory. In this talk, we classify these subcategories using the notions of functorial finiteness and Serre subcategories. As an application, we give a definition of the "$\tau$-cluster morphism category" of an arbitrary finite-dimensional algebra. This definition extends those of Igusa-Todorov (hereditary case) and Buan-Marsh ($\tau$-tilting finite case). This talk is based on joint work with Aslak Bakke Buan.

Website details here:

Math & Stats Department Colloquium

Friday, October 22nd, 2021

Time: 2:30 p.m.  Place: In-person (Jeffery Hall 234) & Online (via Zoom)

Speaker: Michael Brannan (University of Waterloo)

Title: Quantum symmetries of graphs and non-local games

Abstract: Given a finite graph X, a fundamental question that one can ask about the structure of X is: ``What are its symmetries?'' Most of the time, when we think of symmetries of X, the usual automorphism group of X comes to mind. In this talk, I will describe a more general notion of symmetry of graphs, called quantum symmetries. Quantum symmetries of graphs arise quite naturally within the framework of non-commutative geometry and are encoded by a certain universal Hopf algebra (i.e., quantum group) co-acting on the algebra of functions on the vertex set of the graph. Very recently, quantum symmetries of graphs have also been found to arise within the context of two-player non-local games in quantum information theory. More precisely, they encode winning entangled strategies for the so-called graph isomorphism game. I will give a light introduction to all of these ideas and highlight how tools from non-local games and representation theory combine in a very powerful way to elucidate the structure of graphs, their quantum symmetries, and related operator algebra problems.

Michael Brannan is an Associate Professor in the Department of Pure Mathematics at the University of Waterloo. Before moving to Waterloo in 2021, he was an Associate Professor in the Department of Mathematics at Texas A&M University. He obtained his Ph.D. in Mathematics from Queen's University. He is interested in operator algebras, quantum information theory, representation theory, quantum symmetries, non-commutative probability, quantum algebra, and interactions between these fields.

Algebra & Geometry Seminar

Monday, October 18th, 2021

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Christopher Eur (Harvard University)

Title: Tautological classes of matroids.

Abstract: Algebraic geometry has furnished fruitful tools for studying matroids, which are combinatorial abstractions of hyperplane arrangements. We first survey some recent developments, pointing out how these developments remained partially disjoint. We then introduce certain vector bundles (K-classes) on permutohedral varieties, which we call "tautological bundles (classes)" of matroids, as a new framework that unifies, recovers, and extends these recent developments. Our framework leads to new questions that further probe the boundary between combinatorics and geometry. Joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.

Website details here:

Algebra & Geometry Seminar

Monday, October 4th, 2021

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Allan Francis Merino (University of Ottawa)

Title: Transfer of characters in the theta correspondence.

Abstract: For every irreducible reductive dual pair $(G, G')$ in $Sp(W)$, R. Howe proved the existence of an isomorphism between the spaces $R(G)$ and $R(G')$, where $R(G)$ is the set of infinitesimal equivalence classes of irreducible admissible representations of $\tilde{G}$; (preimage of $G$ in the metaplectic group) which can be realized as a quotient of the metaplectic representation. All the representations appearing in the previous duality have a distribution character, and characters are analytic objects completely identifying the irreducible representations. In particular, one natural question is to understand the transfer of characters in the previous duality (or Howe’s duality).

In 2000, T. Przebinda introduced the Cauchy-Harish-Chandra integral and conjectured that the transfer of characters should be obtained via this map. In my talk, after recalling carefully Howe’s duality theorem and the construction of the Cauchy-Harish-Chandra integral, I will explain what is known about the conjecture, explain recent results I got and some ongoing projects for the lift of discrete series representation of $\tilde{G}$ for a general dual pair $(G, G')$.

Website details here:

Math & Stats Department Colloquium

Friday, October 8th, 2021

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Julian Sahasrabudhe (University of Cambridge)

Title: Probabilistic combinatorics at exponentially small scales

Abstract: In this talk I will discuss a set of tools that has recently been developed to think about events of extremely small probability in finite probability spaces. I try to will weave together how this sort of thinking was used to solve two longstanding problems in two different fields: an old problem of Littlewood in harmonic analysis, and an exponential bound on the singularity probability of a random symmetric matrix.

Julian Sahasrabudhe is a university lecturer (assistant professor in the Canadian system) in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge. He was a Junior Research Fellow at Peterhouse, University of Cambridge from 2017 to 2021. He got his Ph.D. in Mathematics at the University of Memphis in 2017. He is interested in extremal and probabilistic combinatorics, and intersections with probability, analysis and combinatorial number theory. Most recently, he has been interested in random polynomials and random matrices. He was awarded the European Prize in Combinatorics in 2021.

Algebra & Geometry Seminar

Monday, September 27th, 2021

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Charles Paquette (Royal Military College & Queen's University)

Title: Simple representations over free products of semi-simple algebras via quivers.

Abstract: This is a report on joint work with Andrew Buchanan, Ivan Dimitrov, Olivia Grace, David Wehlau and Tianyuan Xu. We consider a free product $A$ of semi-simple $\mathbb{K}$-algebras over an algebraically closed field and show how representation theory of a suitable finite acyclic quiver $Q$ can be used to understand the representation theory of $A$. In particular, we show that the simple $A$-modules correspond to stable representations of $Q$ for some stability parameter. We apply this to the representation theory of free products of finite groups.

Website details here:

Math & Stats Department Colloquium

Friday, October 1st, 2021

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: David Sumpter (University of Uppsala)

Title: Are there Ten Equations that rule the world? And if so, what are they?

Abstract: I describe ten key equations you need to know in order to make better decisions, understand the filter created by social media and even to be a better person. Using examples, such as ‘deciding whether your new boss is an idiot?’, 'deciding when to stop watching a Netflix series' and ‘buying new headphones’, I present some new ways of seeing some of our favourite mathematical results. I also describe these equations' role in society: in finance, gambling, social media and artificial intelligence. Surprisingly few people in the general public understand these key equations and the small group that do are becoming increasingly rich and powerful. I think there are (order) ten equations that rule the world and we have to learn how to use them and spread their usage.

David Sumpter is Professor of Applied Mathematics at the University of Uppsala, Sweden. He is the author of Soccermatics and Outnumbered, which have been translated into ten languages, and Collective Animal Behaviour, the leading text in the academic field he helped create. He has worked with a number of the world's biggest football clubs, advising on analytics, as well as consulting on betting.

Video of David Sumpter's talk (mp4, 1.7 GB)

Algebra & Geometry Seminar

Monday, September 20th, 2021

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Kaveh Mousavand (Queen's University)

Title: Module varieties of biserial algebras.

Abstract: Biserial algebras form and important family of tame algebras which contain several interesting families, such as gentle, string and special biserial algebras. In our ongoing joint work with Charles Paquette, we have verified a recent conjecture on the behavior of Schur representations (a.k.a bricks) over biserial algebras. This allows us to better describe the behavior of irreducible components of moduli spaces of representations over biserial algebras and also relate our results to the recent work of Chindris-Kinser-Weyman on the semi-invariants of the ring of polynomials of every such components.

Website details here: