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Algebra & Geometry Seminar

Speaker: Sébastien Labbé (Université de Bordeaux)

Title: $q$-analogs of rational numbers: from Ostrowski numeration systems to perfect matchings

Abstract: We consider the $q$-deformation of rational numbers introduced recently by Morier-Genoud and Ovsienko. We propose three enumerative interpretations of these $q$-rationals: in terms of a new version of Ostrowski's numeration system for integers, in terms of order ideals of fence posets and in terms of perfect matchings of snake graphs. Contrary to previous results which are restricted to rational numbers greater than one, our interpretations work for all positive rational numbers and are based on a single combinatorial object for defining both the numerator and denominator. The proofs rest on order-preserving bijections between posets over these objects. We recover a formula for a $q$-analog of Markoff numbers. We also deduce a fourth interpretation given in terms of the integer points inside a polytope in $\mathbb{R}^k$ on both sides of a hyperplane where $k$ is the length of the continued fraction expansion. This is joint work with Jean-Christophe Aval.

Algebra & Geometry Seminar

Speaker: Rita Fioresi (University of Bologna)

Title: Quantum Principal bundles on non affine quantum spaces and their differential calculi

Abstract:

Sheaf-theoretic methods can be effectively applied to obtain quantum deformations of principal bundles on non-affine quantum spaces. In particular, we study the example of the quantum principal bundle corresponding classically to the projection of an algebraic group onto its quotient by the maximal parabolic subgroup. We then examine the theory of differential calculi on quantum principal bundles.
   
This is a joint work with:
Aschieri (UPO, Alessandria) Latini (UNIBO. Bologna), Weber (Charles U. Prague)

Curves Seminar

Speaker: Chika Oluigbo (Queen's University)

Title: Representations of the general linear group - Irreducible representations of GLmC

Abstract: We shall proceed to finish finding the irreducible representations of GLmC. The general strategy for demonstrating irreducibility shall make use of weights. Furthermore, the Schur module will end up playing an integral role in this process. From these we can further derive all of the irreducible representations of GLmC.

Department Colloquium

Speaker: Mathilde Gerbelli-Gauthier (University of Toronto)

Title: Sphere-Packing, Fourier Interpolation, and the Segal--Shale--Weil Representation

Abstract:
In 2016, Viazovska proved that the E_8​ lattice provides the optimal sphere-packing in dimension 8, and soon after, Cohn--Kumar--Miller--Radchenko--Viazovska proved the analogous result for the Leech lattice in dimension 24. Viazovska's breakthrough came through the solution of a Fourier interpolation problem: she constructed a function f such that f, its Fourier transform, and their first derivatives take on specific values at square roots of natural numbers. Prior to this, Radchenko and Viazovska had solved a "toy case"—a Fourier interpolation result for even Schwartz functions on the real line. In this talk, I will explain how this one-dimensional version can be understood through the lens of the Segal--Shale--Weil representation, an infinite-dimensional representation that originally arose in the context of quantum mechanics.

Algebra & Geometry Seminar

Speaker: David Miyamoto (Queen's University)

Title: A singular Serre-Swan theorem via tepui fibrations

Abstract: The classical Serre-Swan theorem constructs a fundamental bridge between geometry and algebra. Given a vector bundle $E$ over a smooth connected manifold $M$, the space of sections Gamma($E$) is a finitely-generated and projective $C^\infty(M)$-module, and Swan showed that every such $C^\infty(M)$-module $Q$ arises in this way. More precisely, the functor Gamma is an equivalence of categories.

Similarly capturing non-finitely-generated or non-projective modules requires leaving the category of smooth manifolds. I will introduce the notion of a tepui fibration, named for the distinctive mountain ranges in Venezuela, within the framework of diffeological spaces, which are themselves a generalization of smooth manifolds. Tepui fibrations let us directly generalize the Serre-Swan theorem: the section functor is an equivalence of categories between tepui vector bundles, and (locally finitely-generated, fiber-determined, and Frechet) $C^\infty(M)$-modules. We will see several concrete examples of this correspondence, in particular drawing from the theory of singular foliations.

This is joint work with Alfonso Garmendia and Leonid Ryvkin.

Department Colloquium

Speaker: Lulu Kang (University of Massachusetts Amherst)

Title: Building GP Surrogate Model with High-Dimensional Input

Abstract:
Gaussian process (GP) regression is a popular surrogate modeling tool for computer simulations in engineering and scientific domains. However, it often struggles with high computational costs and low prediction accuracy when the simulation involves too many input variables. In this talk, I will present two different approaches to build Gaussian process surrogate model for experiments with high dimensional input. I first introduce an optimal kernel learning approach to identify the active variables, thereby overcoming GP model limitations and enhancing system understanding. This method approximates the original GP model's covariance function through a convex combination of kernel functions, each utilizing low-dimensional subsets of input variables. The second approach is Bayesian bridge GP regression approach, in which we impose shrinkage penalty on the linear regression coefficients of the mean and correlation coefficients in the covariance function. This is equivalent to using certain proper informative priors on these parameters under Bayesian framework. Using Spherical Hamiltonian Monte Carlo, we can directly sample from the constrained posterior distribution without the restrictions on prior distribution as in Bayesian bridge regression.

Algebra & Geometry Seminar

Speaker: Samuel Leblanc (Queen's University)

Title: (Co)limit Computation and Its Application to Representation Theory

Abstract: A fundamental problem of representation theory is to classify functors C --> k-Vect up to isomorphism for a fixed small category C. Notable examples include when C = BG is a group, C = Q is a quiver, and C = P is a poset. Surprisingly, the limit and colimit of such functors provide some information about the decomposition of the functor into indecomposable summands. Motivated by this, in a joint work with T. Brüstle and J. Desrochers, we constructed the minimal full subcategory of a poset P that preserves the (co)limit of every functor P --> k-Vect. In this talk, we will explain these ideas, compute some examples, and discuss a possible generalization to arbitrary small categories.