LACIM Seminar

The LACIM Seminar, created in the early 80s, takes place every Friday at 11h from September to June (and sometimes during the summer) in room PK-4323 of UQAM’s Président-Kennedy Building.

Seminar organizers

Seminar Calendar

  • 2024-06-21
    à 15h
    Matt Satriano, University of Waterloo
    Beyond twisted maps: crepant resolutions of log terminal singularities and a motivic McKay correspondence
    Abstract (click to expand) Crepant resolutions have inspired connections between birational geometry, derived categories, representation theory, and motivic integration. In this talk, we prove that every variety with log-terminal singularities admits a crepant resolution by a smooth stack. We additionally prove a motivic McKay correspondence for stack-theoretic resolutions. Finally, we show how our work naturally leads to a generalization of twisted mapping spaces. This is joint work with Jeremy Usatine.
  • 2024-06-14
    Louis Marin, UQAM
    Énumération de polyominos dans un rectangle $b$ x $h$
    Abstract (click to expand) Dans cet exposé, on s'attaque à un sous-problème de l'énumeration des polyominos qui est un problème étudié depuis son introduction par Goulomb. On énumère les polyominos qui sont inscrits dans un rectangle d'une certaine taille. Si on fixe la base $b$ et varie la hauteur, le nombre de polyominos inscrits dans ces rectangles suit une récurrence linéaire. L'objectif est de construire des automates qui reconnaissent ces polyominos pour obtenir les séries génératrice. En adaptant des méthodes décrites dans des travaux de Zeilberger et de Bousquet-Mélou et Brak, on développe une construction systématique de ces automates pour chaque valeur de $b$.
  • 2024-05-31
    Nicole Lemire, University of Western Ontario (professeure visiteure au LACIM)
    Demazure Models of Algebraic Tori
    Abstract (click to expand) For a field \(k\), algebraic \(k\)-tori are algebraic \(k\)-groups which become isomorphic to a split \(k\)-torus, after base change to the separable closure of \(k\). The possible algebraic \(k\)-tori of dimension n are classified up to isomorphism by finite subgroups of \(GL(n,Z)\) up to conjugacy and hence by integral representations of finite groups. Work of Demazure, Voskresenskii and Klyachko, among others, established, in principle, the existence of a smooth projective model of an algebraic \(k\)-torus which was used to determine its birational properties. Kunyavskii’s birational classification of algebraic \(k\)-tori in dimension \(3\) involved constructions of smooth projective toric models of the algebraic tori corresponding to maximal finite subgroups of \(GL(3,Z)\). We discuss some explicit constructions of toric models of algebraic tori, making connections with the defining integral representations of their splitting groups. The constructions determine smooth projective toric models of the algebraic tori corresponding to maximal finite subgroups of \(GL(4,Z)\). Since projective toric varieties are determined by lattice polytopes, our constructions focus on some highly symmetric families of polytopes, such as root polytopes and duals of central transportation polytopes.
  • 2024-05-24
    Patricia Commins, University of Minnesota
    The descent algebra, the braid arrangement, and extensions to left regular bands
    Abstract (click to expand) The faces of the braid arrangement form a monoid. The associated monoid algebra -- the face algebra -- is well-studied, especially in relation to Markov chains. In this talk, we explore the action of the symmetric group on the face algebra from the perspective of invariant theory. Bidigare proved the invariant subalgebra of the face algebra is (anti)isomorphic to Solomon's descent algebra. We answer the more general question: what is the structure of the face algebra as a simultaneous representation of the symmetric group and Solomon's descent algebra? A key tool in answering this question is the poset topology of the partition lattice. We will say a bit about how poset topology can be applied to a more general setting: groups acting on left regular band algebras.
  • 2024-05-17
    Carl Mautner, University of California, Riverside
    Combinatorics of Hilbert Schur algebras
    Abstract (click to expand) The Schur algebra is a finite dimensional algebra that connects the representation theory of symmetric and general linear groups. In joint work with Tom Braden we introduce new algebra that enhances the Schur algebra and provides a new window into the representation theory of symmetric groups. I will give a conceptual and a diagrammatic description of this new “Hilbert Schur algebra" and discuss some of its combinatorial structure.
  • 2024-04-12
    Yan Lanciault, UQAM
    Une symétrie des mots de Christoffel
    Abstract (click to expand) Dans le cadre de l’étude des mots trapézoïdaux, dont la séquence de la cardinalité des facteurs est symétrique, on introduit une variance bivariée qui reflète cette symétrie dans l’image commutative des facteurs, caractérisant ainsi, sous certaines hypothèses, les mots de Christoffel. Bien que le cas primitif soit contraint aux mots sturmiens, le cas non-primitif se départie de cette hypothèse et, ce faisant, caractérise parfaitement les mots de Christoffel.
  • 2024-04-05
    Maria Gillespie, Colorado State University
    Skewing formulas and a geometric interpretation for the Delta conjecture
    Abstract (click to expand) We show a skewing relation holds between the symmetric functions involved in the Rational Shuffle Conjecture and those of the Delta Conjecture, and use it to derive a geometric interpretation for the Delta polynomials in terms of the Borel-Moore homology of certain affine Springer fibers. Our geometric work simultaneously generalizes that of Hikita and Borho--Macpherson from two special cases, and our algebraic proof of the skewing relation relies on the new work of Blasiak, Haiman, Morse, Pun, and Seelinger that resolved the "Rise" half of the Delta Conjecture. We also note some progress towards a purely combinatorial proof of the relation. This is joint work with Sean Griffin and Eugene Gorsky.