Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Geometry and Representation Theory Seminar

Geometry & Representation - Veronique Bazier-Matte (UQAM)

Monday, December 2nd, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Véronique Bazier-Matte (Université du Québec à Montréal)

Title: Quasi-cluster algebras.

Abstract:  In 2015, Dupont and Palesi defined quasi-cluster algebra from non-orientable surfaces. The goal of this talk is to compare cluster algebras and quasi-cluster algebras and to explain some conjectures about them.

Geometry & Representation - Gregory G. Smith

Monday, November 18th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Gregory G. Smith (Queen's University)

Title: Smooth Hilbert schemes.

Abstract:  In algebraic geometry, Hilbert schemes are the prototypical parameter spaces: their points correspond to closed subschemes in a projective space with a fixed Hilbert polynomial. After surveying some of their known features, we will present new numerical conditions on the polynomial that completely characterize when the associated Hilbert scheme is smooth. In this smooth situation, our explicit description of the subschemes being parametrized also provides new insights into the global geometry of Hilbert schemes. This talk is based on joint work with Roy Skjelnes (KTH).

Geometry & Representation - Chris Brav (Higher School of Economics, Moscow)

Monday, November 11th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Chris Brav (Higher School of Economics, Moscow)

Title: Cartan's magic without the formulas.

Abstract:  The Cartan calculus concerns vector fields on a smooth variety X acting on differential forms via Lie derivative and contraction, with Cartan's magic formula expressing the relation between the two actions: Lie derivative is the graded commutator of the de Rham differential with contraction. On a smooth variety, the magic formula can be checked in local coordinates, while for singular schemes (and more general prestacks) it is necessary to work with the tangent complex, where it is no longer feasible to give explicit local formulas. Interpreting the magic formula as giving Griffiths transversality for the Gauss-Manin connection of the universal infinitesimal deformation of X, we are able to construct a formula-free, chain level Cartan calculus using the tangent complex of a singular scheme, and to establish the compatibility of this calculus with the noncommutative calculus of Hochschild cochains acting on Hochschild chains. This is joint work with Nick Rozenblyum.

Geometry & Representation - Anne Dranowski (University of Toronto)

Monday, November 4th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Anne Dranowski (University of Toronto)

Title: Generalized orbital varieties and MV modules.

Abstract:  Let O be the conjugacy class of a nilpotent matrix, and let C be its closure. By work of Joseph and Spaltenstein, the irreducible components of the subvariety of uppertriangular matrices in C, (aka orbital varieties,) can be labeled by standard Young tableaux. We explain how this labeling generalizes to the intersection of C and and a Slodowy slice, S. This question is motivated by the fact (due to Mirkovic-Vybornov) that such intersections are related to the Mirkovic-Vilonen (MV) construction of a cohomological crystal basis of GL(m). By D., the Mirkovic-Vybornov isomorphism maps the generalized orbital varieties to the MV cycles such that the crystal structure on tableaux matches the crystal structure on MV cycles. Our labeling enables us to determine equations of MV cycles and therefore compare the MV basis to another basis in bijection with tableaux - Lusztig's dual semicanonical basis - under the magnifying glass of the Duistermaat-Heckman measure.

Geometry & Representation - John Michael Machacek (York University)

Monday, October 7th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: John Michael Machacek (York University)

Title: Mutation combinatorics and upper cluster algebras.

Abstract:  Any cluster algebra is contained in an intersection of Laurent polynomial rings known as its upper cluster algebra. There are known cases where this containment is equality as well as cases of strict containment. We will discuss combinatorial approaches to determining if this containment is strict or not. Notions used will include reddening sequences and locally acyclic cluster algebras.

Geometry & Representation - Atabey Kaygun (Istanbul Tech University)

Monday, September 23rd, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Atabey Kaygun (Istanbul Technical University)

Title: Distributive Laws, Smash Biproducts and Hochschild Homology.

Abstract:  In this talk I am going to talk about distributive laws between algebras, resulting smash biproducts and their Hochschild homology. The examples include Ore extensions, Hopf smash products, quantum affine spaces and quantum complete intersection algebras. This is joint work with Serkan Sutlu.

Nonlocal problems in PDEs and geometry

May 20-24, 2019

Prof. Eleonora Cinti (Università di Bologna, Italy) will teach a 5-day mini-course aimed at graduate students and junior researchers at the intersection of Analysis and Geometry.

Dr. Cinti earned her Ph.D in 2010 and since then has worked at the Max Planck Institute in Leipzig, the Weierstrass Institute in Berlin, and various Italian universities (Pavia, Bologna, Torino). Eleonora Cinti's research focuses on nonlocal partial differential equations, geometric measure theory, and calculus of variations.

Schedule

The mini-course will be structured as follows:

  • Lecture 1 (Monday, May 20): Preliminaries: basic facts about the Laplacian and harmonic functions.
  • Lecture 2 (Tuesday, May 21): The fractional Laplacians: motivations and properties.
  • Problem Session (Wednesday, May 22).
  • Lecture 3 (Thursday, May 23): $s$-Harmonic functions and the Caffarelli-Silvestre extension theorem.
  • Lecture 4 (Friday, May 24): Geometry meets PDEs, a nonlocal phase transition model and nonlocal minimal surfaces.

Registration

If you want to attend the mini-course, please email nonlocal19@queensu.ca.

Geometry & Representation - Ba Nguyen (Queen's University)

Monday, April 1st, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Ba Nguyen (Queen's University)

Title: A Combinatorial Model for Detecting Cluster Variables of Type D Cluster Algebras

Abstract:  Perfect Matchings of a Snake Diagram, which was developed by G. Musiker, R. Schiffler and L. Williams, can be employed to compute cluster variables of a type D cluster algebra. Based on that model we have developed a new model called Globally Compatible Sequence. In this talk, we will discuss how to find cluster variables using this model. To have a better understanding of those cluster variables, description of their denominator vectors will also be introduced.

Geometry & Representation - Colin Ingalls (Carleton University)

Monday, March 18th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Colin Ingalls (Carleton University)

Title: McKay quivers

Abstract:  Fix a finite group G and a representation W, the McKay quiver has vertices given by irreducible representations $V_i$ and $\dim\mathrm{Hom}_G(W\otimes V_i,V_j)$ many arrows between $V_i$ and $V_j$. We briefly present the history of McKay quivers and their applications in geometry and representation theory. Then we discuss recent descriptions of McKay quivers of relfection groups by M. Lewis, and work with E. Faber and R. Buchweitz applying results of Lustzig on McKay quivers to understand the relations of the basic model of the skew group ring.

Geometry & Representation - Alistair Savage (University of Ottawa)

Monday, January 7th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Alistair Savage (University of Ottawa)

Title: Universal categories

Abstract:  Universal constructions are ubiquitous in mathematics.  For example, the polynomial ring is uniquely characterized by a universal property for commutative rings.  Other examples include free monoids, free groups, and tensor algebras.  In this mainly expository talk we will discuss an analogous, but somewhat less well known, concept on the level of categories.  In particular, we will see how one can define categories that are determined by universal properties.  Examples include the Temperley--Lieb category (the free monoidal category on a self-dual object), the Brauer category (the free symmetric monoidal category on a self-dual object), and the oriented Brauer category (the free symmetric monoidal category on a pair of dual objects).  We will discuss intuitive diagrammatic descriptions of these categories and how these universal constructions allow one to easily find deep symmetries in a wide range of categories.

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