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 - 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.


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.


If you want to attend the mini-course, please email

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.

Geometry & Representation - Kaveh Mousavand (UQAM)

Monday, December 3rd, 2018

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

Speaker: Kaveh Mousavand (UQAM)

Title: $\tau$-tilting finiteness of special biserial algebras

Abstract:  $\tau$-tilting theory, recently introduced by Adachi-Iyama-Reiten, is an elegant generalization of the classical tilting theory which fixes the deficiency of the tilting modules with respect to the notion of mutation. In this talk, I view $\tau$-tilting finiteness of algebras as a natural generalization of the representation finiteness property. The natural question then becomes: For which families of algebras does $\tau$-tilting finiteness imply representation finiteness?

First I introduce a reductive method that can be applied to certain families of algebras to reduce this, a priori, intractable problem to a subfamily with nice features. Then, as an interesting class of algebras, I consider the special biserial algebras and for every minimal representation infinite member of this family, I give a full answer to the above question and show. As a corollary, we conclude that a gentle algebra is $\tau$-tilting finite if and only if it is representation finite.

Geometry & Representation - Mike Roth (Queen's University)

Monday, November 26th, 2018

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

Speaker: Mike Roth (Queen's University)

Title: Generating Rays for the Eigencone (after Belkale and Piers)

Abstract:  Let G be a semisimple algebraic group. A fundamental question in the representation theory of G is knowing how to decompose the tensor product of two irreducible representations into its irreducible components, or slightly weaker, which irreducible components appear in a tensor product of two irreducible representations. The irreducible representations of G are parameterized by highest weights, vectors in ℕ^{r}, where r is the rank of G. For a highest weight λ the corresponding irreducible representation is denoted V_{λ} If one takes triples (λ, μ, ν) of highest weights such that V_{ν} appears in V_{λ} ⊗ V_{μ} then these triples generate a polyhedral cone in ℚ^{3r}, known as the eigencone (or sometimes the tensor cone). Trying to find explicit equations for the hyperplanes cutting out the eigencone is a problem with a long history, including fundamental contributions by Weyl, Gelfand, Lidskii, and Wielandt. Finally, twenty years ago, Klyachko found a set of hyperplane inequalities cutting out the eigencone in type A. Progress in the last 20 years has included finding hyperplane inequalities for the eigencones in all types, finding minimal hyperplane inequalities in all types, and finally, also finding descriptions of the linear conditions cutting out higher codimensional faces of the eigencone. Dually to their description by hyperplane inequalities, polyhedral cones may also be described by their generating rays. It is of course natural to then ask for the generating rays of the eigencone. This talk will discuss a recent paper of Belkale and Piers giving a recursive method, valid in all types, of finding generating rays for the eigencone.

Geometry & Representation - Yin Chen (NE Normal University, China)

Monday, November 12th, 2018

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

Speaker: Yin Chen (Northeast Normal University, China)

Title: On vector invariant fields for finite classical groups

Abstract:  In the recent work ( with David Wehlau, we found a minimal polynomial generating set for the vector and covector invariant field of the general linear group over finite fields. Our method relied on some relations between the generators for the invariant ring of one vector and one covector. The remaining case (without covectors) is more complicated. In this talk, I will present an approach to find polynomial generating sets for the vector invariant fields of the most of finite classical groups. This is a joint work with Zhongming Tang.

Geometry & Representation - Steven Spallone (Indian Institute)

Monday, November 5th, 2018

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

Speaker: Steven Spallone (Indian Institute of Science Education and Research)

Title: Spinoriality of Orthogonal Representations of Reductive Groups

Abstract:  Let G be a connected semisimple complex Lie group and $\pi$ an orthogonal representation of G. We give a simple criterion for whether $\pi$ lifts to the spin group Spin(V), in terms of its highest weights. This is joint work with Rohit Joshi.