PIMS Geometry, Algebra and Physics Seminar

Organized by Alex Weekes, Steve Rayan and Curtis Wendlandt


Schedule (2023-2024)

Talks are normally held on Thursdays at 4:00-5:30pm CST, and each can be attended virtually via Zoom (including those talks held in-person at the University of Saskatchewan) using the following information:

Meeting ID Passcode Link
936 6357 4827 qUSaskGAP Here

The seminar is intended to be informal and accessible to graduate students studying algebra, geometry, or mathematical physics. Speakers typically prepare an hour-long talk, with the remaining half hour dedicated to discussion and questions. 



Title and Abstract

Thursday, October 12th

4:30-6:00PM CST



Abid Ali

Strong Integrality of Inversion Subgroups of Kac-Moody Groups

The question of integrality for semi-simple algebraic groups over the field of rational numbers was esablished by Chevalley in the 1950s as part of his work on associating affine group schemes with groups over integers. For infinite-dimensional Kac-Moody groups, it remains an open problem. To state this problem more precisely, let \(\mathfrak{g}\) be a symmetrizable Kac-Moody algebra over \(\mathbb{Q}\), \(V\) be an integral highest weight \(\mathfrak{g}\)-module and \(V_{\mathbb{Z}}\) be a \(\mathbb{Z}\)-form of \(V\). Let \(G=G(\mathbb{Q})\) be an associated minimal representation-theoretic Kac-Moody group and let \(G(\mathbb{Z})\) be its integral subgroup. Suppose \(\Gamma(\mathbb{Z})\) is the Chevalley subgroup of \(G\), that is, the the subgroup that stabilizes the lattice \(V_{\mathbb{Z}}\) in \(V\). The integrality for \(G\) is to determine if \(G(\mathbb{Z})=\Gamma(Z)\). We will discuss some progress on this problem, which we made in a joint work with Lisa Carbone, Dongwen Liu, and Scott H. Murray. Our results have various applications, including the integrality of subgroups of the unipotent subgroup \(U\) of \(G\) that are generated by commuting real root groups.

Wednesday, November 22nd

4-5:30PM CST



Casey Blacker

Geometric and Algebraic Reduction of Multisymplectic Manifolds

A symplectic Hamiltonian manifold consists of a Lie group action on a symplectic manifold, together with the additional structure of a moment map, which encodes the group action in terms of the assignment of Hamiltonian vector fields. In special cases, the moment map determines a smooth submanifold to which the Lie group action restricts and the resulting quotient inherits the structure of symplectic manifold. In every case, it is possible to construct a reduced Poisson algebra that plays the role of the space of smooth functions on the reduced symplectic manifold.

In this talk, we will discuss an adaptation of these ideas to the multisymplectic setting. Specifically, we will exhibit a geometric reduction procedure for multisymplectic manifolds in the presence of a Hamiltonian action, an algebraic reduction procedure for the associated L-infinity algebras of classical observables, and a comparison of these two construction. This is joint work with Antonio Miti and Leonid Ryvkin.

Thursday, January 11th

4-5:30PM CST



Yuly Billig

Sheaves of AV-Modules Over Projective Varieties

AV-modules are representations of Lie algebra V of vector fields that admit a compatible action of the commutative algebra A of functions. This notion is a natural generalization of D-modules. In this talk we shall start by reviewing the theory of AV-modules over smooth irreducible affine varieties. When variety X is projective, it is necessary to consider sheaves of AV-modules. We describe associative algebras that control the category of AV-modules, and construct a functor from the category of strong representations of Lie algebra of jets of vector fields to the category of AV-modules.   This talk is based on the joint work with Colin Ingalls, as well as the work of Emile Bouaziz and Henrique Rocha.

Wednesday, February 7th

4:30-5:30PM CST



Julius Grimminger

Quivers and their 3d Coulomb branches

I will aim to give a pedagogical introduction to 3d Coulomb branches of quiver gauge theories, which are certain holomorphic symplectic varieties with symplectic singularities. After briefly explaining why I care about such spaces as a physicist, I will focus on the mathematical aspects of Coulomb branches, and discuss in particular how to count their holomorphic functions. I will then show how discrete modifications of a quiver induces a discrete quotient on its Coulomb branch.

Wednesday, February 7th

5:30-6:30PM CST



Antoine Bourget

Decay and fission of magnetic quivers

Over the past 5 years, the stratification of 3d Coulomb branches into symplectic leaves has been the subject of intense work in the physics community. Mathematically, this corresponds to understanding the nested singularity structure of conical symplectic singularities. This class includes Nakajima quiver varieties, slices in affine grassmannians, instanton moduli spaces, and much more. In this talk, I will review the methods that have been developed, centered on the concept of magnetic quivers. I will show how insights from string theory provide easy algorithms that can be implemented on a computer.


PIMS Logo 


Past Seminars: 2021-22 |2022-23