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.
Date/Location |
Speaker |
Title and Abstract |
|---|---|---|
|
Thursday, October 12th 4:30-6:00PM CST (Hybrid) THORV 124 |
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 (Hybrid) PHYSICS 128 |
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. |
|
Thursday, January 11th 4-5:30PM CST (Hybrid) THORV 124 |
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 (Hybrid) THORV 124 |
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 (Hybrid) THORV 124 |
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. |
|
May 31st - June 3rd University of Saskatchewan |
Lisa Carbone, Elizabeth Jurisich, Maryam Khaqan, Scott Murray |
Unveiling Infinite SymmetriesA Mini Course of the 2024 CMS Summer Meeting on Algebraic Structures of Generalized Kac-Moody Type Webpage: https://summer24.cms.math.ca/mini-courses Description: Infinite-dimensional Lie algebras, including Kac-Moody algebras, Monstrous Lie algebras, and more generally, Borcherds algebras, play a crucial role in various branches of mathematics and theoretical physics. The mini-course will cover fundamental concepts related to these infinite-dimensional Lie algebras and their associated groups, share the latest developments on these topics, and discuss potential future projects on the new constructions. |
