Symplectic singularities
at the
University of Bath

Schedule:
Lecture series

• Amihay Hanany, Imperial - Branes and symplectic singularities
  Abstract: I will present the brane system which leads to all the statements one can make from a physical system on the properties of symplectic singularities.
  
  

• Dinakar Muthiah, Glasgow - Coulomb branches and Kac-Moody affine Grassmanian slices
  Abstract:   I will talk about Coulomb branches, which are remarkable symplectic varieties originating in physics.  Recently, Braverman, Finkelberg, and Nakajima have given a rigorous mathematical construction of Coulomb branches. My talks will focus on how Coulomb branches can be used to construct Kac-Moody affine Grassmannian slices. With such a construction, one can hope to understand the Kac-Moody Geometric Satake Correspondence.
  
  

• Travis Schedler, Imperial - Representation theory of symplectic singularities
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  Abstract:  Given a symplectic singularity, one considers non-commutative algebras (quantisations) deforming the ring of functions on the singularity, with first-order non-commutativity governed by the Poisson bracket (analogously to the Heisenberg uncertainty principle). We will survey various aspects of this construction and the categories of modules over these quantisations. In the case of the nilpotent cone, this recovers the representation theory of semisimple Lie algebras. Other well known algebras recovered include Cherednik algebras and W-algebras. I will spend some time on quantised quiver varieties. In the last lecture I will explain how to bound the number of irreducible representations using D-modules on the singularity (joint work from 2009 with Etingof).
  
  

Research seminars

• Pavel Shlykov (x2), Glasgow - The Hikita-Nakajima conjecture

Abstract: I will speak about Hikita-Nakajima conjecture. This is a certain observation in the spirit of symplectic duality which relates equivariant cohomology on one side of  the picture with the schematic fixed points (or, in case of quantizations, B-algebra) of the universal deformation on the dual side. I will try to show some precursors of it, as well as show some relatively basic examples of the conjecture. If time permits, I will explain some ways people took in proving it, and, if time permits (x2) I will say something about the limitations of the original conjecture

• Juan Villarreal, Bath - Associated varieties/Higgs branches

Abstract: We will explain the construction of associated varieties of vertex algebras. From 4D/2D duality they correspond to the Higgs branches of four-dimensional N = 2 superconformal field theories (SCFTs). Finally, we will explain a work in progress where we study the associated variety of modules and their intertwiners.

• Ryo Yamagishi, Bath - Duality involution on single-vertex quiver varieties

Abstract: Motivated by the study of compact symplectic varieties, we consider quiver varieties for single-vertex quivers and their involutions which we call the duality involutions. I will describe the geometric structure of these quiver varieties and of their quotients by the involutions, with a particular focus on the (non)existence of symplectic resolutions.

Organisers:

• Lewis Topley (Bath) - lt803@bath.ac.uk

• Juan Villarreal (Bath) - jjv27@bath.ac.uk

Please enjoy the above artistic rendering of a Roman soldier eating a symplectic singularity, courtesy of generative AI.