Talks will take place fortnightly during terms. Each talk is 2 hours: the first hour is expository, with the goal of explaining a fundamental idea or computation, and is aimed at HDR students. The second hour is a research talk. You may come for one or both hours.
Schedule: Term 1, 2026
Wednesday, February 25, 2-4pm:
Location: Red Centre 3085
Speaker: Anna Romanov (UNSW)
Title: Intersection cohomology without spaces
Abstract: A beautiful theme in representation theory is that combinatorial and representation-theoretic information about a structure (group, algebra, polytope, etc.) is often captured by the intersection cohomology groups of a related algebraic variety. Intersection cohomology is complicated to compute in general, but for the varieties which arise in this way, there is often a straightforward combinatorial method for computing it. A remarkable feature of these stories is that the graded vector spaces arising in the combinatorial construction are part of a larger family, for which there is no associated variety. In other words, they act as “intersection cohomology without spaces”.
In this talk, I will explain my favourite example of this phenomenon - the intersection cohomology groups of Schubert varieties and their incarnations as Soergel bimodules. In this example, the special class of groups are Weyl groups, and the larger family are Coxeter groups.
I will dedicate the first hour to cohomology, with a focus on examples. We will discuss some deep Hodge-theoretic properties of its structure (the hard Lefshetz theorem and the Hodge-Riemann bilinear relations), then I will spend some time motivating intersection cohomology.
In the second hour, I will describe Soergel’s method for computing the intersection cohomology groups of Schubert varieties in a purely algebraic way. This involves a brief tour of reductive algebraic groups, flag varieties, and Weyl groups. In the final minutes, I’ll explain how this topic touches on my own research by proposing another setting in which “intersection cohomology without spaces” might occur - the settling of real Lie groups.
Wednesday, March 11, 2-4pm:
Location: Red Centre 3085
Speaker: Balazs Elek
Title: Crystals, Heaps and preprojective algebra modules
Abstract: A crystal is a colored directed graph associated to a representation of a reductive algebraic group that enables us to understand the structure of the representation in purely combinatorial terms. Crystals are very easy and fun to work with. In the first hour we will give a hands-on introduction to crystals with lots of examples. Then we will study a concrete type independent model for certain special crystals based on heaps, which are configurations of beads on runners.
In the second hour we will shift to geometry and introduce a second model for crystals on the irreducible components of a certain quiver variety. Then we'll see how the Jordan canonical form provides a crystal isomorphism between the two models. This is joint work with Anne Dranowski, Joel Kamnitzer and Calder Morton-Ferguson.
Wednesday, April 8, 2-4pm:
Location: Red Centre 3085
Speaker: Yixuan Li
Title: Homological Mirror Symmetry for Affine Grassmannian Slices in Type A
Abstract: This is joint work with Mina Aganagic, Ivan Danilenko, Vivek Shende and Peng Zhou. In this talk we discuss a homological mirror symmetry result, where both the symplectic side and the algebraic side are related to affine grassmannian slices in type A. In the pre-talk, we will review some basic examples of homological mirror symmetry. In the second part of the talk, we will begin with examples of type A Affine grassmannian slices, which are useful geometric objects for studying representation theory. The primary examples are A_n surfaces, which are the main players of the pure math seminar this week. They also include all type A Slodowy slices. By the geometric Satake theorem, their homologies are related to weight spaces in tensor products of fundamental representations. Since Fukaya category can be loosely viewed as a categorification of the middle homology, this result can be viewed as one step towards two different categorifications of the geometric Satake theorem.
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