Random Polytopes Seminar: The diameter of polytopes and the Hirsch conjecture (1/2)
Geordie Williamson, The University of Sydney
Friday 20th June 2025
Abstract: Can one reasonably bound the diameter of the graph of a simple polytope? This is a fascinating open problem in combinatorial geometry. I’ll outline what is known, and in particular sketch Santos’ remarkable 2012 counter-example to the 50 year-old Hirsch conjecture. With the DeepMind team, we recently attacked this problem using machine learning. We made some progress, but this is not the focus of the talk. I want to communicate a simple problem where the right idea could be revolutionary.
Random Polytopes Seminar: A short survey on random polytopes (2/2)
Renjie Feng, The University of Sydney
Friday 20th June 2025
Abstract: Suppose we take random points in the plane from a Gaussian and look at the polytope they generate. What can I say about it? What is its expected volume, number of vertices, number of edges etc? This talk will provide an introduction to these kinds of questions. The focus is on many points in a fixed dimension, but some striking results where one allows the dimension to grow will also be touched upon.
Special Seminar: Modern Perspectives in Representation Theory, Wild character varieties and braid varieties (6/6)
Masoud Kamgarpour, University of Queensland
Thursday 12th June 2025
Special Seminar: Modern Perspectives in Representation Theory, Singularities of orbit closures in loop spaces of symmetric varieties (5/6)
Tsao-Hsien Chen, University of Minnesota
Tuesday 10th June 2025
Special Seminar: Modern Perspectives in Representation Theory, Hodge theory and unitary representations of real reductive groups (4/6)
Dougal Davis, The University of Melbourne
Tuesday 10th June 2025
Special Seminar: Modern Perspectives in Representation Theory, Catalan from the DAHA lens (3/6)
Arun Ram, The University of Melbourne
Tuesday 10th June 2025
Special Seminar for Modern Perspectives in Representation Theory (2/6)
Calvin Yost-Wolff, The University of Michigan
Wednesday 28th May 2025
Special Seminar for Modern Perspectives in Representation Theory (1/6)
Robert Cass, The University of Michigan
Wednesday 28th May 2025
Short Course: Tensor products and highest weight structures
A short course by Jonathan Gruber
Abstract: In this series of talks, I will discuss connections and interactions between two types of structures that are ubiquitous in Lie theory and representation theory. The first is the formalism of highest weight categories, which provides an axiomatic framework for studying categories with “highest weight modules” (or “standard modules”). The second is the theory of monoidal categories, i.e. of categories equipped with a tensor product bifunctor.
The primordial example of a highest weight category with a monoidal structure is the category of rational representations of a reductive algebraic group, and here the highest weight structure interacts with the monoidal structure by way of the fact that tensor products of standard modules admit filtrations whose successive subquotients are standard modules. This kind of interaction can surprisingly be observed in many other examples, and I will give an explanation for this phenomenon via a monoidal enhancement of Brundan-Stroppel’s semi infinite Ringel duality. As applications, I will present solutions to two open problems: One concerns the existence of monoidal structures on categories of representations of affine Lie algebras at positive levels; the other concerns the existence of highest weight structures on monoidal abelian envelopes of certain “interpolation tensor categories”. All of this is based on joint work with Johannes Flake.
The program for the lecture series is as follows:
Wednesday 22 January 2025 Lecture 1: In the first talk, I will mostly discuss the motivating examples (algebraic groups, affine Lie algebras, interpolation categories), explain how they fit into the framework of “lower finite” or “upper finite” highest weight categories, and state the two aforementioned open problems.
Thursday 23 January 2025 Lecture 2: In the second talk, I will explain how lower finite and upper finite highest weight categories are related via Ringel duality and how a monoidal structure on a lower finite highest weight category gives rise to a monoidal structure on the Ringel dual upper finite highest weight category. This allows us to solve our first open problem: We define a canonical monoidal structure on a parabolic version of the BGG category O for an affine Lie algebra at positive level, and we construct a monoidal functor (a “Kazhdan-Lusztig correspondence”) to a category of representations of a quantum group at a root of unity.
Wednesday 29 January 2025 Lecture 3: The third and fourth lecture will be devoted to explaining the converse, that is, how a monoidal structure on an upper finite highest weight category gives rise to a monoidal structure on the Ringel dual lower finite highest weight category. As an application, we show that a large class of interpolation tensor categories (defined by Knop, generalizing a construction of Deligne) can be embedded as categories of tilting objects in monoidal lower finite highest weight categories. This gives a uniform explanation for the previously mysterious observation that many “abelian envelopes” of these interpolation categories are highest weight categories.
Thursday 30 January 2025 Lecture 4: (As above) The third and fourth lecture will be devoted to explaining the converse, that is, how a monoidal structure on an upper finite highest weight category gives rise to a monoidal structure on the Ringel dual lower finite highest weight category. As an application, we show that a large class of interpolation tensor categories (defined by Knop, generalizing a construction of Deligne) can be embedded as categories of tilting objects in monoidal lower finite highest weight categories. This gives a uniform explanation for the previously mysterious observation that many “abelian envelopes” of these interpolation categories are highest weight categories.
Short Course: Affine Grassmannians and Bun_G: A Parahoric Perspective
A short course by Jiuzu Hong
Abstract: Affine Grassmannians and Bun_G play a central role in geometric representation theory, geometric Langlands and algebraic geometry. Parahoric group schemes originated from Bruhat-Tits theory, and their global counterparts over algebraic curves called parahoric Bruhat-Tits group schemes were introduced more recently by Pappas-Rapoport and Heinloth.
All partial affine flag varieties can be viewed as affine Grassmannians of parahoric group schemes. This perspective has advantage of globalizing their geometry over an algebraic curve in the style of Beilinson-Drinfeld Grassmannians. This approach has led to interesting applications, including Zhu’s proof of the coherence conjecture of Pappas and Rapoport, and the determination of smooth loci of Schubert varieites in twisted affine Grassmannians by Besson and myself.
Moreover, the moduli of bundles over parahoric Bruhat-Tits group schemes generalizes parabolic Bun_G and Prym varieties. In fact, their non-abelian theta functions can be identified with (twisted) conformal blocks. This is my recent work joint with Damiolini, building on my earlier works with Kumar.
Friday 7 February 2025 Lecture 1: Affine Grassmannians and Schubert varieties
Friday 14 February 2025 Lecture 2: Loop groups and parahoric group schemes
Friday 21 February 2025 Lecture 3: Global Schubert varieties
Friday 28 February 2025 Lecture 4: Line bundles on moduli of parahoric bundles