See the calendar below for future seminars and events.
Following every Thursday seminar, attendees are welcome to come to one of our SMRI Afternoon Teas which take place on Thursday afternoons at 2pm on the Quadrangle Terrace, accessed through the entry in Quadrangle Lobby P and via the SMRI Common Room on level 4.
Upcoming and current events: seminars, workshops and courses
Public Lecture, ‘My Mathematical Journey: From Play To Sea!’ by Jordan Pitt
Abstract: Every time I mention that I’m a mathematician to someone new, the most popular response is ‘Oh I was TERRIBLE at maths!‘ and a general vibe that I’m an extreme weirdo for not sharing in this feeling. Honestly, as mathematicians, we are a bit different but I am going to try and explain why we’re not that weird in this talk. To do this I will provide some stories of my own mathematical journey and why I ended up loving it.
In this AMSI Summer School 2025 public lecture, University of Sydney mathematician Jordan Pitt will talk about his journey in mathematics. This will be a free event targeted to the general public, including interested school students. More details and registration.
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:
Details: All lectures will run from 15:30 – 16:30, in the SMRI Seminar Room (Macleay Building A12 Room 301).
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.
Details: All lectures will run on Fridays from 9:30 – 11:00, in the SMRI Seminar Room (Macleay Building A12 Room 301).
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
Modern Perspectives in Representation Theory: Special Semester 2025
Organizers: Charlotte Chan (University of Michigan), Thomas Lam (University of Michigan), and Geordie Williamson (University of Sydney)
A special semester on “Modern Perspectives in Representation Theory” at the Sydney Mathematical Research Institute from May 5 through June 13, 2025. As part of the program, special events are planned for Week 3, May 19 – 23, and a conference running in Week 5, June 2 – 6.