Past meetings: 2025-26

13:30-14:20 Noah Dizep (Heriot-Watt)

Title: Difference Equations, Relative Gromov-Witten theory and the Geometry of the Nekrasov-Shatashvili limit

Abstract: Recently there has been an increased interest in the geometry of the refined topological string on a Calabi-Yau threefold in the Nekrasov-Shatashvili limit. In this talk, building up on the work by Bousseau and Brini-Schueler, we will discuss the enumerative meaning of the quantum corrected B-period of the resolved Conifold. This serves to showcase the curious interplay between difference equations arising from the quantization of mirror curves, relative Gromov-Witten theory and refined sheaf counts.

15:00-15:50 Mikhail Vasilev (Glasgow)

Title: Calogero-Moser systems, integrability and Cherednik algebras

Abstract: I will give a gentle introduction to the phenomenon of integrability in general and in the theory of Calogero-Moser systems. I will introduce both classical and quantum integrable Calogero-Moser systems, including scalar and spin(matrix) versions of this model. On the algebraic side one of the structures, which controls the integrability of the Calogero-Moser systems are Cherednik algebras. I will show how the representation theory of Cherednik algebras can be utilised to prove the integrability of the known Calogero-Moser systems. If time permits, I will briefly talk about our recent work with Misha Feigin and Martin Vrabec about the derivation of the spin deformed Calogero-Moser systems from the Cherednik algebras. 

16:30-17:20 Emanuel Roth (Edinburgh)

Title: The structure of instability of moduli of bundles

Abstract: In the moduli theory of bundles, we impose stability to obtain varieties or schemes parametrizing bundles with nice properties (smoothness, separatedness, etc.). When bundles are unstable, we have less of a grasp on how to classify them. Harder and Narasimhan constructed a filtration of bundles that measures how close a bundle is to semistability, which works for principal bundles (by Ramanathan), as well as for Higgs bundles and parahoric torsors. Similarly, Jordan-Hölder filtrations measure how close a semistable bundle is to being stable. I will talk about constructing complementary polyhedra (from Kai Behrend's PhD thesis), a root-theoretic object that records the (semi)-stability of bundles and provides a unified approach to Harder-Narasimhan/Jordan-Hölder filtrations. I will explain how I hope to use this to prove the normality of the stack of parahoric Higgs torsors in a future project.

João Camarneiro (Edinburgh)

Title: Infinite symplectic staircases and where to find them

Abstract: In 1985, Gromov proved the famous non-squeezing theorem, giving new fundamental insights into the nature of symplectic geometry. Since then, there has been a lot of interest in various symplectic embedding problems. In this talk, I will tell you about the "infinite staircases" that sometimes appear when embedding 4-dimensional symplectic ellipsoids into various targets, and the surprisingly intricate structure governing their existence.

Theresa Ortscheidt (Glasgow)

Title: Combinatorics of Structure Coefficients in the Fusion Ring

Abstract: The fusion or Littlewood-Richardson ring is a quotient of the ring of symmetric functions. While much is known about the Littlewood-Richardson coefficients, which are the structure coefficients of the ring of symmetric functions with respect to the Schur basis, the fusion coefficients are not yet well understood. In particular, one open problem is the lack of a strictly positive combinatorial formula to compute fusion coefficients.

In my talk I will recap some of the combinatorics of Littlewood-Richardson coefficients and then present some of the results of Goodman and Wenzl, who found a (not strictly positive) combinatorial formula to compute fusion coefficients for certain representations of Hecke algebras at roots of unity. I will then briefly discuss some of my progress in trying to find a strictly positive algorithm.

Giorgio Mangioni (Heriot-Watt)

Title: Cyclic splittings of Artin groups

Abstract: Atari founder Nolan Bushnell once said that "every good game is easy to learn but hard to master". As an instance of this principle, it is straightforward to define Artin groups, which arise naturally in representation theory and topology, but most basic questions about this class of groups have been open for over a century. We don't even have a unified solution to the isomorphism problem, which asks to determine when two Artin groups are isomorphic!

In this talk we characterise which Artin groups admit a splitting over Z (that is, an action on a tree with infinite cyclic edge stabilisers), and we use this to shed more light on the isomorphism problem. In the process, we also construct a JSJ splitting for any Artin group, which roughly "refines" every splitting over Z. If time allows, we shall also mention some applications to the study of automorphism groups of Artin groups, and further directions of investigation. 

This talk is based on joint work with Oli Jones and Giovanni Sartori.