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.

Alexandra Ciotau (Edinburgh)

Title: Growth and Noetherianity of U(W_+).

Abstract: Motivated by the question of whether any infinite-dimensional Lie algebra admits a Noetherian universal enveloping algebra, we study one of the best-understood cases, U(W_+).

We introduce Gelfand–Kirillov dimension as a measure of algebraic growth and work through examples of algebras of different GK-dimension, before showing that the partition-number growth rate (Hardy–Ramanujan’s formula) forces "GKdim"(U(W_+))=∞. Despite this, U(W_+) has a controlled two-sided ideal structure: too large to be Noetherian but more structured than a free algebra. We present the theorem of Iyudu and Sierra that U(W_+) has “just infinite” GK-dimension: every nonzero proper quotient has polynomial growth, making U(W_+) infinite-dimensional in the most minimal way.

Simeon Hellsten (Glasgow)

Title: Cobordism and Concordance of Surfaces in 4-Manifolds

Abstract: The goal of a mathematician is often to classify phenomena up to the “natural” notion of equivalence. Unfortunately, said phenomena are often impossible to classify, and so we force ourselves to be happy with a classification up to weaker equivalence relations.

One such phenomenon is embedded surfaces in 4-manifolds, where we are forced to weaken the natural notion of isotopy in various ways to obtain a complete classification. In this talk, I will discuss my completion of this classification up to cobordism, and up to concordance in simply-connected 4-manifolds. I will also discuss the status of the concordance classification in general 4-manifolds, and how much actually seems within reach.

Ervin Hadžiosmanović (Scuola Normale Superiore)

Title: Bounded cohomology of negatively curved manifolds and differential forms

Abstract: Bounded cohomology is a functional-analytical variant of singular cohomology developed by Gromov in the 80s in his work on Riemannian geometry. Since then, it has developed into an independent and rich research field. As we will see, the definition may seem innocuous, but it turns out that it behaves very differently from ordinary cohomology, leading to some strange phenomena (for example, it can be infinite-dimensional even for closed manifolds).

A general principle is that it tends to vanish for "flat" manifolds and be very big for "negatively curved" ones, even if its definition is purely topological. In this talk, I will focus on the second part of this principle by discussing a result of Barge and Ghys, which allows one to build quite concretely an infinite-dimensional subspace of the second-degree bounded cohomology of closed hyperbolic surfaces via differential forms.

If time permits, I will also discuss some possible related questions and answers.