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.

Talia Shlomovich (Heriot-Watt)

Title: Non-planarity of Group Boundaries

Abstract: When can you draw a boundary of a group on a sphere? And when can you not?

In this talk, I aim to convince you that this property is a useful tool for describing and distinguishing between boundaries of groups. This is especially useful for boundaries which are topological fractals, such as Cantor spaces and Menger curves. 

Although non-planar boundaries are common among hyperbolic groups, it is surprisingly difficult to construct them. I will discuss how graph-theoretic characterisations of non-planarity are used to do so, and present certain classes of Coxeter groups as a source of examples. I will then share my progress towards constructing a right-angled Coxeter group with non-planar Morse boundary.

Cassia Edwards (Edinburgh)

Title: Motivating Factorisation Algebras in Quite a Lot of Generality

Abstract: Factorization algebras have been defined in multiple contexts: the manifold setting of Costello-Gwilliam; the chiral algebra setting of Beilinson-Drinfeld and the locally constructible setting of Karlsson-Scheimbauer-Walde. Recently, Barwick has constructed a formalism that seeks to encompass these approaches through the use of twofold symmetric monoidal categories. 

In this talk we will use the definition by Beilinson-Drinfeld to motivate the setting constructed by Barwick, and introduce the starting concepts of his theory - the combinatorics of isolation and isolability spaces.

Barbara Santana Muniz (Jagiellonian University, Krakow)

Title: Atomic decomposition of affine Kazhdan–Lusztig polynomials

Abstract: The spherical Hecke algebra is a deformation of the ring of dominant weights Z[X+], which appears as the decategorification of the categories in the geometric Satake equivalence. Its associated Kazhdan–Lusztig polynomials are fundamental combinatorial objects with multiple incarnations across representation theory. Chiefly, they admit interpretation as quantum analogues of weight multiplicities, although realizing this interpretation for all types is a long-standing problem in algebraic combinatorics.

In this talk, we will introduce these Hecke algebras in accessible terms and discuss the existence of an atomic decomposition, a strengthening of the monotonicity condition on the KL polynomials that is closely related to this realization problem.

Julie Tavernier (University of Bath)

Title: When the McKay correspondence goes wild

Abstract: A theorem by Batyrev says that for a finite group G, the Euler characteristic of a crepant resolution of the quotient of affine space by the action of G is equal to the number of conjugacy classes of G. This is part of the McKay correspondence.
But does this result also hold over fields of prime characteristic?
In this talk, we explore what happens to this correspondence when |G| is divisible by the characteristic of the base field, known as the wild McKay correspondence. We attempt to recover a version of Batyrev's statement for certain metacyclic groups, and discuss the difficulties occurring in the wild case. This is joint work with Takehiko Yasuda.

Xinyang Liu (University of Newcastle)

Title: Representation Theory of Very Non-Standard Quantum 𝔰𝔬(2N)

Abstract: The classical Gelfand–Tsetlin basis yields explicit constructions associated with a chain of nested orthogonal Lie algebras. To study its quantum analogue, standard Drinfeld–Jimbo quantum groups are insufficient.

In this talk, I will introduce a quantum analogue of 𝔰𝔬(2N) within the framework of Letzter’s quantum symmetric pairs and classify its finite-dimensional representations over arbitrary fields, where q is not a root of unity. These algebras depend on an additional parameter c. Under mild restrictions on c, we show that all such representations are deformations of classical 𝔰𝔬(2N)-modules. Furthermore, we classify these representations by their highest weights and demonstrate how these weights deform the classical notion of a dominant integral weight.

This talk is based on joint work with Stefan Kolb and Martina Balagovic.

Diana Bergerova (Edinburgh)

Title: Joyce's construction of the Donaldson-Thomas sheaf

Abstract: In simple terms, the Donaldson-Thomas (DT) invariant counts stable coherent sheaves on Calabi-Yau threefolds. It turns out that this count is closely related to taking the weighted Euler characteristic of the threefold itself. In some special cases we actually can express the weights using certain perverse sheaves called the vanishing cycles sheaves. We thus arrive to a possible construction of a categorified DT invariant, i.e. a sheaf whose cohomology gives the DT invariants. In this talk, I aim to explain this idea and, in particular, which problems arise in wanting to define such an invariant.

Emmanouil Sfinarolakis (Herriot Watt)

Title: Shannon Entropy to KL Divergence: Operadic Derivations Away from the Origin

Abstract: Bradley showed that Shannon entropy is a derivation of the operad of topological simplices, and that every derivation of this type recovers a scalar multiple of Shannon entropy when evaluated at the origin. This talk explains how this picture extends to relative entropy. The positive product operad O°_{KL} = Δ° × Δ° is introduced as the natural operadic setting for pairs of probability distributions, and the chain rule for Kullback--Leibler divergence is shown to be an operadic Leibniz rule. In this setting, KL divergence becomes a scalar derivation for the natural P--expectation coefficient structure.

The main focus is what happens beyond the scalar origin trace. The talk describes the scalar coefficient bimodule, its realization inside constant endomorphisms, and the full conditional endomorphism bimodule. While the scalar theory recovers the classical KL chain rule, the operator--valued theory reveals additional structure away from the origin: the affine theory admits a rigid finite--dimensional classification, while the nonlinear theory reveals a much larger family of derivations, including familiar dispersion laws as special cases. Thus, the framework generalises Bradley’s theorem from Shannon entropy to KL divergence and uncovers a richer operator--valued theory of information and dispersion.

Willoughby Seago (Glasgow)

Title: Alternating Vertex Models and Super Symmetry

Abstract: The six vertex model is a combinatorial tool with connections to many areas of mathematics, including the theory of quantum groups and crystals, where the combinatorics of Young tableaux play a key role through the RSK correspondence. In this talk I will introduce these topics, and a generalisation of the six vertex model, Young tableaux, and RSK, to the supersymmetric setting. Time permitting, I will discuss ongoing work relating these to crystals for quantum groups of Lie superalgebras.