GLEN

Schedule

12-12.45 Lunch (registration required)

12.45-13.30 Lotte Hollands: Liouville blocks from spectral networks

13.30-13.45 Break

13.45-14.30 Nick Rekuski: Bogolomov--Gieseker type inequalities via flat families

14.30-14.45 Break

14.45 - 15.30 Svala Sverrisdóttir: Eigenvector varieties

15.30-16.00 Colloquium Tea Break

16.00-17.00 Federico Ardila Mantilla (joint with the St Andrews Colloquium): Intersection theory of matroids: variations on a theme

There will also be a social dinner in the evening, details TBC.

Registration

Registration deadline: 5th November 2026

Registration is required to attend this event, the registration form will open soon. To attend, please register using this form by Thursday 5th November.

We have a limited amount of travel funding to contribute towards travel for early career researchers - please apply for this via the registration form above.

Titles and Abstracts

Federico Ardila Mantilla (Queen Mary University of London and San Francisco State University)

Title: Intersection theory of matroids: variations on a theme

Abstract: Chow rings of toric varieties feature a rich combinatorial structure, which was one of the earliest glimpses into what is now called tropical geometry. We will begin by surveying four different ways of computing in these rings, due to Billera, Brion, Fulton–Sturmfels, and Allermann–Rau. We will illustrate the beauty and power of these methods by sketching four proofs of Huh and Huh–Katz’s formula μ_k(M) = deg_M(α*{r−k}β^k) that interprets the coefficients of the characteristic polynomial of a matroid M as intersection numbers in the Chow ring of M.  Each of these proofs sheds light on a different aspect of matroid combinatorics, and provides a framework for more recent developments in the intersection theory of matroids.

Lotte Hollands (Heriot-Watt University)

Title: Liouville blocks from spectral networks

Abstract: Spectral networks are networks of trajectories on a Riemann surface, determined by a branched covering of the surface. Gaiotto, Moore and Neitzke introduced them to compute the BPS spectra of certain supersymmetric field theories, and to describe how these spectra jump across walls of marginal stability. In this talk I will argue that spectral networks also play an important role in constructing conformal blocks. After introducing conformal blocks in Liouville conformal field theory, I will sketch how they can be built by abelianisation with respect to a spectral network, reducing the problem on the Riemann surface to an abelian one on its spectral cover.

Nick Rekuski (University of Liverpool)

Title: Bogolomov--Gieseker type inequalities via flat families

Abstract: Stability allow us to study vector bundles in moduli: if we fix topological data (i.e. Chern classes) then there is an algebraic variety parameterizing stable vector bundles with that data. A fundamental difficulty is determining whether this moduli is nonempty. In short, does there exist a stable vector bundle with prescribed topological invariants?

It is fruitful to consider a generalization of stability called a weak prestability condition. Broadly, a weak prestability condition is a notion of stability for chain complexes which "acts" like stability for vector bundles. For a fixed weak prestability condition, a Bogomolov--Gieseker type inequality is any algebraic obstruction to the existence of stable objects.

In this talk we show that if a weak prestability condition admits sufficiently large flat families of stable objects then it must also satisfy an explicit Bogomolov--Gieseker type inequality. As an application, we completely describe the polarized Bridgeland stability manifold of an Abelian variety of any dimension.

This is joint work with Franco Rota.

Svala Sverrisdóttir (Merton College, Oxford)

Title: Eigenvector Varieties

Abstract: Eigenvalue problems for NxN matrices arise throughout mathematics and applications. While conceptually elementary, solving these problems exactly, or even numerically, can become computationally infeasible for large values of N. At the same time, the matrices of interest often belong to structured families whose eigenvectors lie on low-dimensional algebraic varieties. I present a systematic study of these varieties, with a focus on Lie algebras and Hamiltonians of quantum systems.

Practical Information

We suggest arriving in St Andrews by taking the train to Leuchars Station, followed by a bus or taxi to the St Andrews Bus Station.

Lunch and the first three talks will be held in the Purdie Building, Room 314.

The colloquium tea break will be held in the common room on the ground floor of the Mathematics Institute, which is next door to Tutorial Room 1A.

The final talk will be held in Lecture Theatre D of the Mathematics Institute.

Organisers

Hannah Dell

Jeff Hicks

Yoav Len

Sponsorship

GLEN is supported by a Scheme 3 Grant of the London Mathematical Society, and the Edinburgh Mathematical Society.

This edition is also supported by EPSRC Project EP/Z53528X/1 and the School of Mathematics and Statistics at the University of St Andrews.