Past Meetings: 2024-25

13.30 – 14.20: Tomasso Faustini (Warwick)

Title: Classical Projective Geometry Across Characteristics 

Abstract: In this talk, we examine key differences in algebraic geometry between characteristic zero and positive characteristic. Although foundational notions like varieties, schemes, and tangent spaces persist across characteristics, their behavior can differ substantially due to arithmetic features of the base field. 

We focus on classical examples involving inflection points and the count of bitangent lines to smooth plane quartic curves, emphasizing the role of the Frobenius morphism. In characteristic zero, these bitangents relate to Del Pezzo surfaces of degree two and their $(−1)$-curves, with rich connections at the level of moduli spaces. 

We conclude by illustrating how these structures degenerate in characteristic two, where classical correspondences break down and new phenomena appear.

14.40 — 15.30: Ruth Raistrick (Glasgow)

Title: Galois module structures of Unit Groups of Number Fields 

Abstract:  Given a number field $K$ of unit rank 1 and an extension $L/K$ of degree 2 and unit rank 2 what is the Galois module structure of the lattice given by the unit group of $L$ quotiented by its torsion? Precisely, given we know said lattice can only take two possible structures as a $\text{Gal}(L/K)$-module, as $L$ varies how often do we see each structure? We answer this question but find it has rather unsatisfying answer which reveals we should ask another question entirely. This talk will assume no knowledge of algebraic number theory and all objects will be defined. 

16.00 — 16.50: Chun-Yu Bai (Edinburgh)

Title: Derived G-skein module of 3-Torus

Abstract: I will introduce an algebraic definition of derived G-skein module of 3-manifold M=Sigma_g\times S^1, where G is a reductive algebraic group. I will in particular focus on 3-torus case, and compute the example when G is multiplicative group.

14.00 – 14.50: Gabriel Frey (Glasgow)

Title: Equivariant Cohomology and Applications in Geometric Invariant Theory

Abstract: Equivariant cohomology is a cohomology theory from algebraic and differential topology which generalizes the notion of cohomology of spaces endowed with a group action. In this talk, we introduce the notion of equivariant differential forms and cohomology, explain some examples such as its relation with moment maps in symplectic geometry and show how they can be used to compute the cohomology of geometric invariant theory (GIT) quotients, a result known as Kirwan's surjectivity theorem.

14.55 – 15.45: Xiang Li (Edinburgh)

Title: TBC

Abstract:  TBC

15.45- 16.10: coffee break

16.10-17.00: Antau Yang (Glasgow)

Title: Continued Fractions in the Field of p-adic Numbers

Abstract: Continued fractions have been extensively explored over the reals. But there are still a lot of openproblems for the ones over the 𝑝-adic field. A natural attempt is to find any analogue results. For many years, people have developed new algorithms and have analyzed their performance. I will quickly summarize the known results and share what I have tried so far.

12.30 – 13.20: Siao Chi Mok (Cambridge)

Title: Logarithmic Fulton—MacPherson configuration spaces

Abstract:  The Fulton—MacPherson configuration space is a well-known compactification of the configuration space of a projective variety. We extend the construction to the logarithmic setting: it is a compactification of the configuration space of points on a projective variety X away from a simple normal crossings divisor D, and is constructed via techniques in combinatorial algebraic geometry. This new construction enables us to make sense of a Fulton—MacPherson space of a non-compact variety. Time permitting, we will describe, given a simple normal crossings degeneration of X, how we could construct a degeneration of the Fulton—MacPherson space of X.

13.25 – 14.15: Adrian Cook (Edinburgh)

Title: Introduction to Polyptych Lattices

Abstract: As many of their algebraic and geometric properties are encoded in various combinatorial objects associated with them, toric varieties provide a bridge between algebraic geometry and combinatorics.  Because of this so called “toric dictionary” it is desirable to approximate an arbitrary projective variety with a toric one, so we may apply such combinatorial results more generally. One well-known way of obtaining such toric degenerations is through the theory of Newton-Okounkov (NO) bodies. Kaveh and Manon explored the theory of valuations to semifields of piecewise linear functions and used these findings to construct families of toric degenerations.  Harada and Escobar then described a “wall-crossing” phenomenon where different NO bodies of the same variety are related via piecewise linear mutation maps. Inspired by these findings, Escobar, Harada, and Manon proposed the theory of polyptych lattices as a combinatorial framework to unify and generalize these results. In joint work with Escobar, Harada, and Manon, we construct an explicit family of polyptych lattices to illustrate the theory more concretely. In this talk, I will introduce the listener to the basic definitions and results of this construction and give examples.

14.15- 15:00: coffee break

15.00-15.50: Lucia Nölle (Glasgow)

Title: What do the modules look like? Or : A representation theorist's quest.

Abstract: Life is full of mysteries, one of its greater ones being the representation theory of the cyclotomic rational Cherednik algebra. Setting out with a non-standard presentation and lots of Wanderlust we investigate modules of the wrapped cyclotomic rational Cherednik. This particular subalgebra allows for a satisfying classification of simple modules. We make some interesting discoveries along the way and then observe how they break down when considering the whole cyclotomic rational Cherednik algebra. Join me for a gentle stroll along some of the lesser-travelled yet scenic craigs of Representation theory.

15.55-16.45: Giovanni Sartori (Heriot-Watt)

Title: Groups acting on trees and Wise's power alternative

Abstract: A group is said to satisfy Wise's power alternative if every pair of elements, possibly after taking suitable non-zero powers of them, generates an abelian or non-abelian free subgroup of rank at most two. We give a sufficient condition for a group to satisfy Wise's power alternative and, as an application, we show that the family of (2,2)-free triangle-free Artin groups satisfy it.

13:30-14:20: Julia Bierent 

Title: Irregular flat connections and Stokes structures

Abstract: Systems of solutions of irregular flat connections are in 1-1 correspondence with Stokes local systems. This is an irregular version of the regular Riemann-Hilbert correspondence between regular flat connections and local systems. In order to open it to irregular flat connections, we need to define Stokes structures. This will lead to the definition of Stokes filtered local systems, which are local systems on the punctured surface with filtrations that are locally constant when they don't cross some particular directions (the Stokes directions). The stack of Stokes local systems is then very interesting to study. We will motivate and explain the irregular Riemann-Hilbert correspondence and the Stokes structures it implies.

14:30-15:20: Gabriel Corrigan

Title: Realising virtual cohomological dimension of automorphism groups of RAAGs

Abstract: In 1986, Culler & Vogtmann introduced 'Outer space' - a complex upon which Out(F_n), the outer automorphism group of a free group, acts properly. This had many applications; one is that the dimension of the so-called 'spine' of Outer space is precisely the virtual cohomological dimension (VCD) of Out(F_n). More recently, Charney-Stambaugh-Vogtmann constructed an 'untwisted Outer space' - an analogous space used for studying the group of untwisted automorphisms of a right-angled Artin group. However, in a departure from the free group case, sometimes the dimension of this rather natural 'untwisted spine' is larger than the VCD of the corresponding group of outer automorphisms! In this talk I will present work examining this phenomenon. I present graph-theoretic conditions under which we can perform an equivariant deformation retraction of the untwisted spine to produce a new complex which geometrically realises the VCD of the group of untwisted automorphisms. As a corollary, this proves that the gap between the VCD of the untwisted subgroup and the dimension of the untwisted spine can be arbitrarily large.

16:00-16:50: Francesco Tesolin 

Title: Generalizing the non-commutative Stone Duality to a wider class of morphisms.

Abstract:  There are many examples of dualities between categories of topological spaces and partially ordered sets. Generalizing the original Stone representation theorem of Boolean algebras, Kudryavtseva and Lawson have shown that Boolean inverse monoids are equivalent to Boolean groupoids. In joint work with Lawson, based on that of Kudryavtseva,  we generalize this duality to include a wider class of morphisms. These include the unusual coherent relational covering morphisms between Boolean groupoids. 

I will include an introduction to semigroups, topological groupoids, and the classical Stone duality.