Talk titles and abstracts

Eloïse Hamilton (Thursday 13:30) - Stability within instability

Abstract: Given the linear action of a reductive group G on a projective variety X, Geometric Invariant Theory (GIT) produces an open locus inside X, called the semistable locus, which has two important properties. Firstly, it admits a quotient which is projective. Secondly, the cohomology of the quotient can be computed inductively in terms of that of X. This talk is guided by the following question: within the unstable locus, namely the complement of the semistable locus, is there a suitable notion of "stability" such that the associated "stable" locus also satisfies these same two properties? I will explain how a recent generalisation of GIT, called Non-Reductive GIT, can be used to give a positive answer to this question, and illustrate the resulting notions of "stability" in the case of rank 2 Higgs bundles.

Frances Kirwan (Thursday 14:30) - GIT and Chow quotients

Abstract: When a reductive group G acts linearly on a projective variety X, Mumford's geometric invariant theory (GIT) provides a 'GIT quotient' which depends on the linearisation of the group action with respect to an ample line bundle on X. The 'Chow quotient' (due to Kapranov and others) is independent of the choice of linearisation but is often more difficult to understand geometrically. The aim of this talk is to explore the relationship between these constructions and related constructions for non-reductive group actions.

Jesus Martinez-Garcia (Thursday 16:00) - Computational Geometric Invariant Theory via Sagemath

Abstract: I will discuss a (pseudo-)algorithmical way to describe (semi/poly-)stable points in GIT problems. I will describe the method and apply it to the study of moduli problems whose elements are naturally represented as points in projective space and the group acting on it is simple. However, the method is easily generalised to other settings (such as products of grassmannians, semi-simple groups). In our particular situation of groups on projective space, I will demonstrate the algorithmical nature by running some Sagemath code. I will also discuss applications to the description of the K-moduli of Fano 3-folds. This is joint work with Patricio Gallardo, Han-Bom Moon and David Swinarski.

Gerhard Röhrle (Friday 9:00) - Overgroups of regular unipotent elements in reductive groups

Abstract: There is a long and remarkable history of the study of the subgroup structure of reductive algebraic groups. This in particular involves overgroups of special elements. I shall report on recent joint work with Michael Bate and Ben Martin where we study reductive subgroups H of a reductive linear algebraic group G such that H contains a regular unipotent element of G. We show that under suitable hypotheses, such subgroups are G-irreducible in the sense of Serre; this means such H are not contained in a proper parabolic subgroup of G. This work generalizes previous results of Malle, Testerman and Zalesski. Time permitting I shall indicate analogous results for Lie algebras and for finite groups of Lie type.

Theodoros Papazachariou (Friday 10:00) - Computational Variational Geometric Invariant Theory and applications to K-stability

Abstract: An important category of geometric objects in algebraic geometry is smooth Fano varieties. These have been classified in 1, 10 and 105 families in dimensions 1, 2 and 3 respectively, while in higher dimensions the number of Fano families is yet unknown. An important problem is compactifying these families into moduli spaces via K-stability. A more interesting setting occurs in the case of pairs of varieties and a hyperplane section where the K-moduli compactifications tessellate depending on a parameter. In this case it has been shown recently that the K-moduli decompose into a wall-chamber decomposition depending on a parameter, but wall-crossing phenomena are still difficult to describe explicitly. In this talk I will describe an algorithmic way to describe non-stable/unstable elements of Variational GIT quotients of pairs of complete intersections and hyperplanes. Consequently, I will describe a method to obtain (poly-)stable elements of these quotients. I will apply this algorithm to the VGIT quotient of complete intersections of two quadrics in P3 and P4. I will then demonstrate how these GIT quotients can provide compactifications for the K-moduli of the Fano threefold family 2.25 (in the Mori-Mukai classification), and an explicit description of K-moduli wall crossing phenomena for log pairs of a del Pezzo surface of degree 4 and an anticanonical divisor.

Ben Martin (Friday 11:30) - Rigid representations of triangle groups

Abstract:Given positive integers $a$, $b$ and $c$, the {\em triangle group} $T(a,b,c)$ is the group defined by the presentation $\langle x,y,z\,|\,x^a = y^b = z^c = xyz = 1\rangle$. Let $G$ be a simple algebraic group over a finite field ${\mathbb F}_q$ and for $d\in {\mathbb N}$, let $j_d$ denote the largest dimension of a conjugacy class of elements of $G$ of order $d$. Claude Marion conjectured that if $j_a+ j_b+ j_c= 2{\rm dim}(G)$ then there are only finitely many values of $r$ such that the finite group of Lie type $G(q^r)$ is a quotient of $T(a,b,c)$. I will discuss a proof of Marion's conjecture using some ideas from the theory of representation varieties.
This is joint work with Alastair Litterick.

Alastair Litterick (Friday 12:30) - Complete Reducibility and Subgroup Structure of Reductive Groups

Abstract: The concept of G-complete reducibility generalises the representation-theoretic notion of 'completely reducible' module to all reductive algebraic groups. Initially defined in terms of parabolic subgroups and their Levi factors, the notion can also be characterised by the closure of G-orbits in the direct product G^n acting by conjugacy. Thus G-complete reducibility unifies group theory, representation theory and geometric invariant theory. This proves a powerful tool in classifying reductive subgroups of reductive groups, and in this talk I will give an overview of my and others' work in this direction.