Talk titles and abstracts
Eloïse Hamilton (Thursday 13:30)  Stability within instabilityAbstract: 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 NonReductive 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 quotientsAbstract: 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 nonreductive group actions. 
Jesus MartinezGarcia (Thursday 16:00)  Computational Geometric Invariant Theory via SagemathAbstract: 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, semisimple 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 Kmoduli of Fano 3folds. This is joint work with Patricio Gallardo, HanBom Moon and David Swinarski. 
Gerhard Röhrle (Friday 9:00)  Overgroups of regular unipotent elements in reductive groupsAbstract: 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 Girreducible 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 KstabilityAbstract: 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 Kstability. A more interesting setting occurs in the case of pairs of varieties and a hyperplane section where the Kmoduli compactifications tessellate depending on a parameter. In this case it has been shown recently that the Kmoduli decompose into a wallchamber decomposition depending on a parameter, but wallcrossing phenomena are still difficult to describe explicitly. In this talk I will describe an algorithmic way to describe nonstable/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 Kmoduli of the Fano threefold family 2.25 (in the MoriMukai classification), and an explicit description of Kmoduli 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 groupsAbstract: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.

Alastair Litterick (Friday 12:30)  Complete Reducibility and Subgroup Structure of Reductive GroupsAbstract: The concept of Gcomplete reducibility generalises the representationtheoretic 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 Gorbits in the direct product G^n acting by conjugacy. Thus Gcomplete 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. 