In the Geometric Invariant Theory construction of the moduli space of semistable sheaves, one has a priori two measures of the failure of a sheaf to be semistable. The first, Harder-Narasimhan type, is sheaf-theoretic, while the second, the Hesselink-Kempf-Kirwan-Ness stratification, comes purely from GIT. Naturally enough, as proved by Hoskins, these two notions coincide in an appropriate 'asymptotic' sense, a fact that has recently been applied to construct moduli of unstable sheaves using new techniques from Non-reductive GIT.

This motivates the analogous questions for another major GIT success story: the moduli of curves. Namely, does there exist some asymptotic HKKN stratification in the GIT construction of the moduli space of stable curves? If so, what meaning does it have, and can it be used to construct new moduli spaces classifying singular curves?

After explaining the necessary background, I will report on joint work with Dave Swinarski answering the first of these questions in the affirmative for a special case, using techniques from convex optimisation.

Let X be a fixed abelian variety. In this talk I will outline a framework to classify semi-homogeneous vector bundles on X using a suitably modified Fourier-Mukai transform that generalizes the well-known homogeneous situation. This will allow us to explicitly construct moduli spaces of semi-homogeneous vector bundles on $X$. Using the perspective of non-Archimedean uniformization, we can use our insights to understand the tropicalization of semi-homogeneous vector bundles.

This talk is based on joint work with Andreas Gross, Inder Kaur, and Annette Werner.

We introduce a new notion of stability on holomorphic vector bundles that relates the slope stability of a vector bundle with the K-stability of its projectivisation. We use this notion of stability to construct Kähler metrics with constant scalar curvature on the projectivisation, in adiabatic classes.

This is joint work with Lars Martin Sektnan.

Flag varieties can be recognized via conic structures in all characteristics; in characteristic zero this was done by Occhetta et al. On the way we are led to a characterization of Coxeter systems. This is joint work with Ian Grojnowski.

There are three underlying classes of algebraic varieties: General Type varieties, Calabi-Yau varieties, and Fano varieties. One of the key goals in complex geometry is to study when the above classes of varieties admit canonical metrics. An important example of such metrics the Kähler-Einstein metrics. General type and Calabi-Yau varieties always admit a unique Kähler-Einstein metric. Fano varieties, however, are more complicated since it is known that some Fano varieties do not admit a Kähler-Einstein metric. It was first conjectured by Yau-Tian-Donaldson and then proven in the works of Chen-Donaldson-Sun-Li-Wang-Xu-Spotti-Yao for Fano varieties with smoothable singularities that the existence of such metrics would be equivalent to algebraic conditions called K-stability conditions. For smooth del Pezzo surfaces Tian and Yau proved that a smooth del Pezzo surface is admit a Kähler-Einstein metric if and only if it is not a blow-up of a projective plane in one or two points. A lot of research was done for Fano threefolds however, not everything is known and often the problem can be reduced to computing stability thresholds (delta-invariants) of possibly singular del Pezzo surfaces.

In my talk, I will describe the status of the problem, present an example of computation of the delta-invariant, show the example of the application of this result for a singular Fano threefold, and explain a possible direction for future research.

Tannaka Duality refers to the reconstruction of a compact group from its representations, while Serre's GAGA theorem relates coherent sheaves on an algebraic variety to its analytification.  This talk will explore various incarnations of these two classical theorems in algebraic geometry with applications to moduli theory.