# Past Algebraic Geometry Seminar

The (total) Steenrod square is a ring homomorphism from the cohomology of a topological space to the Z/2-equivariant cohomology of this space, with the trivial Z/2-action. Given a closed monotone symplectic manifold, one can define a deformed notion of the Steenrod square for quantum cohomology, which will not in general be a ring homomorphism, and prove some properties of this operation that are analogous to properties of the classical Steenrod square. We will then link this, in a more general setting, to a definition by Seidel of a similar operation on Floer cohomology.

The Hitchin fibration is a natural tool through which one can understand the moduli space of Higgs bundles and its interesting subspaces (branes). After reviewing the type of questions and methods considered in the area, we shall dedicate this talk to the study of certain branes which lie completely inside the singular fibres of the Hitchin fibrations. Through Cayley and Langlands type correspondences, we shall provide a geometric description of these objects, and consider the implications of our methods in the context of representation theory, Langlands duality, and within a more generic study of symmetries on moduli spaces.

Combining work of Galkin, Christopherson-Ilten, and Coates-Corti-Galkin-Golyshev-Kasprzyk we see that all smooth Fano threefolds admit a toric degeneration. We can use this fact to uniformly construct all Fano threefolds: given a choice of a fan we classify reflexive polytopes which break into unimodular pieces along this fan. We can then construct closed torus invariant embeddings of the corresponding toric variety using a technique - Laurent inversion - developed with Coates and Kaspzryk. The corresponding binomial ideal is controlled by the chosen fan, and in low enough codimension we can explicitly test deformations of this toric ideal. We relate the constructions we obtain to known constructions. We study the simplest case of the above construction, closely related to work of Abouzaid-Auroux-Katzarkov, in arbitrary dimension and use it to produce a tropical interpretation of the mirror superpotential via broken lines. We expect the computation to be the tropical analogue of a Floer theory calculation.

I will talk about some basic facts about slope stable sheaves and the Bogomolov inequality. New techniques from stability conditions will imply new stronger bounds on Chern characters of stable sheaves on some special varieties, including Fano varieties, quintic threefolds and etc. I will discuss the progress in this direction and some related open problems.

Geometric Invariant Theory is a central tool in the construction of moduli spaces, and shares the property ubiquitous among such tools that certain so-called 'unstable' objects must be excluded if the moduli space is to be well behaved. However, instability in GIT is a structured phenomenon: after making a choice of a certain invariant inner product, one has the HKKN stratification of the parameter space which, morally, sorts the objects according to how unstable they are. I will explain how one can use recent results of Berczi-Doran-Hawes-Kirwan in Non-Reductive GIT to perform quotients of these unstable strata as well, extending the classifications given by classical moduli spaces. This can be carried out, at least in principle, for any moduli problem that can be posed using GIT, and I will discuss two examples in particular: unstable (i.e. singular) curves, and coherent sheaves of fixed Harder-Narasimhan type. The latter of these is joint work with Gergely Berczi, Victoria Hoskins and Frances Kirwan.

I will explain how the definition of Bridgeland stability condition on a triangulated category C can be generalised to allow for massless objects. This allows one to construct a partial compactification of the stability space Stab(C) in which each `boundary stratum' is related to Stab(C/N) for a thick subcategory N of C, and has a neighbourhood which fibres over (an open subset of) Stab(N). This is joint work with Nathan Broomhead, David Pauksztello, and David Ploog.

Enumerative invariants since 1995 are defined as integrals of cohomology classes over a particular homology class, called the virtual fundamental class. When there is a torus action, the virtual fundamental class is localized to the fixed points and this turned out to be the most effective technique for computation of the virtual integrals so far. About 10 years ago, Jun Li and I discovered that when there is a cosection of the obstruction sheaf, the virtual fundamental class is localized to the zero locus of the cosection. This also turned out to be quite useful for computation of Gromov-Witten invariants and more. In this talk, I will discuss a generalization of the cosection localization to real classes which provides us with a purely topological theory of Fan-Jarvis-Ruan-Witten invariants (quantum singularity theory) as well as some GLSM invariants. Based on a joint work with Jun Li at arXiv:1806.00116.