Lagrangian Floer cohomology groups are extremely hard compute in most situations. In this talk I’ll describe two ways to extract information about the self-Floer cohomology of a monotone Lagrangian possessing certain kinds of symmetry, based on the closed-open string map and the Oh spectral sequence. The focus will be on a particular family of examples, where the techniques can be combined to deduce some unusual properties.

# Past Algebraic and Symplectic Geometry Seminar

The goal of this talk will be to give an overview of recent work, joint with Kim Moore, on a short time existence problem in Lagrangian mean curvature flow. More specifically, we consider a compact initial Lagrangian submanifold with a finite number of singularities, each asymptotic to a pair of transversely intersecting planes. We show it is possible to construct a smooth Lagrangian mean curvature flow, existing for positive times, that attains the singular Lagrangian as its initial condition in a suitable weak sense. The construction uses a family of smooth solutions whose initial conditions approximate the singular Lagrangian. In order to appeal to compactness theorems and produce the desired solution, it is necessary to first establish uniform curvature estimates on the approximating family. As time allows I hope to focus in particular on the proof of these estimates, and their role in the proof of the main theorem.

Coisotropic submanifolds arise naturally in symplectic geometry as level sets of moment maps and in algebraic geometry in the context of normal crossing divisors. In examples, the Marsden-Weinstein quotient or (Fano) complete intersections are often uniruled.

We show that under natural geometric assumptions on a coisotropic submanifold, the symplectic quotient of the coisotropic is always geometrically uniruled.

I will explain how to assign a Lagrangian and a hypersurface to a fibered, stable coisotropic C. The Lagrangian inherits a fibre bundle structure from C, the hypersurface captures the generalised Reeb dynamics on C. To derive the result, we then adapt and apply techniques from Lagrangian Floer theory and symplectic field theory.

This is joint work with Jonny Evans.

Given a Calabi-Yau threefold X, one can count curves on X using various approaches, for example using stable maps or ideal sheaves; for any curve class on X, this produces an infinite sequence of invariants, indexed by extra discrete data (e.g. by the domain genus of a stable map). Conjecturally, however, this sequence is determined by only a finite number of integer invariants, known as Gopakumar-Vafa invariants. In this talk, I will propose a direct definition of these invariants via sheaves of vanishing cycles, building on earlier approaches of Kiem-Li and Hosono-Saito-Takahashi. Conjecturally, these should agree with the invariants as defined by stable maps. I will also explain how to prove the conjectural correspondence for irreducible curves on local surfaces. This is joint work with Yukinobu Toda.

With the purpose of examining some relevant geometric properties of the moduli space of sheaves over an algebraic surface, Le Potier conjectured some unexpected duality between the complete linear series of certain natural divisors, called Theta divisors, on the moduli space. Such conjecture is widely known as Strange Duality conjecture. After having motivated the problem by looking at certain instances of quantization in physics, we will work in the setting of surfaces. We will then sketch the proof in the case of abelian surfaces, giving an idea of the techniques that are used. In particular, we will show how the theory of discrete Heisenberg groups and fiber wise Fourier-Mukai transforms, which might be applied to other cases of interest, enter the picture. This is joint work with Alina Marian, Dragos Opera and Kota Yoshioka.

Every finite regular CW complex is, ipso facto, a cohomologically stratified space when filtered by skeleta. We outline a method to recover the canonical (i.e., coarsest possible) stratification of such a complex that is compatible with its underlying cell structure. Our construction proceeds by first localizing and then resolving a complex of cosheaves which capture local cohomology at every cell. The result is a sequence of categories whose limit recovers the desired strata via its (isomorphism classes of) objects. As a bonus, we observe that the entire process is algorithmic and amenable to efficient computations!

Given a Mori Dream Space X, we construct via tropicalisation a model dominating all the small Q-factorial modifications of X. Via this construction we recover a Minkowski basis for the Newton-Okounkov bodies of Cartier divisors on X and hence generators of the movable cone of X.

This is joint work with Stefano Urbinati.

We consider two types of actions on moduli spaces of quiver representations over a field k and we decompose their fixed loci using group cohomology. First, for a perfect field k, we study the action of the absolute Galois group of k on the points of this quiver moduli space valued in an algebraic closure of k; the fixed locus is the set of k-rational points and we obtain a decomposition of this fixed locus indexed by the Brauer group of k. Second, we study algebraic actions of finite groups of quiver automorphisms on these moduli spaces; the fixed locus is decomposed using group cohomology and each component has a modular interpretation. If time permits, we will describe the symplectic and holomorphic geometry of these fixed loci in hyperkaehler quiver varieties. This is joint work with Florent Schaffhauser.