Past Algebraic and Symplectic Geometry Seminar

Given a Lagrangian submanifold invariant under a Hamiltonian loop, we partially compute the image of the loop's Seidel element under the closed-open string map into the Hochschild cohomology of the Lagrangian. This piece captures the homology class of the loop's orbits on the Lagrangian and can help to prove that the closed-open map is injective in some examples. As a corollary we prove that $\mathbb{RP}^n$ split-generates the Fukaya category of $\mathbb{CP}^n$ over a field of characteristic 2, and the same for real loci of some other toric  varieties.

I will speak about weak Fano 3-folds, K3 surfaces and their Picard lattices, and explain how to solve the matching problem in various situations.

In recent years, some powerful tools for computing semi-orthogonal decompositions of derived categories of algebraic varieties have been developed: Kuznetsov's theory of homological projective duality and the closely related technique of VGIT for LG models. In this talk I will explain how the latter works and how it can be used to understand the derived categories of complete intersections in Sym^2(P^n). As a consequence, we obtain a new proof of result of Hosono and Takagi, which says that a certain pair of non-birational Calabi-Yau 3-folds are derived equivalent.

The goal of this talk is to discuss a link between the Homogeneous Monge Ampere Equation in complex geometry, and a certain flow in the plane motivated by some fluid mechanics.   After discussing and motivating the Dirichlet problem for this equation I will focus to what is probably the first non-trivial case that one can consider, and prove that it is possible to understand regularity of the solution in terms of what is known as the Hele-Shaw flow in the plane. As such we get, essentially explicit, examples of boundary data for which there is no regular solution, contrary to previous expectation.  All of this is joint work with David Witt Nystrom.

Donaldson-Thomas invariants are fundamental deformation invariants of Calabi-Yau threefolds. We describe a recent conjecture of Oberdieck and Pandharipande which predicts that the (three variable) generating function for the Donaldson-Thomas invariants of K3xE is given by the reciprocal of the Igusa cusp form of weight 10. For each fixed K3 surface of genus g, the conjecture predicts that the corresponding (two variable) generating function is given by a particular meromorphic Jacobi form. We prove the conjecture for K3 surfaces of genus 0 and genus 1. Our computation uses a new technique which mixes motivic and toric methods.

Rationally and polynomially convex domains in ${\mathbb C}^n$ are fundamental objects of study in the theory of functions of several complex variables. After defining and illustrating these notions, I will explain joint work with Y.Eliashberg giving a complete characterization of the possible topologies of such domains in complex dimension at least three. The proofs are based on recent progress in symplectic topology, most notably the h-principles for loose Legendrian knots and Lagrangian caps.