Algebraic and Symplectic Geometry Seminar

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.

 In a previous work, we introduced a refinement of Juhasz’s sutured Floer homology, and constructed a minus theory for sutured manifolds, called sutured Floer chain complex. In this talk, we introduce a new description of sutured manifolds as “tangles” and describe a notion of cobordism between them. Using this construction, we define a cobordism map between the corresponding sutured Floer chain complexes. We also discuss some possible applications. This is a joint work with Eaman Eftekhary.

Hamiltonian reduction arose as a mechanism for reducing complexity of systems in mechanics, but it also provides a tool for constructing complicated but interesting objects from simpler ones. I will illustrate how this works in representation theory and algebraic geometry via examples. I will describe a new structure theory, motivated by Hamiltonian reduction (and in particular the Morse theory that results), for some categories (of D-modules) of interest to representation theorists. I will then explain how this implies a modified form of "hyperkahler Kirwan surjectivity" for the cohomology of certain Hamiltonian reductions. The talk will not assume that members of the audience know the meaning of any of the above-mentioned terms. The talk is based on joint work with K. McGerty.

The goal of this second talk is to study the existence of global jet differentials. Thanks to the algebraic Morse inequalities, the problem reduces to the computation of a certain Chern number on the Demailly tower of projectivized jet bundles. We will describe the significant simplification due to Berczi consisting in integrating along the fibers of this tower by mean of an iterated residue formula. Beside the original argument coming from equivariant geometry, we will explain our alternative proof of such a formula and we will particularly be interested in the interplay between the two approaches.

Hyperbolicity is the study of the geometry of holomorphic entire curves $f:\mathbb{C}\to X$, with values in a given complex manifold $X$. In this introductary first talk, we will give some definitions and provide historical examples motivating the study of the hyperbolicity of complements $\mathbb{P}^{n}\setminus X_{d}$ of projective hypersurfaces $X_{d}$ having sufficiently high degree $d\gg n$.

Then, we will introduce the formalism of jets, that can be viewed as a coordinate free description of the differential equations that entire curves may satisfy, and explain a successful general strategy due to Bloch, Demailly, Siu, that relies in an essential way on the relation between entire curves and jet differentials vanishing on an ample divisor.