# Seminars & Events for Joint PU/IAS Symplectic Geometry Seminar

##### On Singularities With Rational Homology Disk Smoothings

##### Qualitative Properties of Gromov-Witten Invariants

Over 15 years ago, di Francesco and Itzykson gave an estimate on the growth (as the degree increases) of the number of plane rational curves passing through the appropriate number of points. This provides an example of an upper bound on (primary) Gromov-Witten invariants. Physical considerations suggest that primary GW-invariants of Calabi-Yau threefolds, of any given genus, grow at most exponentially in the degree. For the genus 0 and 1 GW-invariants of projective complete intersections, this can be seen immediately from the known mirror formulas. Maulik and Padharipande expect that such a bound in higher genera can be deduced from a suitable bound on the genus 0 descendant GW-invariants of $P^3$.

##### Hodge Structures in Symplectic Geometry

I will explain how essential information about the structure of symplectic manifolds is captured by algebraic data, and specifically by the non-commutative (mixed) Hodge structure on the cohomology of the Fukaya category. I will discuss how mirror symmetry leads to Hodge theoretic symplectic invariants arising from twist functors, and from geometric extensions. I will also explain how the instanton-corrected Chern-Simons theory fits in the framework of normal functions in non-commutative Hodge theory and will give applications to explicit descriptions of quantum Lagrangian branes. This is a joint work with L. Katzarkov and M. Kontsevich.

##### Stochastic Twist Maps and Symplectic Diffusions

I discuss two examples of random symplectic maps in this talk. As the first example consider a stochastic twist map that is defined to be a stationary ergodic twist map on a planar strip. As a natural question, I discuss the fixed point of such maps and address a Poincare-Birkhoff type theorem. As the second example I consider stochastic flows associated with diffusions and discuss those diffusions which produce symplectic maps only in average sense. Using stochastic diffusions, it is possible to derive Iyer-Constantin Circulation Theorem for Navier-Stokes Equation.

##### Orientability and open Gromov-Witten invariants

I will first discuss the orientability of the moduli spaces of J-holomorphic maps with Lagrangian boundary conditions. It is known that these spaces are not always orientable and I will explain what the obstruction depends on. Then, in the presence of an anti-symplectic involution on the target, I will give a construction of open Gromov-Witten disk invariants. This is a generalization to higher dimensions of the works of Cho and Solomon, and is related to the invariants defined by Welschinger.

##### Morse Theory and Invariants of (almost) Symplectic Manifolds

I will discuss two of my current projects which have different aims but use similar techniques. The first aims to understand the equivariant K-theory of symplectic orbifolds. The second is about the topology of toric origami manifolds. Neither space is a symplectic manifold, but each is almost.

##### Ramussen's s-Invariant via Instantons and Other Remarks on Khovanov Homology as Seen by Instanton Floer Homology

##### Bihermitian metrics and Poisson deformations

A bihermitian metric (or generalized Kahler structure) involves a pair of complex structures and a metric compatible with both. The study of these is closely related to that of holomorphic Poisson structures on a manifold. A deformation theorem for such complex Poisson manifolds leads to a concrete description of certain pairs of complex structures and we show how these can generate bihermitian metrics.

##### Bubbles and Onis

This is joint work with Peter Albers. We study a Floer gradient equation on a very negative line bundle over a symplectic manifold. If the line bundle is negative enough there are generically no holomorphic spheres. We explain the metamorphosis of bubbles to Onis and then point out a weakness of the Onis - namely a free involution on them...

##### A Mathematical Theory of Quantum Sheaf Cohomology

TBA