Symplectic Geometry

I will discuss joint work with Luya Wang on the other direction of the Donaldson 4-6 problem. Specifically, we show that any two simply-connected symplectic 4-manifolds, whose products with S^2 are deformation equivalent, have the same Gromov-Witten invariants. The proof relies on a…

Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products…

Arnold conjecture says that the number of 1-periodic orbits of a Hamiltonian diffeomorphism is greater than or equal to the dimension of the Hamiltonian Floer homology. In 1994, Hofer and Zehnder conjectured that there are infinitely many periodic orbits if the equality doesn't hold. In this…

Abstract: The moduli space of closed holomorphic curves in a closed symplectic manifold can be compactified using stable maps. However, even in the nicest of situations (e.g., degree d curves of genus g in a complex projective space, with d .. g), counting dimensions shows that most stable maps…

Abstract: This is joint work in progress with Dan Cristofaro-Gardiner. We explore the topological dynamics of Reeb flows beyond periodic orbits and find the following rather general phenomenon. For any Reeb flow for a torsion contact structure on a closed 3-manifold, any point is arbitrarily…

We define an equivariant Lagrangian Floer theory on compact symplectic toric manifolds for the subtorus actions. We prove that the set of Lagrangian torus fibers (with weak bounding cochain data) with non-vanishing equivariant Lagrangian Floer cohomology forms a rigid analytic space. We can…

Distinct Hamiltonian isotopy classes of Lagrangian tori in $\mathbb{CP}^2$ can be associated to Markov triples. With two exceptions, each of these tori are symplectomorphic to exactly three Hamiltonian isotopy classes of tori in the ball (the affine part of $\mathbb{CP}^2$). We investigate…

Abstract: In his seminal 2001 paper, Biran introduced the concept of Lagrangian Barriers, a symplectic rigidity phenomenon coming from obligatory intersections with Lagrangian submanifolds which don't come from mere topology.

In this joint work with Richard Hind and Yaron Ostrover, we…

Given a positive factorisation of the identity in the mapping class group of a surface S, we can associate to it a Lefschetz fibration over the sphere with S as a regular fiber. Its total space X is a symplectic 4-manifold, so it is a natural question to ask what kind of invariants of X can be…

We will discuss co-dimension 2 embeddings of manifolds in  smooth and symplectic category.