Symplectic Geometry

Just as Heegaard-Floer (HF) theory is about Fukaya categories for symmetric product of Riemann surfaces, HF theory with coefficient is about  Fukaya categories of horizontal Hilbert scheme (possibly with fiberwise superpotential). This is used to give a symplectic realization of…

To a symplectic algebraic variety or stack, we should be able to associate a 3d A-model and B-model, which are equivalent to the B-model and A-model associated to some dual space. Boundary conditions for these are expected to form 2-categories whose objects are holomorphic Lagrangian…

Transversality does not play well with symmetry as symmetric objects are typically not in "general position". As a result, one may not be able to achieve transversality on orbifolds. This feature is often accompanied with another feature of orbifolds: Poincar\'e duality only holds over rational…

In this talk, based on a joint work with Erman Cineli and Basak Gurel, we discuss the multiplicity problem for prime closed orbits of dynamically convex Reeb flows on the boundary of a star-shaped domain. The first of our two main results asserts that such a flow has at least n prime closed Reeb…

The ellipsoid embedding function generalizes symplectic ball packing problems. For a symplectic manifold, this function determines the minimum scaling factor required for a standard ellipsoid with a given eccentricity to embed symplectically into the manifold. If the function has infinitely many…

One of the earliest achievements of mirror symmetry was the prediction of genus zero Gromov-Witten invariants for the quintic threefold in terms of period integrals on the mirror. Analogous predictions for open Gromov-Witten invariants in closed Calabi-Yau threefolds can be …

Let W be a complete finite type Liouville manifold. One can associate to each closed subset K of W that is conical at infinity an invariant SH_W(K). I will first explain the construction of SH_W(K) and note how it recovers known invariants through special choices of K. Then, I will prove a big…

I will talk about a series of joint papers with Alexander Ritter, where we examine a large class of non-compact symplectic manifolds, including semiprojective toric varieties, conical symplectic resolutions, Higgs moduli spaces, etc.These manifolds admit a Hamiltonian circle action which is…

In this talk, I will survey several notions of dimension for (pre-)triangulated categories naturally arising from topology and symplectic geometry. I will discuss joint work with Andrew Hanlon and Jeff Hicks in which we prove new bounds on these dimensions and raise several questions for…

A  recent article by Ben-Zvi, Sakellaridis, and Venkatesh proposes a framework for duality between automorphic periods and L-functions via a duality between Hamiltonian G- and G^-spaces, where G is a reductive group and G^ is its Langlands dual. In this talk, I will review some of this…