There is growing interest in looking at operations on quantum
cohomology that take into account symmetries in the holomorphic
spheres (such as the quantum Steenrod powers, using a
Z/p-symmetry). In order to prove relations between them, one needs
to...
We discuss interactions between quantum mechanics and symplectic
topology including a link between symplectic displacement energy, a
fundamental notion of symplectic dynamics, and the quantum speed
limit, a universal constraint on the speed of...
We show that any two birational projective Calabi-Yau manifolds
have isomorphic small quantum cohomology algebras after a certain
change of Novikov rings. The key tool used is a version of an
algebra called symplectic cohomology, which is...
For given a Lagrangian in a symplectic manifold, one can
consider deformation of A-infinity algebra structures on its Floer
complex by degree 1 elements satisfying the Maurer-Cartan equation.
The space of such degree 1 elements can be thought of as...
Consider a Calabi-Yau manifold which arises as a member of a
Lefschetz pencil of anticanonical hypersurfaces in a Fano variety.
The Fukaya categories of such manifolds have particularly nice
properties. I will review this (partly still conjectural)...
(Joint work with Chris Woodward) Consider a Lagrangian
submanifold $\bar L$ in a GIT quotient $\bar X = X//G$. Besides the
usual Fukaya $A_\infty$ algebra $Fuk(\bar L)$ defined by counting
holomorphic disks, another version, called the quasimap...
Ribbon graphs capture the topology of open Riemann surfaces in
an elementary combinatorial form. One can hope this is the first
step toward a general theory for open symplectic manifolds such as
Stein manifolds. We will discuss progress toward such...
Knot contact homology is a knot invariant derived from counting
holomorphic curves with boundary on the Legendrian conormal to a
knot. I will discuss some new developments around the subject,
including an enhancement that completely determines the...
In joint work with Buryak, Pandharipande and Tessler (in
preparation), we define equivariant stationary descendent integrals
on the moduli of stable maps from surfaces with boundary to
$(\mathbb{CP}^1,\mathbb{RP}^1)$. For stable maps of the disk...
We will explain a definition of open Gromov-Witten invariants on
the rational elliptic surfaces and explain the connection of the
invariants with tropical geometry. For certain rational elliptic
surfaces coming from meromorphic Hitchin system, we...
