Tag - Symplectic geometry
I will explain recent joint work proving that the group of compactly supported area preserving homeomorphisms of the two-disc is not a simple group; this answers the 'Simplicity Conjecture' in the affirmative. Our proof uses new spectral invariants, defined via periodic Floer homology, that I will introduce: these recover the Calabi invariant of monotone twists.
The purpose of the talk is to discuss a class of pseudo-metrics that can be defined on the set of objects of a triangulated category whose morphisms are endowed with a notion of weight. In case the objects are Lagrangian submanifolds (possibly immersed) there are a some natural ways to define such pseudo-metrics and, if the class of Lagrangian submanifolds is unobstructed, these pseudo-metrics are non-degenerate and extend in a natural way the Hofer distance.
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 quantum-mechanical processes.
For given a Lagrangian in a symplectic manifold, one can consider deformation of A∞-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 giving a local chart of the mirror. In this talk, I will explain how to glue local charts from different Lagrangians using isomorphisms between Lagrangians in the Fukaya category.
As an application, we will discuss the mirror construction for Gr(2,4) that recovers its Lie-theoretical mirror.
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 constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace.
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) picture, and how it constrains the field of definition of the Fukaya category.
(Joint work with Chris Woodward) Consider a Lagrangian submanifold L̅ in a GIT quotient X̅=X//G. Besides the usual Fukaya A∞-algebra Fuk(L̅) defined by counting holomorphic disks, another version, called the quasimap Fukaya algebra FukK(L), is defined by counting holomorphic disks in X modulo group action. Motivated from the closed string quantum Kirwan map studied by Ziltener and Woodward, as well as the work of Fukaya-Oh-Ohta-Ono, Chan-Lau-Leung-Tseng, we construct an open string version of the quantum Kirwan map. This is an A∞-morphism from FukK(L) to a bulk deformation of Fuk(L̅). The deformation term is defined by counting affine vortices (point-like instantons) in the gauged sigma model, while the A∞-morphism is defined by counting point-like instantons with Lagrangian boundary condition.
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 a higher-dimensional theory (joint work with Alvarez-Gavela, Eliashberg, and Starkston), and in particular, what kind of topological spaces might generalize graphs. We will also discuss applications to the calculation of symplectic invariants.
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 knot (joint work with Tobias Ekholm and Vivek Shende) and recent progress in the circle of ideas connecting knot contact homology, recurrence relations for coloured HOMFLY polynomials, and topological strings (joint work in progress with Tobias Ekholm).
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