# Mini-Workshop on Symplectic Geometry, Utrecht University, March 16, 2018

## Titles and abstracts for the talks

### Agnès Gadbled:

Categorical action of the braid group of the cylinder: symplectic aspect

Khovanov and Seidel gave in 2000 an action of the classical braid group on a category of algebraic nature that categorifies the Burau representation. They proved the faithfulness of this action through the study of curves in a punctured disk (while Burau representation is not faithful for braids with five strands or more). In a recent article with Anne-Laure Thiel and Emmanuel Wagner, we extended this result to the braid group of the cylinder.

The work of Khovanov and Seidel also had a symplectic aspect that we now generalize. In this talk, I will explain the strategy and tools to get a symplectic monodromy in our case and prove its faithfulness. If time permits, I will explain how this action lifts to a symplectic categorical representation on a Fukaya category that should be related to the algebraic categorical representation.

This is a joint work in progress with Anne-Laure Thiel and Emmanuel Wagner.

### Dietmar Salamon:

Complex structures, moment maps, and the Ricci form.

This talk is based on joint work with Oscar Garcia-Prada and Samuel Trautwein. We prove that the Ricci form is a moment map for the action of the group of exact volume preserving diffeomorphisms on the space of almost complex structures. As a consequence the Weil--Petersson symplectic form on the Teichmueller space of isotopy classes of complex structures with real first Chern class zero and nonempty Kähler cone can be obtained via infinite-dimensional symplectic reduction.

### Darko Milinković:

Lagragian spectral invariants for open subsets

To a given open subset V in a smooth manifold M, we can associate its conormal set in T^*M. R. Kasturirangan and Y-G. Oh constructed a Lagrangian Floer homology for this set. I will talk about the spectral invariants constructed by this Floer homology and about some of their properties. This is a joint work with Jelena Katic and Jovana Nikolic.

### Michael Entov:

First steps of the symplectic function theory (colloquium talk)

Symplectic function theory studies the space of smooth (compactly supported) functions on a symplectic manifold. This space is equipped with two structures: the Poisson bracket and the C^0 (or uniform) norm of the functions. Interestingly enough, while the Poisson bracket of two functions depends on their first derivatives, there are non-trivial restrictions on its behavior with respect to the C^0-norm. We will discuss various results of this kind and their applications to Hamiltonian dynamics.

### Michael Entov:

Lagrangian isotopies and symplectic function theory

I'll discuss two new related invariants of Lagrangian submanifolds. The invariants are functions on the first cohomology of the submanifold. The first invariant is of topological nature and is related to the study of Lagrangian isotopies with a given Lagrangian flux. It is also related to a well-known symplectic invariant called ``symplectic shape". The second invariant is of analytical nature, comes from symplectic function theory and has a dynamical interpretation. This is a joint work with Yaniv Ganor and Cedric Membrez.