Mini-Workshop on Symplectic Geometry, Utrecht University, June 24, 2022

Titles and abstracts for the talks

Umberto Hryniewicz, colloquium talk: Symplectic methods applied to systolic geometry

Systolic geometry was born in the late 1940’s with Loewner’s inequality on the two-torus, asserting that there is an explicit sharp upper bound for the length of the shortest non-contractible closed geodesic in terms of the area. It developed into a rich and active field, at the crossroads of many different mathematical disciplines. The focus of this talk is to explain how methods from symplectic dynamics were recently used to study systolic inequalities on Riemannian surfaces, and more generally for contact forms on closed 3-manifolds. As an example, we will focus on results obtained in collaboration with Alberto Abbondandolo (RUB-Bochum), Barney Bramham (RUB-Bochum) and Pedro Salomão (USP-São Paulo and NYU-Xanghai), which prove the local systolic maximality of the round 2-sphere, as well as its global systolic maximality among spheres of revolution.

Umberto Hryniewicz: Existence of global surfaces of section, and applications

A global surface of section (GSS) reduces the study of a non-singular flow in 3D to that of a surface diffeomorphism. In this talk I will present some existence statements within the class of Reeb flows. Firstly, I will focus on the question of finding a GSS spanned by a given collection of periodic orbits, with no genericity assumptions, and state results obtained in collaboration with Salomão and Wysocki. These statements have applications in Celestial Mechanics. Secondly, I will present recent existence statements for rational GSS’s (Birkhoff sections) under genericity assumptions, and explain how to use them to prove that generically a Reeb flow on a closed 3-manifold has positive topological entropy. This is fruit of joint work with Colin, Dehornoy and Rechtman, and is based on broken book decompositions.

Sobhan Seyfaddini: Quasimorphisms and area-preserving homeomorphisms of the sphere

I will discuss a recent work constructing quasimorphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. The existence of these quasimorphisms answers a question of Entov, Polterovich, and Py. As an immediate corollary, we learn that the commutator length is unbounded, sharply contrasting a result of Tsuboi regarding the group of homeomorphisms that do not preserve area. A key role is played by a new family of spectral invariants, called "link spectral invariants", associated to certain Lagrangian links in the sphere.

This is joint work with Cristofaro-Gardiner, Humilière, Mak and Smith.

Sara Tukachinsky: Bounding chains and deformations

Bounding (co)chains were developed as a tool in Lagrangian Floer theory, and their use has since expanded to homological mirror symmetry and open Gromov-Witten theory. In their book, Fukaya, Oh, Ohta, and Ono show that certain cohomological assumptions ensure that a Lagrangian submanifold admits a bounding chain after deformation. I will talk about their result and some of their applications for Lagrangian intersections. Then I will introduce a "globalized" version of this theorem and discuss its application in open Gromov-Witten theory.

Federica Pasquotto: Differential topology for orbifolds

In this talk I hope to discuss recent work with T. Rot which aims at extending basic notions and results in differential topology (degree of a map between manifolds, transitivity of the diffeomorphism groups) to the orbifold setting.

I intend to devote part of the talk to describing how my interest for orbifolds originates from symplectic geometry, for instance in the study of questions about symplectic fillability.