Location: IHP, room 201
The talks are broadcasted via Zoom:
https://zoom.us/j/89445897442?pwd=QmVZWHBiM2Nwb09yUVJERnYrSHJrUT09
10:45 Marcelo Alves (Bruxelles)
From curve shortening to Birkhoff sections of geodesic flows.
Abstract: In this talk, based on joint work with Marco Mazzucchelli, I will present some new results on the dynamics of geodesic flows of closed Riemannian surfaces, proved using the curve shortening flow. The first result is a forced existence theorem for orientable closed Riemannian surfaces of positive genus, asserting that the existence of a contractible simple closed geodesic \gamma forces the existence of infinitely many closed geodesics in every primitive free homotopy class of loops and intersecting \gamma. I will then explain how this type of result can be used to show the existence of Birkhoff sections for the geodesic flow of any closed orientable Riemannian surface.
14:00 Vincent Colin (Nantes)
Birkhoff sections and Reeb chords.
Abstract: I will explain how the existence of a Birkhoff section implies the
existence of infinitely many Reeb chords for every Legendrian curve in a
3-dimensional contact manifold. I will also show that for a generic
Reeb vector field, given any finite collection of periodic orbits of its
Reeb flow and finite collection of Legendrian curves, there exists an
embedded Birkhoff section for the Reeb flow that contains the periodic
orbits in its boundary and a C^0-small Legendrian deformation of the
Legendrians in its interior.
This is joint work with Umberto Hryniewicz and Ana Rechtman, together with a contribution of Pierre Dehornoy..
15:45 Vera Vertesi (Vienna)
Heegaard Splittings and the 3-Dimensional Giroux Correspondence.
Abstract: Open book decompositions—known in various contexts as global Poincaré–Birkhoff sections, relative mapping tori, Milnor fibrations, fibered links, and spinnable structures—have arisen independently across several areas of mathematics. Introduced by Thurston and Winkelnkemper, they became a central tool in 3-dimensional contact topology through the groundbreaking work of Giroux, who established a one-to-one correspondence between contact structures up to isotopy and open book decompositions up to positive stabilization.
The strength of this correspondence lies in its combinatorial nature: an open book is determined by a mapping class group element of a surface with boundary, which in turn can be expressed in terms of Dehn twists along simple closed curves. As a result, problems in contact topology can be translated into combinatorial questions about curves on surfaces. This perspective enables explicit computations and offers a powerful framework for proving structural results.
In this talk, I will sketch a proof of the Giroux correspondence using the interplay between open book decompositions and Heegaard splittings. This is joint work with Joan Licata.
Other symplectic activity in Paris (and in France):
- Séminaire Nantes-Orsay- Symplectic Zoominar (every Fridays at 15:15 (except Symplectix' Fridays) Paris time)
