Research Seminar: Symplectic Geometry (SS 2022)

Eliashberg and Thurston discovered in '98 a surprising relationship between foliations and contact structures in dimension 3: (most) C^2 foliations can be C^0-approximated by contact structures; there are moreover interesting connections between their properties. In higher dimensions, the best candidates for such type of results seem to be (codimension 1) foliations equipped with a conformal symplectic leafwise structure, according to recent results by Bertelson and Meigniez on manifolds with non-empty boundary. In this talk, I will give an introduction to this and related problematics, and present a work joint with Lauran Toussaint where we prove the existence of conformal symplectic foliations on closed manifolds of odd dimension at least 7 in any given almost contact class using the h-principle of Bertelson--Meigniez. Time permitting, I will also discuss the existence of conformal symplectic and symplectic foliations on simply connected 5-dimensional manifolds.

In the first talk of this two-part series, I will give a rough outline of the algebraic structure of (full and rational) symplectic field theory, and recount some of the history of its use for constructing invariants that obstruct symplectic fillings and cobordisms between contact manifolds.

In the second talk, I will sketch the recent contributions to this subject by Moreno and Zhou, who have introduced the first (as far as I'm aware) geometric application based specifically on rational SFT.