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.

Part 1: We define genus zero Gromov-Witten type invariants that count pseudo-holomorphic spheres that approximate a fixed local codimension-2 submanifold up to a finite order. We explain that the count does not depend on the choice of the local submanifold and generic almost complex structure. We give examples of symplectic manifolds where these invariants don't vanish in certain cases. (Reference: Cieliebak–Mohnke)
Part 2: Using neck-stretching and obstruction bundle gluing, we explain that in dimension four the invariants defined in part 1 are equal to genus zero Gromov-Witten type invariants that count punctured pseudo-holomorphic spheres with a negative end on the surface of a small ellipsoid. (Reference: § 3.1, 4.1 in McDuff–Siegel)

I’ll describe a new version of Legendrian rational SFT which generalizes the Chekanov-Eliashberg algebra. It reformulates Ekholm’s RSFT using ideas from Chas-Sullivan's ``chord diagram'' formalism for string topology and Hutchings’ ``q-variables only’’ version of closed-orbit RSFT. The result is a non-commutative DGA with a special filtration. If time permits, I’ll demo software which computes associated augmentations and bilinearized homologies for Legendrian links in R3.

In this talk, we will discuss a very celebrated theorem of Eliashberg that Stein fillings of connected sum of contact 3-manifolds can be obtained by taking the boundary connected sum of individual fillings. Stein fillings and stein cobordisms arise when 0, 1 or 2-handles are attached to a 4-Ball. This, along with the fact that any contact 3-manifold can be obtained via dehn (-1)-surgery along a legendrian Link make Stein manifolds the most natural fillings that interests mathematicians.

Contact homology is a Floer-homology-type invariant for contact manifolds and is a part of Symplectic Field Theory. One of its first applications was to show the existence of exotic contact structures on spheres.
Originally, contact homology was defined only for closed contact manifolds. We will describe how to extend it to open contact manifolds that are "convex" in a suitable sense. This mainly involves proving a suitable compactness statement for the relevant moduli spaces of holomorphic curves, as well as surmounting a few other interesting technical hurdles (and a gigantic mountain of much more serious technical hurdles, which have fortunately already been solved elsewhere).
As an application, we prove the existence of (infinitely many) exotic contact structures on R^2n-1 for all n>2.
This is joint with François-Simon Fauteux-Chapleau.

We’ll give an overview of Perelman’s proof of the Poincaré conjecture. We’ll discuss the history of the problem, Hamilton’s roadmap to prove the conjecture using the Ricci flow, some of the main technical difficulties in his program and some of the techniques introduced by Perelman to overcome those.

In this talk I will study fillings of contact structures compatible with planar open books by analyzing positive factorizations of their monodromy. I will explain this method, which is based on Chris Wendl's theorem on symplectic fillings of planar open books, and show how Plamenevskaya and Van Horn-Morris used it to prove that certain contact structures on the lens spaces L(p,1) have a unique filling.

M. Abouzaid and Y. Imagi recently proved that any closed, immersed, unobstructed( in Floer theory) special Lagrangian (SL) which is very close to an embedded SL whose fundamental group has no nonabelian free subgroup is unbranched. In this talk we go thorough the proof of this theorem. The main idea is to use Thomas-Yau uniqueness theorem with inputs coming from Fukaya category of the cotangent bundle of the embedded SL. After a very brief introduction to Fukaya category in general, we talk about the necessary facts from the Fukaya category of the cotangent bundle that are required to apply Thomas-Yau uniqueness theorem.