This is a working group seminar run by Klaus Mohnke, Chris Wendl, and Thomas Walpuski on recent developments in symplectic geometry. It generally runs every semester on Mondays. Participants are expected to be familiar with the basics of symplectic geometry, including some knowledge of holomorphic curves and/or Floer-type theories. The seminar is conducted in English.

The seminar takes place this semester on Mondays, 13:00-15:00 (c.t.) via Zoom starting 9 November 2020. The Zoom link and password can be found on this seminar's Moodle page (key: gromov).

Note for new students: If you think you might be interested in this seminar but have neither attended before nor spoken with Profs. Mohnke, Wendl, or Walpuski about it, it is a good idea to get in touch with one of us ahead of time!


09.11.2020
Chris Wendl (HU Berlin)
Spinal open books and symplectic fillings with exotic fibers

We consider in this talk the following natural question in contact topology: how many different ways can a given contact manifold arise as the convex boundary of a compact symplectic manifold (i.e. a symplectic filling)? One way to attack this question is by considering certain geometric decompositions that reduce the dimension of the problem, e.g. a Lefschetz fibration on a symplectic 4-manifold presents it as a 2-parameter family of symplectically embedded surfaces, which can then be turned into J-holomorphic curves. The natural structure arising on the boundary of a 4-manifold with a Lefschetz fibration is called a spinal open book. In a joint paper with Sam Lisi and Jeremy Van Horn-Morris, we proved that for "most" contact 3-manifolds admitting spinal open books with genus zero pages, all possible symplectic fillings come from Lefschetz fibrations whose fibers are J-holomorphic curves, including finitely many nodal curves (Lefschetz singular fibers). But there is one caveat: for more complicated spinal open books, the holomorphic curves one obtains on the filling can also include finitely many so-called "exotic fibers", a new type of degeneration that involves the breaking off of a multiple cover of a trivial cylinder and cannot be described in terms of Lefschetz critical points. My goal in this talk will be to define all these notions in general terms, illustrate them with examples, and then state some interesting questions that I do not currently know how to answer.

Chris' notes
16.11.2020
Sushmita Venugopalan (IMSc Chennai)
Tropical Fukaya algebras (part 1)

A multiple cut operation on a symplectic manifold produces a collection of cut spaces, each containing relative normal crossing divisors. We explore what happens to curve count-based invariants when a collection of cuts is applied to a symplectic manifold. The invariant we consider is the Fukaya algebra of a Lagrangian submanifold that is contained in the complement of relative divisors.

In the first talk, I will discuss the homotopy equivalence between the ordinary Fukaya algebra in the unbroken manifold and a "broken Fukaya algebra" whose structure maps count "broken disks" associated to rigid tropical graphs.

In the second talk, I will discuss a further degeneration, which shows that the broken Fukaya algebra is homotopy equivalent to a "tropical Fukaya algebra" whose structure maps are sums of products over vertices of tropical graphs. This is joint work with Chris Woodward.

Sushmita's notes
23.11.2020
Sushmita Venugopalan (IMSc Chennai)
Tropical Fukaya algebras (part 2)

A multiple cut operation on a symplectic manifold produces a collection of cut spaces, each containing relative normal crossing divisors. We explore what happens to curve count-based invariants when a collection of cuts is applied to a symplectic manifold. The invariant we consider is the Fukaya algebra of a Lagrangian submanifold that is contained in the complement of relative divisors.

In the first talk, I will discuss the homotopy equivalence between the ordinary Fukaya algebra in the unbroken manifold and a "broken Fukaya algebra" whose structure maps count "broken disks" associated to rigid tropical graphs.

In the second talk, I will discuss a further degeneration, which shows that the broken Fukaya algebra is homotopy equivalent to a "tropical Fukaya algebra" whose structure maps are sums of products over vertices of tropical graphs. This is joint work with Chris Woodward.

Sushmita's notes
30.11.2020
Alexander Fauck (HU Berlin)
Rabinowitz–Floer homology — an introduction
We will define the Rabinowitz-Floer homology (RFH) of an embedded contact hypersurface in an exact symplectic manifold. We will discuss some of the technical difficulties involved in the definition. Moreover, we will discuss the relation between RFH and the symplectic homology of the ambient manifold and how RFH provides an obstruction for embedding contact manifolds.
07.12.2020
Fabio Gironella (HU Berlin)
Some high dimensional applications of spinal open book decompositions

This talk goes in the same direction of Chris' one on 09.11, by explaining how (sufficiently nice) spinal open book decompositions can also be used in higher dimensions in order to study tightness and fillability questions.

More precisely, I will talk about a joint work with Jonathan Bowden and Agustin Moreno, where we studied an explicit construction of high dimensional manifolds on products with the 2-torus due to Bourgeois '02. In the first part of the talk I will motivate and recall such construction, explaining in particular how this is naturally supported by an explicit high dimensional spinal open book decomposition with pages of codimension >1.

Then, I will explain how, for certain specific Bourgeois contact manifolds, such decomposition can be used to prove uniqueness up to diffeomorphism of symplectically aspherical strong fillings, or even to obstruct their existence. Lastly, I will conclude with some interesting open questions about existence and properties of spinal open books in high dimensions.

Fabio's notes
14.12.2020
Douglas Schultz (HU Berlin)
Rabinowitz–Floer homology toward leaf-wise intersections
This will be a continuation of Alex’s talk from 30.11. Following Albers-Frauenfelder, we will introduce a perturbed version of the Rabinowitz action functional that is generically Morse. This will enable us to prove one of their results, which says that the non-vanishing of RFH gives the existence of a "leaf-wise" intersection for an exact convex hypersurface inside of a Liouville domain.
Douglas' notes
04.01.2021
Thomas Walpuski (HU Berlin)
Castelnuovo's genus bound in symplectic geometry
Castelnuovo's theorem gives an sharp upper bound for the genus of an irreducible, non-degenerate curve of degree d in Pn. It is an interesting question to ask whether analogues of this result can hold in symplectic geometry. It was observed by McDuff, that a direct analogue holds in dimension four. In higher dimensions, the situation is complicated due to Gromov's h–principle. However, the index formula gives strong generic genus bounds in dimension at least eight. This leaves (in particular) the case of symplectic Calabi–Yau 3–folds. Using ideas from geometric measure theory and De Lellis–Spadaro–Spolar's regularity results for semi-calibrated 2–currents, one can prove a non-effective genus bound for super-rigid almost complex structures. I suspect that super-rigidity might be a red herring for this question. Indeed, at least for primitive classes, the non-effective genus bound holds for generic 1–parameter families (for which super-rigidity cannot hold). This is joint work with Aleksander Doan (partially in progress).
Thomas' notes
11.01.2021
Pavel Hájek (Universität Hamburg)
Chain models of string topology coming from symplectic geometry I — Introduction
I will recall loop spaces, natural structures on their homology and the relation to symplectic geometry of the cotangent bundle, specifically to chain level structures defined by counting holomorphic curves. I will then introduce a chain model of equivariant string topology based on de Rham forms and Chern-Simons theory and explain how it is supposed to fit in the picture.
18.01.2021
Johan Apslund (Uppsala universitet)
Fiber Floer cohomology and conormal stops
Let M be a closed orientable spin manifold, and let K be a submanifold of M. In this talk we show that there is a chain level A-infinity isomorphism between the wrapped Floer cochains of a cotangent fiber in T*M stopped by the unit conormal of K, and chains of a Morse-theoretic model of the based loop space of the complement of K. The isomorphism is constructed by counts of holomorphic semi-infinite strips, and was already studied by Mohammed Abouzaid in case K is empty.
Johan's notes
25.01.2021
Pavel Hájek (Universität Hamburg)
Chain models of string topology coming from symplectic geometry II — Computations in the equivariant case
I will present some results and computations for the chain model of equivariant string topology based on de Rham forms and Chern-Simons theory. I will introduce a BV formalism, explain its applications and sketch relations to other structures in mathematical physics like string field theory in the operadic formalism and topological conformal field theories.
01.02.2021
Marc Kegel (HU Berlin)
Heegaard Floer homology and the isomorphism with embedded contact homology I — Introduction to Heegaard–Floer homology and Lipshitz's cylindrical reformulation
We will discuss a sketch of the general picture of Heegaard Floer homology and Lipshitz's cylindrical reformulation (AKA "HF as a special case of SFT") and briefly discuss some of its applications.