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!*

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.

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.

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.

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.