# Research Seminar: Symplectic Geometry (WS 2022/23)

This is a working group seminar run by Klaus Mohnke, Chris Wendl, and Thomas Walpuski on recent developments in symplectic geometry and related areas. 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.

**Time:** Mondays 13:15-14:45 (starting on 2022-10-24)

**Location:** Johann von Neumann-Haus (Rudower Chaussee 25) Room 1.114

Moodle page (key: gromov)

Firstly, I'll show that simple holomorphic curves are generically Fredholm regular; i.e. "transversality holds for simple curves". The proof is standard, but will become much more interesting in the equivariant case. Secondly, we will review the slice theorem for smooth Lie group actions on Banach manifolds --- with an eye towards infinite-dimensional applications. If time permits, we might see how these are related.

*G*acting symplectically. What can we say about the moduli space of holomorphic curves on

*M*, w.r.t. a generic

*G*-equivariant almost complex structure? We should not expect it to be a manifold (after all, transversality and symmetry are famously incompatible). However, we can hope for a clean intersection condition: the moduli space decomposes into disjoint strata which are smooth manifolds of explicitly computable dimension. I will focus on the case of simple curves (which is already surprisingly interesting): I'll explain how to decompose the moduli space into iso-symmetric strata and prove that each stratum is smooth.

*S*.

^{3}*D*, they provide a method to obtain a space whose stable homotopy type is a link invariant, and whose cohomology is isomorphic to the Khovanov homology of the link represented by

*D*. In this talk we will present some simplifications of the Khovanov spectrum for some specific gradings. As a consequence, we will obtain properties of the Khovanov homology of several families of links.

*M*, for every Morse function

*f*on

*M*a generic Riemannian metric

*g*satisfies that

*(f,g)*is Morse-Smale, i.e. stable and unstable manifolds intersect transversally for every pair of critical points. If we impose an additional finite symmetry on

*M*, i.e. if a finite group

*G*acts on

*M*by diffeomorphisms such that

*f*and

*g*are

*G*-invariant, then this statement fails: there are Morse functions

*f*such that

*(f,g)*is never Morse-Smale, for any

*g*. We will show that nontheless there exist for any manifold

*M*and any group

*G*always

*G*-invariant Morse functions

*f*such that generic

*G*-invariant Riemannian metrics give Morse-Smale pairs

*(f,g)*, and we will show that whenever the action by

*G*is not free, then there also always exist

*G*-invariant Morse functions such that

*(f,g)*is for no

*G*-invariant metric Morse-Smale.

*References:*

- Miguel Pereira, On the Lagrangian capacity of convex or concave toric domains
- Kai Cieliebak and Klaus Mohnke, Punctured holomorphic curves and Lagrangian embeddings

**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!*