# 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*.

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**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!*