Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

Zhen Gao (Universität Augsburg) : Morse-Bott Floer Homology and Rectangular Pegs

The rectangular peg problem, an extension of the square peg problem, is easy to outline but challenging to prove through elementary methods. In this talk, I discuss how to show the existence and a generic multiplicity result assuming the Jordan curve is smooth, utilizing Morse-Bott Floer homology. In particular, we obtain a convenient formula for computing the algebraic intersection number of cleanly intersecting Lagrangian submanifolds, which is well consistent with the Euler characteristic of Morse-Bott Floer homology in the spirit of "categorification''.

Zihong Chen (Massachusetts Institute of Technology) : The Exponential Type Conjecture for Quantum Connection

The (small) quantum connection is one of the simplest objects built out of Gromov-Witten theory, yet it gives rise to a repertoire of rich and important questions such as the Gamma conjectures and the Dubrovin conjectures. There is a very basic question one can ask about this connection: what is its formal singularity type? People's expectation for this is packaged into the so-called exponential type conjecture, and I will discuss a proof in the case of closed monotone symplectic manifolds. My approach follows a reduction mod p argument, by combining Katz's classical result on differential equations and the more recent quantum Steenrod operations.

Jonghyeon Ahn (University of Illinois Urbana-Champaign) : $S^1$-Equivariant Relative Symplectic Cohomology and Relative Symplectic Capacities

In this talk, I will construct an $S^1$-equivariant version of the relative symplectic cohomology developed by Varolgunes. As an application, I will construct a relative version of Gutt-Hutchings capacities and a relative version of symplectic (co)homology capacity. We will see that these relative symplectic capacities can detect the diplaceability and the heaviness of a compact subset of a symplectic manifold. We compare the first relative Gutt-Hutchings capacity and the relative symplectic (co)homology capacity and prove that they are equal to each other under a convexity assumption.