Symplectic Topology Student Seminar (Fall 2025)

Abstract: In the span of two talks, I will describe the construction of the Hofer-Zehnder capacity, which gives an alternative proof of Gromov's celebrated non-squeezing theorem. The main technique is a variational principle tailored for the action functional.

Abstract: I will introduce the notion of a Liouville sector and outline its basic properties. The first part of the talk will be focused on motivating the sectorial condition, which arises naturally from Kontsevich's cosheaf conjecture and symplectic field theory. One particularly nice consequence of this condition is that a Liouville sector admits a family of almost complex structures for which holomorphic disks stay away from the boundary, which is the key geometric property which makes wrapped Floer theory well-defined and functorial. I will also give many examples of Liouville sectors and (time-permitting) discuss ways of decomposing a Liouville manifold into a union of sectors.

Abstract: A general feature of Fukaya categories is that geometric “surgery” constructions correspond to algebraic operations in the Fukaya category. There are many instances of this idea, e.g. by Seidel, FOOO, Biran-Cornea, and GPS. In this talk, I will define the partially wrapped Fukaya category and discuss GPS's formulation of this idea. Roughly, their result states that attaching a "Lagrangian cobordism at infinity" to disjoint Lagrangians \(L, K\) results in the mapping cone of a morphism \(L \to K\). The key to the proof of this fact is an action argument.

A version of a famous problem of Arnold asks whether a Lagrangian plane inside Euclidean space is smoothly unknotted (rel boundary). I will explain joint work with Yash Deshmukh and Alex Pieloch where we prove the answer is yes for some Weinstein manifolds. I will assume no familiarity with Floer homotopy theory.