Berlin-Hamburg-Seminar am 10.11.2023

Alex Oancea (Strasbourg) Morse and Floer theory with DG coefficients

Building on a 2004 construction of Barraud-Cornea, I will explain how Morse and Floer theory can be formulated in order to accommodate differential graded local coefficients, also known under the name of infinity local systems. This construction allows, for example, to speak of and compute the homology with coefficients in chains on the fibers of a fibration over the base space. This is joint work with Jean-François Barraud, Mihai Damian and Vincent Humilière, partly accomplished and partly in progress. I will also discuss symplectic applications.

Yash Deshmukh (MPI Bonn) Operations in symplectic cohomology

A chain-level action of the framed E₂ operad on symplectic cohomology was constructed by Abouzaid-Groman-Varolgunes. In this talk I will report on an ongoing project about extending this to a chain-level action of the PROP of framed Riemann surfaces. An algebraic result about this PROP provides predictions regarding algebraic structures underlying some variants of symplectic cohomology, including positive symplectic cohomology and Rabinowitz Floer cohomology. I will mention this result and outline how the existence of a chain-level action of the PROP would allow us to realize these predictions. Time permitting, I will also indicate how this chain-level action provides the first step towards comparing the categorical and geometric enumerative invariants.