Soft construction of Floer Homology

12 March 2020
12:00
Michele Stecconi
Abstract

Invented by Andreas Floer in 1988 to solve Arnold's
conjecture, (symplectic) Floer Homology is a machinery to relate the
existence of periodic trajectories of an Hamiltonian flow on a
symplectic manifold M with the homology groups of M, analougously to
Morse Homology. Indeed this is done by developing an infinite
dimensional Morse theoretic framework adapted to a certain functional
(the action functional) on the loop space of M, whose critical points
are the periodic trajectories of the given hamiltonian flow.
Despite the topological nature of the results, the construction is
technically quite heavy, involving hard analysis and elliptic systems of
PDEs.

Together with Andrei Agrachev and Antonio Lerario we are developing a
method to construct such infinite dimensional homology invariants using
only soft and essentially finite dimensional tools. In my talk I will
present our approach.
The main feature consists in approximating the loop space with finite
dimensional submanifolds of increasing dimension, we do this with the
language of control theory, and then interpret asymptotically the
information provided by classical Morse theory

  • PDE CDT Lunchtime Seminar