Barcodes and bubbles: The role of asphericity in Hamiltonian persistence modules

Abstract

Barcodes and bubbles: The role of asphericity in Hamiltonian persistence modules. This TFM concerns itself with a presentation of the theory of persistence modules associated to Hamiltonian Floer theory. We concentrate on the case of symplectically aspherical manifolds and present a proof of the nondegeneracy of the Hofer metric and its connection to a stability theorem for Floer persistent homology, as well as describe the invariant known as boundary depth. We also compare this aspherical theory to the symplectically monotone case and discuss the relevance of the asphericity assumption. In order to do this, we first describe the mathematical theory of Hamiltonian dynamics, with special attention paid to the development of Floer homology from Morse homology, and we briefly introduce the ideas behind persistent homology and its usefulness.