Colloque des sciences mathématiques du Québec

5 février 2021 de 15 h 00 à 16 h 00 (heure de Montréal/Miami)

Symmetry, barcodes, and Hamiltonian dynamics

Colloque par Egor Shelukhin (Université de Montréal, Canada)

Lauréat 2020 du prix de mathématiques André-Aisenstadt

In the early 60s Arnol'd has conjectured that Hamiltonian diffeomorphisms, the motions of classical mechanics, often possess more fixed points than required by classical topological considerations.  In the late 80s and early 90s Floer has developed a powerful theory to approach this conjecture, considering fixed points as critical points of a certain functional.  Recently, in joint work with L.  Polterovich, we observed that Floer theory filtered by the values of this functional fits into the framework of persistence modules and their barcodes, originating in data sciences.  I will review these developments and their applications, which arise from a natural time-symmetry of Hamiltonians.  This includes new constraints on one-parameter subgroups of Hamiltonian diffeomorphisms, as well as my recent solution of the Hofer-Zehnder periodic points conjecture.  The latter combines barcodes with equivariant cohomological operations in Floer theory recently introduced by Seidel to form a new method with further consequences.