Mike Usher. A longstanding technique in symplectic topology has involved using the real-valued filtrations on symplectic Floer complexes associated to the action functional to obtain information about quantitative questions such as how much energy is required to generate a given Hamiltonian diffeomorphism, or how much one domain needs to be dilated in order to contain a symplectically embedded copy of a different domain. In the mid-2010s it came to be appreciated that this information is best organized in terms of the theory of persistence modules, as studied in the field of topological data analysis. This has led to fruitful interactions between these fields, as constructions from the persistent homology literature have been adapted and generalized and persistence-type barcodes have become a core tool in quantitative symplectic topology.
