UC Berkeley

This thesis contains polyfold constructions with applications to symplectic topology, as well as Künneth formulas for persistent homology of various filtrations with applications to Topological Data Analysis.

In symplectic topology, we provide a polyfold version of the Piunikhin-Salamon-Schwarz proof of the Arnold conjecture -- our proof holds for general closed symplectic manifolds. This proof relies on the polyfold regularization of moduli spaces with a finite dimensional constraint imposed on evaluation maps, for which we provide multiple polyfold constructions that can be applied as black boxes in general situations. One of these black boxes can be viewed as an implicit function theorem: we construct a polyfold structure on the subset of a polyfold cut out by a submersive finite dimensional constraint, and then we prove the sc-Fredholm property for the restriction of a sc-Fredholm section to this subset. We go on to further investigate when an implicit function theorem holds in polyfold theory: We give explicit counterexamples to a general implicit function theorem for sc-smooth maps, and we show how the novel notion of a sc-Fredholm map overcomes this difficulty, justifying the technical complexity of polyfold theory.

In Topological Data Analysis, we prove a Künneth formula in low homological dimensions for the persistent homology of a Cartesian product of finite metric spaces equipped with the sum metric. In all homological dimensions, we bound the interleaving distance between the prediction from the Künneth formula and the true persistent homology. As preliminary results of independent interest, we prove Künneth formulas in all homological dimensions for the persistent homology of the Cartesian product of R_+-filtered simplicial sets and also for the homology of the graded tensor product of simplicial k[R_+]-modules.