Principal Investigator: Peter Albers
This project aims to investigate three sets of problems. First, the principal investigator's PhD thesis laid the foundations of a novel approach to functoriality for Floer homology, and the next steps in this direction are to remove some restrictive hypotheses and to confirm the compatibility of this theory with the pair-of-pants product on Floer homology. Second, Frauenfelder and the PI have recent established Floer homology for negative line bundles, that is, for a class of non-compactly supported Hamiltonian functions in non-compact symplectic manifolds, and this theory should be made more robust by removing symplectic asphericity hypotheses and should also be extended to a Morse-Bott situation. Third, the PI will apply the Fredholm theory in polyfolds developed by Hofer, Wysocki, and Zehnder to explore newly emerging invariants from the theory of polyfolds with operations.
Symplectic geometry is the study of spaces equipped with the basic structure that underlies Hamiltonian mechanics, for which preferred coordinates give both position and momentum of a moving object and in which conservation of energy is natural and inevitable. The Floer homology theory referred to in the first paragraph studies such manifolds through the critical point behaviors of certain auxiliary spaces; this kind of analysis of critical points goes back to work of Marston Morse circa the 1930s, but its application in the modern context requires new tools such as the polyfold theory cited above.
