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This seminar talk explores the development of Integral Floer Homology Theory, presented by Guangbo Xu from Rutgers University as part of the Symplectic Geometry Seminar series at the Institute for Advanced Study. Discover how Xu and collaborator Shaoyun Bai utilize a novel FOP transversality condition on orbifolds to construct Hamiltonian Floer theory over integers for all compact symplectic manifolds. Learn about the formal structures in this theory, including flow categories, bimodules, homotopies, and their lifts to refined categories such as derived orbifolds. Understand how the FOP perturbation method enables the derivation of standard Floer theory objects like Floer chain complexes, continuation maps, PSS maps, pair-of-pants products, and Steenrod-type operations. The talk concludes with two significant dynamical applications for compact symplectic manifolds: how a non-contractible periodic orbit of a Hamiltonian implies infinitely many such orbits, and how the existence of Hamiltonian pseudorotation implies geometric uniruledness.
