Joint IAS/Princeton University Symplectic Geometry Seminar

Arnold's famous conjecture on the numbers of fixed points of Hamiltonian diffeomorphisms on symplectic manifolds has motivated numerous important developments in geometry and topology, most notably the invention of Floer homology. In this talk I will present the recent proof of the integral version of the Arnold conjecture: on any closed symplectic manifold, the number of fixed points of a nondegenerate Hamiltonian diffeomorphism is bounded from below by a version of total Betti number over ${\mathbb Z}$ which takes accounts of torsions of all characteristics. This result strengthens the rational version proved by Fukaya-Ono and Liu-Tian as well as the finite field version proved recently by Abouzaid-Blumberg. The most crucial inputs of the proof are 1) Fukaya-Ono's idea of integral counting of holomorphic curves, the detail of which has been worked out recently by a previous joint work with Shaoyun Bai, and 2) the construction of global Kuranishi charts for the Hamiltonian Floer flow category which generalizes the recent work of Abouzaid-McLean-Smith. This talk is based on the joint work with Shaoyun Bai (arxiv: 2209.08599).