Arnold's famous conjecture on the numbers of fixed points of Hamiltonian diffeomorphisms on symplectic manifolds is widely regarded as the major driving force for the emergence of the subject called “symplectic topology”; it also motivated numerous important developments in geometry and topology, most notably the invention of Floer homology. Recently Shaoyun Bai and I proved a version of Arnold conjecture for all compact symplectic manifolds which is stronger than all previously proved versions. Our proof relies on our earlier technical breakthrough based on the original idea of Fukaya-Ono, which allows one to define integer-valued counts of pseudoholomorphic curves. This new technique applies not only to Floer theory and Arnold conjecture, but also to other directions such as Gromov-Witten theory, leading to new integer-valued curve counting invariants.

In this talk I will review the background and history of Arnold conjecture, the prototypical argument using Floer homology, and the subtlety about counting pseudoholomorphic curves with symmetries. Then I will talk about our technical breakthrough about integral counts of pseudoholomorphic curves and applications, including both the definition of integer-valued Gromov-Witten invariants and the proof of the integral version of Arnold conjecture. Some prospective works will also be sketched at the end of the talk. This talk is based on the joint works with Shaoyun Bai (arxiv: 2201.02688, 2209.08599).