As a virtual enumeration of (pseudo-)holomorphic curves,
Gromov-Witten invariants are generally rational-valued due to the
presence of non-trivial symmetries of the curves. Realizing a proposal
of Fukaya-Ono back in the 1990s, I will explain how to define
integer-valued Gromov-Witten type invariants for all closed symplectic
manifolds. I will also discuss how this construction fits into a
larger program on refining curve-counting invariants initiated by
Joyce, Pardon, and Abouzaid-McLean-Smith. This is based on joint work
with Guangbo Xu.
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