Distinguished Lectures in Geometry
|Speaker:||Mohammed Abouzaid (Columbia University & Stanford University)|
|Title:||Gromov-Witten theory and generalised cohomology II: Counting beyond numbers.|
|Time:||2023-06-21 (Wed.) 10:00 - 12:00|
|Place:||Auditorium, 6 Floor, Institute of Mathematics (NTU Campus)|
Gromov-Witten theory has played a central point in organising enumerative invariants in algebraic and symplectic geometry, as well as an entry point for relating Floer theoretic invariants to geometric ones. One disadvantage of the classical formulation of Gromov-Witten invariants is that they are in general formulated as rational counts. Work of Givental and Lee earlier this century gave, in the algebro-geometric context, a lift of these invariants to (complex) K-theory, thus removing the need to introduce denominators.
In this lecture series, I will describe a parallel development on the symplectic side, as well as generalisations to other cohomology theories, and point out some applications to symplectic topology. These lectures will largely describe joint work with Mark McLean and Ivan Smith, some of which is forthcoming.
Lecture 2: Counting beyond numbers.
After recalling the notion of a generalised (co)-homology theory, I will explain a formal framework where homology classes give rise to counts valued in the homology of a point. After illustrating the theory with classical cobordism, I will describe a formalism that is adapted to the Kuranishi spaces that arise in Gromov-Witten theory. Finally, I will present a formalism for stating the consistency of enumerative invariants, using the notion of an operad.
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