Distinguished Lectures in Geometry
|Speaker:||Mohammed Abouzaid (Columbia University & Stanford University)|
|Title:||Gromov-Witten theory and generalised cohomology I: What does it mean to count curves?|
|Time:||2023-06-20 (Tue.) 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 1: What does it mean to count curves?
The Deligne-Mumford compactification of the moduli space of Riemann surfaces has an analogue for the moduli space of holomorphic maps from Riemann surfaces to symplectic manifolds (or algebraic varieties), which is sometimes called the Kontsevich compactification. I will begin by formulating the basic features of this moduli space, including the fact that it is a smooth manifold in the simplest situations, and then describe the geometric issues that prevent the simplest picture from applying in general, and allude to the notion of Kuranishi space (or derived orbifold), which is a generalisation of the notion of manifolds that will be the geometric basis of the enumerative theory in these lectures. The plan for this first lecture will be kept flexible in order to take audience background into account.
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