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Symplectic Geometry Seminar

Complex Cycles and Symplectic Topology (Spencer Cattalani)

Spencer Cattalani

Location:  Hill 705
Date & time: Thursday, 18 April 2024 at 1:15PM - 2:15PM

Title: Complex Cycles and Symplectic Topology

Abstract: Among all almost complex manifolds, those which are tamed by symplectic forms are particularly well studied. What geometric properties characterize this class of manifolds? That is, given an almost complex manifold, how can one tell whether it is tamed by a symplectic form? By a 1976 result of D. Sullivan, this question can be answered by studying complex cycles. I will explain what complex cycles are and their role in two recent results, which confirm speculations posed by M. Gromov in 2000 and 1985, respectively. The first is that an almost complex manifold admits a taming symplectic structure if and only if it satisfies a certain bound on the areas of coarsely holomorphic curves. The second is that an almost complex 4-manifold which has many pseudoholomorphic curves admits a taming symplectic structure. This leads to an almost complex analogue of D. McDuff’s classification of rational symplectic 4-manifolds.