Speaker: Spencer Cattalani
Location: Math Tower 5-127
Abstract:
Gromov revolutionized symplectic geometry by connecting it with (almost) complex geometry. This connection, centered on the notion of pseudoholomorphic curve, has proven to be incredibly fruitful for both fields. We aim to expand it through a study of complex cycles, a generalization of pseudoholomorphic curves due to Sullivan. In particular, we extend the positivity of intersection between pseudoholmorphic curves to include complex cycles and approximate complex cycles by ``coarsely'' holomorphic curves. Using these results, we prove two geometric criteria proposed by Gromov for an almost complex manifold to be tamed by a symplectic form. We also study a special class of complex cycles, called Ahlfors currents. We construct Ahlfors currents using a continuity method, show that they control the asymptotic behavior of pseudoholomorphic curves, and prove that the set of Ahlfors currents is convex. Using these results, we are able to characterize when a product of surfaces contains a complex line, generalizing a theorem of Bangert and answering a question of Ivashkovich and Rosay. These results indicate a parallel to the theory of rational curves and provide tools for the nascent study of symplectic non-hyperbolicity envisioned by Gromov.
