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Title: Ahlfors Currents and Symplectic Non-Hyperbolicity

Abstract: The study of rational curves has revealed deep connections between symplectic and algebraic geometry. Complex lines are a more general class of curve that has the potential to connect symplectic and complex analytic geometry. Remarkably, as shown by Bangert in 1998, every almost complex structure on a 2n-torus tamed by a linear symplectic structure contains a complex line. These curves are non-compact, which presents a serious difficulty in understanding their symplectic aspects. In this talk, I will explain how Ahlfors currents can be used to resolve this difficulty and produce a theory parallel to that of rational curves.