The fifteenth Michael Brin Prize in Dynamical Systems has been awarded to Dzmitry Dudko recognizing his important contributions to renormalization techniques in complex dynamics with applications to the conjecture of local connectivity of the Mandelbrot set (MLC), and for his work on Thurston Maps.  Congratulations to Dima for this well-deserved award!

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.

Speaker: Matthew Huynh

Location: Math 5-127

Abstract:

Let X be a smooth, complex projective variety, let V be a vector bundle of rank d on X, and let G be a connected, reductive group.
First, we introduce the Hitchin morphism from the moduli space of V-valued G-Higgs bundles on X to the Hitchin base. Next, we explain the Chen-Ngô Conjecture, which predicts that the image of the Hitchin morphism when V is the bundle of holomorphic 1-forms is the so-called space of spectral data.
We verify the Chen-Ngô Conjecture whenever X is a ruled surface, or a smooth modification of a non-isotrivial elliptic fibration with only reduced fibers. Furthermore, we show that if in addition G is a classical group, then the space of spectral data is surjected upon by the moduli space of semiharmonic G-Higgs bundles.