Complex dynamics and moduli spaces

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This project concerns the interplay between dynamical systems, complex analysis, and the algebra, geometry and topology of moduli spaces. The settings for this investigation include(i) The moduli space of Riemann surfaces M_g, its complex geodesics, and the bundle of holomorphic 1-forms;(ii) Billiards in rational polygons;(iii) Real and complex K3 surfaces, their automorphisms and their moduli as encoded by Hodge structures;(iv) Automorphisms of rational surfaces and the moduli space of point configurations in the projective plane;(v) Lattices in R^n and flows on their moduli space SL_n(R)/\SL_n(Z);(vi) The moduli spaces of iterated polynomials and rational maps in one complex variable; and(vii) The moduli space of vector bundles on a Riemann surface.We focus on problems of rigidity, the statistics and topology of orbits (are all invariant measures algebraic? Is there a spectral gap?), deformation spaces and their compactifications, and ramifications for Diophantine approximation. Can we know the future? Models for planetary motion, evolution of species, climate change and a host of other dynamical systems suggest the answer is yes. But the concerted mathematical study of even the simplest models reveals engines of unpredictability, coexisting with complete knowledge of the underlying laws of change. This project brings analysis, geometry and number theory to bear on the study of mathematical dynamical systems, with the goal of comprehending their core behaviors and what can and cannot be predicted.