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.
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 brought 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. The main scientific advances of this project were in two areas. The first is the theory of dynamical systems in 2 variables (surfaces). Here we discovered the simplest, interesting algebraic dynamical system. Its "entropy", a measure of its complexity, is about 0.162. In contrast the "Baker's transformation", used to knead bread, has complexity about 0.69. The discovery and enumeration of these low-complexity dynamical systems provides a census of the models for 2-dimensional behavior. The second area is the theory of billiards. Since the sides of a billiard table are 1-dimensional, so is this type of dynamical system. Here we discovered that self-similar cascades of periodic billiard dynamics arise when the shape of the table is changed in a simple way. We also found that, using number theory, some of these cascades give rise to deterministic procedures to visits all the points on an infinite chess board. Finally we developed connections with the classical theory of functions of complex variables. The broader impact of this project included numerous lectures to the general public on recent scientific breakthroughs in mathematics. This included talks in Chicago, Stanford and Paris on geometry, the solution of the Poincare' conjecture and quantum topology. Within the academic realms this project fostered the education and research of several Ph. D. students and postdocs, who are en route to becoming leaders in mathematical research themselves.
