Complex Dynamics and Moduli Spaces

Harvard University, Cambridge MA

Investigators

Abstract

Moduli spaces of lattices, Riemann surfaces, rational maps and other algebraic structures exhibit rich geometry, often accompanied by rigidity and a connection with arithmetic. These spaces also have a dynamical nature -- they support natural flows or group actions with complicated orbits, or they classify such actions. This projects aims to investigate moduli spaces from a dynamical perspective. In the setting of Riemann surfaces we study Teichmueller curves, Jacobians with real multiplication, volumes of complex hyperbolic cone manifolds and representations of the braid group developed by Picard, Deligne-Mostow and Thurston. We also investigate dynamical systems of low entropy, ranging from pseudo-Anosov maps to automorphism of K3 surfaces, via constructions adapted to small Salem numbers and Coxeter groups. In the setting of lattices we aim to investigate packing constants of ideals in number fields, and to demonstrate an abundance of bounded orbits for rank one flows, to complement the dearth of such orbits predicted in higher rank by conjectures of Littlewood and Margulis. Finally, in the setting of iterated rational maps in one complex variable, we study the natural bifurcation measure on moduli spaces attached to individual critical points, and aim to show their independence and transversality, with an eye towards arithmetic applications. We also intend to link the theory of Berkovitch spaces to the degeneration of hyperbolic manifolds to trees. From particle physics to finance, from evolution to climate change, the world is full of dynamical systems. Simple algebraic transformations already exhibit many of the features of these natural phenomena, such as phase transitions and tipping points that signal the onset of new regimes. These universal patterns may be revealed through the rigorous study of moduli spaces, their compactifications and their stratifications by dynamical invariants. This project appeals to a broad range of mathematical disciplines to both deepen our understanding of dynamical systems and to sharpen our mathematical and computational methods.