This will be a broad program of research in geometric aspects of holomorphic dynamics, Teichmuller theory, laminations, and related areas. The project addresses central problems in these fields: The Local Connectivity Problem for the Mandelbrot set and for Julia sets would help to give a thorough understanding of dynamics for the complex quadratic family. The Renormalization and Universality Conjectures concern fundamental rigidity features of the phase and parameter domains for dynamical systems. The Regular or Stochastic Conjecture would give a complete measure-theoretic picture of the dynamics of unimodal maps. The Ehrenpreis Conjecture asserts that any two compact Riemann surfaces have almost isomorphic compact coverings. The project would explore further interplay between holomorphic dynamics, hyperbolic geometry, Teichmuller theory, and the theory of laminations, as well as the interplay between real and complex dynamics in one and two variables.
Dynamical systems theory studies evolution of various systems described by differential equations or by the iteration of a single map. It has numerous applications in celestial mechanics, statistical physics, fluid dynamics, biology, and other branches of natural science. Holomorphic dynamics is the part of dynamical systems theory that deals with iterates of complex analytic maps. It has proved to be a powerful tool in understanding important models of real low-dimensional dynamics. There are numerous interconnections between holomorphic dynamics, geometric analysis, hyperbolic geometry, and the theory of foliated spaces. Holomorphic dynamics also produces beautiful fractal objects, such as Julia sets and the Mandelbrot set, whose intricate structure has fascinated scientists for decades. All of this structure and its applications will be further explored by the Stony Brook dynamics group.
