Geometric Problems in Conformal Analysis, Dynamics, and Probability

Conformal and holomorphic maps are fundamental to many areas of mathematics, physics, engineering and probability. These are two special kinds of maps that preserve angles and complex structures. For example, Brownian motion (a mathematical model of continuous random paths) can be studied using conformal analysis and the project will use this approach to investigate some well known problems, e.g., to determine what types of simple sets are covered by Brownian motion. Another focus of the project is the iteration theory of holomorphic functions. The PI will study the size, shape and behavior of the Julia and Fatou sets associated to entire functions (holomorphic functions defined on the whole plane). Traditionally, most work has focused on the special case of polynomials; the case of more general entire functions is known as "transcendental dynamics", but has been much less studied so far. The project will also consider several problems in analysis that can be attacked using ideas from discrete and computational geometry. Successful completion of this work could result in faster algorithms related to meshing, as used in graphics, medical imaging, and a wide variety of design and manufacturing applications. The project divides into several broad areas: transcendental dynamics, Brownian motion, and the interactions of analysis with computational geometry. Continuing earlier work, the PI will study problems such as computing the dimension of Julia sets of entire functions, investigating the existence of wandering domains with bounded orbits, and studying the escaping set (points that iterate to infinity). The PI will also work on related geometric problems for random sets such as Brownian motion, considering both well known questions (does Brownian motion cover any rectifiable arcs?) as well as more novel ones (is Brownian motion removable for conformal maps? Does the set of cut-points lie on a rectifiable curve?) Finally, the PI will work on extending his earlier algorithms for optimal meshing, triangulation and conformal mapping using ideas from analysis, geometry and topology. The most interesting generalization would be to three dimensions, where no rigorous complexity bounds are known, but where the most important applications lie. The proposal also considers problems in "pure" analysis that might be solved using strengthened versions of known results in computational geometry; one such problem is the connectedness of the space of chord-arc curves in the BMO topology; another is the factoring of general bi-Lipschitz maps into bi-Lipschitz maps with small constants.