Geometric Problems in Conformal Analysis, Dynamics, and Probability

共形和全纯映射是数学、物理、工程和概率论许多领域的基础。这是两种特殊的地图，保留了角度和复杂的结构。例如，布朗运动（连续随机路径的数学模型）可以使用保角分析来研究，该项目将使用这种方法来研究一些众所周知的问题，例如，确定布朗运动涵盖了哪些类型的简单集。该项目的另一个重点是全纯函数的迭代理论。PI将研究与整个函数（在整个平面上定义的全纯函数）相关的Julia集和Fatou集的大小、形状和行为。传统上，大多数工作都集中在多项式的特殊情况上；更一般的整个函数的情况被称为“先验动力学”，但迄今为止研究得很少。该项目还将考虑分析中的几个问题，这些问题可以使用离散几何和计算几何的思想来解决。这项工作的成功完成可能会导致与网格相关的更快的算法，如用于图形、医学成像和各种设计和制造应用。该项目分为几个广泛的领域：先验动力学，布朗运动，以及分析与计算几何的相互作用。继续早期的工作，PI将研究诸如计算整个函数的Julia集的维数，研究有界轨道的漫游域的存在性，以及研究转义集（迭代到无穷远的点）等问题。PI还将研究随机集合的相关几何问题，如布朗运动，考虑众所周知的问题（布朗运动是否涵盖任何可校正的弧线？）以及更新颖的问题（布朗运动是否可以在保角映射中移除？）截断点是否位于可校正曲线上？)最后，PI将利用分析、几何和拓扑学的思想扩展他早期的算法，用于优化网格划分、三角测量和保角映射。最有趣的泛化是在三维空间，在那里没有严格的复杂性界限，但最重要的应用就在那里。该提案还考虑了“纯”分析中的问题，这些问题可以使用计算几何中已知结果的强化版本来解决；其中一个问题是BMO拓扑中弦弧曲线空间的连通性问题；另一个是将一般的双lipschitz映射分解成具有小常数的双lipschitz映射。

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.