Geometric Problems in Conformal Analysis, Dynamics, and Probability

共形分析、动力学和概率中的几何问题

• 批准号: 1608577
• 负责人: Christopher Bishop
• 金额: $ 22.16万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1608577&HistoricalAwards=false
• 关键词: Geometric; Problems; Conformal; Analysis; Dynamics

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.

共形和全态图是数学，物理，工程和概率的许多领域的基础。这是保留角度和复杂结构的两种特殊地图。例如，可以使用保形分析研究布朗运动（连续随机路径的数学模型），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Brownian Motion涵盖了哪些类型的简单集。该项目的另一个重点是跨形功能的迭代理论。 PI将研究与整个功能相关的Julia和FATOU集的大小，形状和行为（整个平面上定义的全态函数）。传统上，大多数工作都集中在多项式的特殊情况下。更一般的整个功能的情况被称为“先验动力学”，但到目前为止研究得多。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和计算几何形状的想法攻击。成功完成这项工作可能会导致与网格划分相关的更快算法，如图形，医学成像以及各种各样的设计和制造应用。该项目分为多个广泛的领域：先验动力学，布朗运动，布朗运动以及分析与计算几何形状的相互作用。继续进行较早的工作，PI将研究诸如计算整个功能的朱莉娅集合的维度，研究具有有界轨道的徘徊域的存在，并研究逃逸集（迭代到无限的点）。 PI还将在相关的几何问题上处理诸如布朗尼运动之类的几何问题，考虑到众所周知的问题（布朗运动是否涵盖了任何可纠正的弧度？使用分析，几何和拓扑的想法。最有趣的概括将是三个维度，在三个维度上，没有严格的复杂性界限，而是最重要的应用所在。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形状的已知结果的增强版本来解决。一个问题是BMO拓扑中和弦曲线空间的连接性。另一个是将一般的Bi-Lipschitz图解到具有小常数的Bi-Lipschitz地图中。