Mathematical Science: Absolutely Continuous Conjugations

数学科学：绝对连续共轭

• 批准号: 9400975
• 负责人: David Hamilton
• 金额: $ 4万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-01 至 1996-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400975&HistoricalAwards=false
• 关键词: Mathematical; Science; Absolutely; Continuous; Conjugations

9400975 Hamilton This award supports mathematical research into problems arising in the modern theory of functions. A central theme of this theory is the deformation of functions by conjugation with bijections. Using tools of classical analysis, potential theory, quasiconformal mapping, boundary values of analytic functions, and conformal mapping two new approaches to old questions in the areas of Riemann surfaces and Complex Dynamics have been developed. The project will continue to exploit the ideas to prove rigidity results on Julia sets of rational functions. For example, if the Julia set is a curve then it has the Fatou property, that is, it is a circle or has dimension greater than one. For Riemann surfaces work will follow on a recent proof of the general version of the Bers Uniformization Theorem to prove a conjecture of Gromov on the uniqueness of the representation. Other work will focus on n-dimensional bilipschitz mappings in the context of quasiconformal mapping. In particular efforts will be made to characterize those which arise as reflections across boundaries of domains. Classical function theory has introduced geometric techniques which have proved to have far wider applicability than first imagined. The subject exploits the interplay between geometric interpretation with powerful analytic techniques to yield some of the most complete mathematical results in the field of analysis. In this project, variational techniques and geometric reasoning are expected to lead to results of valuable physical significance. ***

9400975汉密尔顿这个奖支持对现代函数理论中出现的问题进行数学研究。这个理论的中心主题是函数的变形与双射共轭。利用经典分析、势理论、拟共形映射、解析函数的边值和共形映射等工具，在黎曼曲面和复杂动力学领域提出了两种解决老问题的新方法。该项目将继续利用这些思想来证明有理函数Julia集上的刚性结果。例如，如果Julia集合是一条曲线，那么它具有Fatou属性，也就是说，它是一个圆或具有大于1的维度。对于黎曼曲面，工作将遵循最近对伯斯均匀化定理的一般版本的证明，以证明Gromov关于表示的唯一性的猜想。其他工作将集中在拟共形映射背景下的n维bilipschitz映射。特别是，将努力描述那些作为跨域边界反射产生的反射。经典函数理论引入了几何技术，这些技术已被证明具有比最初想象的更广泛的适用性。该学科利用几何解释与强大的分析技术之间的相互作用，在分析领域产生一些最完整的数学结果。在这个项目中，变分技术和几何推理有望导致有价值的物理意义的结果。＊＊＊