Mathematical Science: Absolutely Continuous Conjugations

9400975汉密尔顿该奖项支持对现代函数理论中出现的问题进行数学研究。这一理论的一个中心主题是函数通过与双射的共轭而变形。利用经典分析、位势理论、拟共形映射、解析函数边值和保角映射等工具，发展了黎曼曲面和复杂动力学领域中的两种新方法。该项目将继续利用这些想法来证明Julia有理函数集上的刚性结果。例如，如果Julia集是一条曲线，则它具有Fatou性质，即它是一个圆或具有大于一的维度。对于Riemann曲面，工作将遵循最近的Bers统一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，将努力将那些出现的反射定性为跨域边界的反射。经典函数理论引入了几何技术，事实证明，这些技术的适用性比最初想象的要广泛得多。本课程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。***

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. ***