Mathematical Science: Absolutely Continuous Conjugations

9400975汉密尔顿 该奖项支持数学研究在现代函数理论中出现的问题。 这个理论的一个中心主题是变形的功能共轭双射。 利用经典分析、势理论、拟共形映射、解析函数的边界值和共形映射等工具，对黎曼曲面和复动力学领域的老问题提出了两种新的解决方法。 该项目将继续利用这些想法来证明有理函数的Julia集上的刚性结果。 例如，如果Julia集是一条曲线，则它具有Fatou属性，即它是一个圆或具有大于1的维度。 对于黎曼曲面的工作将遵循最近的证明一般版本的Bers一致化定理，以证明一个猜想格罗莫夫的唯一性表示。 其他工作将集中在n维bilipschitz映射的背景下拟共形映射。 特别是，将努力描述那些作为跨越域边界的反射而出现的特征。 经典函数论引入了几何技术，这些技术已被证明具有比最初想象的更广泛的适用性。 该科目利用强大的分析技术与几何解释之间的相互作用，产生一些分析领域最完整的数学结果。在这个项目中，变分技术和几何推理预计将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. ***