Mathematical Sciences: Problems in Analysis Related to Symmetrization

数学中对称性的价值被普遍认为是用于分析复杂问题的伟大概念之一。无论是在本质上还是在抽象上，最对称的解决方案选择往往是所有候选方案中正确的或最有利的。在这个项目上要做的工作是继续一个扩展的计划，发展一个统一的对称化理论，这个理论是从1974年提出和证明的某个大师不等式推导出来的(至少要追溯到Pappus的第五本书)。重点介绍复变函数论、偏微分方程组和拟共形映射中的极值问题。例如，在Landau的复盖定理中找到最佳常数，该复盖定理给出了全纯函数的像的直径与其导数在某一点上的最佳不等。在偏微分方程组中，要确定使热核最小的固定直径区域的形状。拟共形映射在任何给定方向上的变形都有很好的文献记载。关于这些地图如何扭曲区域的局限性，人们知之甚少。工作将集中在这个问题上。数学中不言而喻的是，通过提出正确的问题，一个人往往可以相对容易地得到一个困难问题的答案。通常，正确的问题涉及对对称性的新见解。这项研究的目的是继续扩大具有广泛适用性的一般对称原则的范围。

The value of symmetry in mathematics is universally recognized as one of the great concepts used in the analysis of complex problems. In nature as well as in abstraction, the most symmetric choice for a solution often turns out to be the correct one or the most favorable among all candidates. Work to be done on this project continues an extended program in the development of a unified theory of symmmetrization deriving from a certain master inequality formulated and proved in 1974 (with antecedents going back to the fifth book of Pappus, at least) . Particular emphases will be placed on extremal problems from complex function theory, partial differential equations and quasiconformal mapping. Examples include finding the best constant in Landau's covering theorem giving the best inequality between the diameter of the image of a holomorphic function and its derivative at some point. In partial differential equations, work will be done in determining the shape of those domains of fixed diameter which minimize the heat kernel. The distortion of quasiconformal maps in any given direction is well documented. Very little is known on the limitations of how these maps can distort area. Work will focus on this question. It is axiomatic in mathematics that by asking the right questions, one is often led to the answer of a difficult question with comparative ease. Often right questions involve new insights into symmetry. The purpose of this research is to continue expanding the scope of general symmetry principles which have the potential for broad applicability.