Mathematical Sciences: Problems in Analysis Related to Symmetrization

数学科学：与对称化相关的分析问题

• 批准号: 9206319
• 负责人: Albert Baernstein
• 金额: $ 10.3万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1995-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9206319&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Problems; Analysis; Related

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.

对称在数学中的价值被普遍认为是用于分析复杂问题的伟大概念之一。在自然和抽象中，最对称的解决方案往往是正确的，或者是所有候选方案中最有利的。在这个项目上要做的工作是继续发展统一的对称理论的扩展计划，该理论是从1974年制定并证明的某个主不等式推导出来的（至少可以追溯到Pappus的第五本书）。特别的重点将放在从复函数理论，偏微分方程和拟共形映射的极值问题。例子包括寻找朗道覆盖定理中的最佳常数，给出全纯函数的像的直径与其导数在某一点上的最佳不等式。在偏微分方程中，要做的工作是确定使热核最小的固定直径区域的形状。准共形映射在任何给定方向上的畸变是有充分记录的。我们对这些地图如何扭曲区域的局限性知之甚少。工作将集中在这个问题上。数学中有一个不言自明的道理：只要提出正确的问题，往往就能相对容易地找到难题的答案。通常正确的问题涉及到对对称性的新见解。本研究的目的是继续扩大具有广泛应用潜力的一般对称原理的范围。