Determining Analytic Properties of Maps from Non-Analytic Data

从非分析数据确定地图的分析属性

• 批准号: 0400636
• 负责人: Sergiy Merenkov
• 金额: $ 7.5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2006-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400636&HistoricalAwards=false
• 关键词: Determining; Analytic; Properties; Maps; Non

The PI proposes to study how certain non-analytic information determines analytic properties of maps. A conformal map between two surfaces is one that preserves angles at each point. This is a very restrictive assumption, and it is interesting to understand to what extent properties of such maps can be deduced from other topological, combinatorial, algebraic, set-theoretic, or analytic assumptions. An example is determining the conformal type of surfaces, namely, whether a surface is conformally equivalent to to a model surface. In the higher dimensional case, biholomorphic equivalence is used in place of conformal equivalence. Some specific topics that the PI proposes to study are: surfaces whose conformal type is determined by combinatorial properties of the associated net, a question raised by EB Vinberg, and two topics related to questions raised by L. Rubel, namely, maximal growth functions that have finitely many critical and asymptotic values, and the study of complex manifolds whose biholomorphic type can be recovered from the knowledge of associated semigroups of analytic endomorphisms and the semiconjugation of analytic functions.. These questions have possible applications in several branches of Mathematics. While these questions fall squarely within the framework of Geometric Function Theory. This subject has its roots in applications to the natural sciences and engineering. Particular examples are Conformal Field Theory and Statistical Mechanics. In the latter subject, Percolation Theory uses heavily the concept of conformal invariance and its generalizations, to which this proposal is partly dedicated.

PI建议研究某些非分析信息如何确定地图的分析特性。两个表面之间的共形图是一个在每个点保持角度的。这是一个非常限制的假设，很有趣的是，可以从其他拓扑，组合，代数，设定理论或分析假设中推论出此类地图的性质在多大程度上。一个例子是确定表面的保形类型，即表面是否与模型表面相同。在较高维度的情况下，使用生物形态等效性代替了形式等效性。 PI提议研究的一些具体主题是：相结合类型的表面由相关网的组合特性确定，EB Vinberg提出的一个问题以及与L. Rubel提出的问题相关的两个主题，即最大生长功能，这些问题是最大的生长功能具有有限的批判性和渐近值有限，并且可以从分析性质内态的相关半群和分析功能的半缀合的知识中回收复杂歧管的研究。这些问题可能在数学的多个分支中应用。尽管这些问题直接属于几何函数理论的框架。该主题源于自然科学和工程的应用。 特定的例子是保形场理论和统计力学。在后一个主题中，渗透理论在很大程度上利用了保形不变性及其概括的概念，该提议部分专门。