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建议研究某些非解析性信息如何决定映射的解析性。两个表面之间的保角映射是在每个点保持角度的映射。这是一个非常严格的假设，了解这种映射的性质在多大程度上可以从其他拓扑、组合、代数、集合论或分析假设中推断出来是很有趣的。一个例子是确定曲面的共形类型，即曲面是否与模型曲面共形等效。在高维情况下，用生物全纯等价代替保形等价。PI建议研究的一些具体课题是：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，即具有有限多个临界值和渐近值的极大生长函数，以及复流形的研究，其生物全纯类型可以从解析自同态的相关半群和解析函数的半共轭的知识中恢复。这些问题可能应用于数学的几个分支。而这些问题完全属于几何函数理论的范畴。这门学科植根于自然科学和工程学的应用。具体的例子是共形场论和统计力学。在后一个主题中，渗透理论大量使用共形不变性的概念及其推广，本提案部分致力于此。