Determining Analytic Properties of Maps from Non-Analytic Data

PI建议研究某些非分析信息如何决定映射的分析性质。两个曲面之间的保角映射是指保持每个点的角度不变的映射。这是一个非常严格的假设，理解从其他拓扑、组合、代数、集合论或分析假设中可以在多大程度上推导出这类映射的性质是很有趣的。例如，确定曲面的共形类型，即曲面是否与模型曲面共形等价。在高维情形下，用双全纯等价代替共形等价。PI建议研究的一些具体主题是：由伴随网的组合性质确定其保形类型的曲面，EB Vinberg提出的一个问题，以及与L.Rubel提出的问题有关的两个主题，即具有有限多个临界值和渐近值的极大增长函数，以及其双全纯型可以从解析自同态的伴随半群的知识和解析函数的半共轭中恢复的复流形的研究。这些问题可能在数学的几个分支中有应用。而这些问题正好落在几何函数理论的框架内。这门学科植根于自然科学和工程学的应用。具体的例子有保形场理论和统计力学。在后一门学科中，渗流理论大量使用了共形不变性的概念及其推广，本提案部分致力于此。

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.