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地图的分析属性。两个曲面之间的保角映射是保持每个点的角度不变的映射。这是一个非常严格的假设，有趣的是要理解在何种程度上可以从其他拓扑、组合、代数、集合论或分析假设推导出这样的映射的性质。一个例子是确定曲面的保形类型，即曲面是否与模型曲面保形等价。在高维情形下，用双全纯等价代替共形等价。PI提出要研究的一些具体主题是：由相关网络的组合性质确定其共形类型的曲面，EB Vinberg提出的一个问题，以及与L. Rubel，即，最大增长函数，有许多临界值和渐近值，以及研究复流形的双全纯型可以恢复从知识的相关半群的解析自同态和共轭的解析函数。这些问题在数学的几个分支中都有可能应用。而这些问题正好属于几何函数论的框架。这门学科起源于自然科学和工程学的应用。 特别的例子是共形场论和统计力学。在后一个主题中，渗流理论大量使用共形不变性及其推广的概念，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.