Determining Analytic Properties of Maps from Non-Analytic Data

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

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.