Loewner theory and holomorphic mappings in several variables

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The theory of univalent functions (i.e. one-to-one complex-analytic functions on the unit disc in the complex plane) is one of the most beautiful topics in one complex variable. There are many remarkable theorems dealing with extremal problems for this class of functions. Perhaps the most famous is the Bieberbach Conjecture for the Taylor series coefficients of univalent functions, solved in 1985 by Louis de Branges, but still sometimes known as the Bieberbach Conjecture. One of the most important techniques used to solve the Bieberbach Conjecture is a variational technique known as the Loewner method, in which a univalent function is embedded in an expanding flow of such functions governed by a differential equation. ******The main objective of my research is to see which aspects of univalent function theory in one variable can be generalized to several complex variables, and which cannot. In cases where a generalization is possible, it is frequently the case that new methods of proof are needed. This is certainly the case with the Loewner method in several variables, one of my main interests. Results of this type contribute to a deeper understanding of univalent function theory in one variable, and to the variety of methods available to study mapping questions in several complex variables. ******The results will contribute to maintaining a knowledge base which is available to solve technical problems when needed by society. They will have particular interest for mathematicians working in complex analysis.