Reflection Principle in Higher Dimensions: Geometric, Analytic and Algebraic Approaches

Indiana University, Bloomington IN

Investigators

Abstract

A B S T R A C T of the NSF proposal DMS-0070462 " Reflection Principle in Higher Dimensions: Geometric, Analytic and Algebraic Approaches" Principal Investigator - Sergey Pinchuk The proposal is focused on the following problems of analytic continuation: (i) Continuation of proper holomorphic mappings between domains with real analytic boundaries; (ii) Analyticity of continuous CR mappings between real analytic manifolds; (iii) Propagation of holomorphic and CR mappings along manifolds. The existence of biholomorphic and/or proper holomorphic mappings between certain domains ( or CR mappings between manifolds) imposes significant restrictions on these domains ( manifolds ) as well as on the mappings. These restrictions give rise to various, sometimes unexpected, phenomena of analytic continuation in several complex variables. The goal of the proposed research is to solve some concrete old problems and to provide a further progress in the study of analytic continuation and related areas. The main method of investigation is the reflection principle, which will be combined with other methods from analysis, geometry, algebra and differential equations. Complex analysis has been used as a powerful tool in mathematics and its applications for a long time and has a tendency to a larger role in certain areas. For example, the famous "edge of the wedge" theorem, which is now one of the main ingredients of the reflection principle, was discovered by a physicist N. Bogolyubov with respect to his research in quantum field theory. Another important object in theoretical physics - the Heisenberg group - is closely connected with the group of holomorphic automorphisms of the unit ball in the 2-dimensional complex space. I believe that this research program will be useful not only for complex analysis but also for the strengthening its ties with other areas of mathematics and sciences by means of potential applications. It will also influence mathematical education at Indiana University via involvement of graduate students.