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

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A B S T R A C Tof the NSF proposal DMS-0070462" Reflection Principle in Higher Dimensions: Geometric,Analytic and Algebraic Approaches"Principal Investigator - Sergey PinchukThe 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 betweencertain domains ( or CR mappings between manifolds) imposessignificant restrictions on these domains ( manifolds ) as well as on themappings. These restrictions give rise to various, sometimes unexpected,phenomena of analytic continuation in several complex variables. The goal ofthe proposed research is to solve some concrete old problems and to providea further progress in the study of analytic continuation and relatedareas. The main method of investigation is the reflection principle, whichwill be combined with other methods from analysis, geometry, algebra anddifferential equations.Complex analysis has been used as a powerful tool in mathematics and itsapplications for a long time and has a tendency to a larger role in certainareas. For example, the famous "edge of the wedge" theorem, which is now oneof the main ingredients of the reflection principle, was discovered by aphysicist N. Bogolyubov with respect to his research in quantum fieldtheory. Another important object in theoretical physics - the Heisenberggroup -is closely connected with the group of holomorphic automorphisms of theunit ball in the 2-dimensional complex space.I believe that this research program will be useful not only for complexanalysis but also for the strengthening its ties with other areas ofmathematics and sciences by means of potential applications. It will alsoinfluence mathematical education at Indiana University via involvement ofgraduate students.