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

美国国家科学基金会DMS-0070462号提案的A B S T R A C T”高维反射原理：几何、解析和代数方法“的主要研究者 - Sergey Pinchuk该建议集中于以下解析延拓问题：（i）具有以下条件的域之间的真全纯映射的延拓 真实的解析边界;（ii）真实的解析流形之间的连续CR映射的解析性;（iii）全纯和CR映射沿着流形的传播.某些区域之间的双全纯和/或真全纯映射（或流形之间的CR映射）的存在性对这些区域（流形）以及这些映射施加了重要的限制.这些限制引起了各种各样的，有时意想不到的，现象的解析延续在几个复杂的变量。本文的目的是解决一些具体的老问题，为解析延拓及相关领域的研究提供一个新的进展。 复变函数的研究方法主要是反射原理，并与分析、几何、代数、微分方程等方法相结合，复变函数分析作为一种强有力的工具在数学及其应用中已经有很长的历史，并有在某些领域发挥更大作用的趋势.例如，著名的“楔边”定理，现在是反射原理的主要成分之一，是由应用物理学家N。博戈柳博夫在量子场论方面的研究。理论物理中的另一个重要对象--海森伯群--与二维复空间中单位球的全纯自同构群有着密切的联系，我相信这一研究计划不仅对复分析有用，而且通过潜在的应用加强了它与数学和科学其他领域的联系。它也将通过研究生的参与影响印第安纳州大学的数学教育。

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.