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

NSF提案DMS-0070462“高维反射原理：几何、解析和代数方法”的主要研究者- Sergey pinchuk。该提案主要研究了解析延拓的以下问题：(i)具有实解析边界的域间固有全纯映射的延拓；（ii）实解析流形间连续CR映射的可分析性；（iii）全纯映射和CR映射沿流形的传播。某些域之间的生物全纯和/或固有全纯映射（或流形之间的CR映射）的存在对这些域（流形）以及映射施加了重要的限制。这些限制导致了一些复杂变量的解析延拓的各种，有时是意想不到的现象。本研究的目的是解决一些具体的老问题，并为分析延拓及相关领域的研究提供进一步的进展。研究的主要方法是反射原理，它将与分析、几何、代数和微分方程等方法相结合。长期以来，复杂分析一直是数学及其应用中的一种强有力的工具，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组成部分之一，是由物理学家N. Bogolyubov在量子场论研究中发现的。理论物理中的另一个重要对象——海森堡群——与二维复空间中单位球的全纯自同构群密切相关。我相信这个研究项目不仅对复杂分析有用，而且通过潜在的应用，加强它与其他数学和科学领域的联系。它还将通过研究生的参与影响印第安纳大学的数学教育。

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.