Mathematical Sciences: Complex Integral Geometry and Analysis at Flag Domains

数学科学：复积分几何和标志域分析

• 批准号: 9706836
• 负责人: Simon Gindikin
• 金额: $ 8.3万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706836&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Complex; Integral; Geometry

Absract Gindikin At the focus of the project there are 2 aspects of integral geometry. First, it is the analysis in flag domains which extends the analysis in noncompact Hermitian symmetric spaces. The first step is the definition of analogs of elementary functions which we call the determinant functions. The well known examples of such functions are the norm-functions for symmetric domains (or Jordan algebras), but for flag domains there is a richer collection of new functions. Using these functions we hope to obtain several explicit formulas and results: descriptions of Stein neighborhood of Riemann symmetric spaces where it might be possible to holomorphically extend solutions of the Schmid equations, parametrizations of complex cycles in flag domains, generalizations of the Hua - Poisson integrals and the Hua equations for them, their computations on the language of integral geometry and multidimensional residues, the generalized Penrose transform etc. The essential role in these constructions are played by the boundary values of the cohomology in nonconvex tube domains. Another direction in this project is an axiomatic of the method of horospheres and its applications. Several years ago in the process of solving the Gelfand problem I gave some axiomatic conditions on a family of submanifolds of a complex manifold providing an explicit local inversion formula of the corresponding problem of integral geometry. These conditions are satisfied for the horospheres on complex semisimple Lie groups, and it is the way to invert the horospherical transform without using group structures. Now we extend this axiomatic in such a way that it becomes possible to invert the horospherical transform for some nonsymmetric homogeneous manifold. The integral geometry is a direction of geometric analysis which connects analysis on manifolds with geometrical structures on them. The philosophy of integral geometry is that there are geometrical structures more general than group invariance which give a base for the development of a rich multidimensional analysis with important applications to analysis on homogeneous manifolds, complex analysis, nonlinear differential equations , mathematical physics, etc. Integral geometry is a theoretical base of computer tomography.

摘要 Gindikin 该项目的重点是积分几何的两个方面。首先，标志域中的分析扩展了非紧埃尔米特对称空间中的分析。第一步是定义基本函数的类似物，我们称之为行列式函数。此类函数的众所周知的示例是对称域（或 Jordan 代数）的范数函数，但对于标志域，有更丰富的新函数集合。使用这些函数，我们希望获得几个明确的公式和结果：黎曼对称空间的 Stein 邻域的描述，其中可能全纯扩展 Schmid 方程的解，标志域中复杂循环的参数化，Hua - Poisson 积分的推广以及它们的华方程，它们对积分几何语言和多维留数的计算，广义彭罗斯变换等。这些构造中的重要作用是由非凸管域中的上同调。该项目的另一个方向是星球方法及其应用的公理化。几年前，在解决格尔凡德问题的过程中，我给出了复流形的子流形族的一些公理条件，提供了相应的积分几何问题的显式局部反演公式。复半单李群上的星球面满足这些条件，是不使用群结构的星球面变换的逆方法。现在我们以这样的方式扩展这个公理，使得对于某些非对称齐次流形的星球变换进行逆变换成为可能。积分几何是将流形分析与流形上的几何结构联系起来的几何分析方向。积分几何的哲学是存在比群不变性更普遍的几何结构，这为丰富的多维分析的发展奠定了基础，在齐次流形分析、复分析、非线性微分方程、数学物理等方面具有重要的应用。几何学是计算机断层扫描的理论基础。