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.

Absract Gindikin 该项目的重点是积分几何的两个方面。第一，旗域上的分析是对非紧埃尔米特对称空间上的分析的推广。第一步是定义初等函数的类似物，我们称之为行列式函数。这类函数的著名例子是对称域（或约当代数）的范数函数，但对于旗域，有更丰富的新函数集合。利用这些函数，我们希望得到几个显式的公式和结果：Riemann对称空间的Stein邻域的描述，Schmid方程的全纯延拓的可能性，旗域上复圈的参数化，Hua-Poisson积分及其Hua方程的推广，积分语言的计算 几何和多维留数，广义Penrose变换等，在这些构造中起着重要作用的是非凸管域上同调的边值。本项目的另一个方向是半球方法的公理化及其应用。几年前在解决Gelfand问题的过程中，我给出了复流形的子流形族的一些公理化条件，为相应的积分几何问题提供了一个显式的局部反演公式。这些条件对于复半单李群上的单球面是满足的，这是不利用群结构而反演单球面变换的方法。现在我们扩展这个公理，使得对某些非对称齐次流形反演时球面变换成为可能。积分几何是几何分析的一个方向，它将流形上的分析与流形上的几何结构联系起来。积分几何的哲学是，存在比群不变性更一般的几何结构，这为丰富的多维分析的发展提供了基础， 在齐次流形分析、复分析、非线性微分方程、数学物理等方面有重要应用。积分几何是计算机层析成像的理论基础。