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.

在项目的焦点上，ab骨gindikin有两个整体几何形状的方面。首先，是在国旗域中的分析，它扩展了非紧密组合的Hermitian对称空间中的分析。第一步是我们称为“行为函数”的基本功能的类似物的定义。此类功能的众所周知的示例是对称域（或Jordan代数）的规范功能，但是对于Flag域而言，有更丰富的新功能集合。使用这些功能，我们希望获得几个明确的公式和结果：Riemann对称空间的Stein邻域的描述，可以在其中塑性地扩展Schmid方程的解决方案，旗帜域中复杂周期的复杂周期的参数化，HUA的概括 - Poisson积分和HUA集成的概述，以及它们在其上的集成界及其计算的群体，以下内容及其计算的群体，计算的群体属性，普遍化的penrose变换等。这些结构中的重要作用是由非凸形管域中的共同体的边界值发挥的。该项目的另一个方向是公理的holosphers及其应用方法。几年前，在解决gelfand问题的过程中，我给了一个复杂流形的子手族的家族，提供了一个显式的局部反转公式的相应几何学问题。对于复杂的半神经谎言基团的霍斯赛，满足了这些条件，这是倒入h鼠转变而无需使用组结构的方法。现在，我们以某种方式扩展了这种公理，使得某些非对称均匀歧管的halosphical变换可能会倒转。积分几何形状是几何分析的方向，它将歧管与几何结构的分析联系起来。积分几何形状的理念是，几何结构比组不变性更一般，这为发展丰富的多维分析提供了基础，具有重要的应用，用于分析均质流形，复杂分析，非线性微分方程，数学物理学等。积分几何学是计算机整体量表的理论基础。