Mathematical Sciences: Complex Integral Geometry and Analysis at Flag Domains

在项目的重点有两个方面的积分几何。首先，标志域上的分析扩展了非紧厄密对称空间上的分析。第一步是定义初等函数的类似物我们称之为行列式函数。这类函数的著名示例是对称域（或Jordan代数）的规范函数，但对于标志域，则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：关于可以全纯扩展Schmid方程解的Riemann对称空间的Stein邻域的描述，复环在flag域上的参数化，Hua - Poisson积分和它们的Hua方程的推广，它们在积分几何语言和多维残数上的计算，广义Penrose变换等。非凸管域上同调的边值在这些构造中起着重要的作用。这个项目的另一个方向是星座及其应用方法的公理化。几年前在求解Gelfand问题的过程中，我给出了复流形的一组子流形的一些公理化条件，并给出了相应的积分几何问题的显式局部反演公式。对于复半单李群上的星球，这些条件都满足，这是不使用群结构反演星球变换的一种方法。现在我们扩展这个公理，使它可以对一些非对称齐次流形的全球变换求反。积分几何是将流形的分析与流形上的几何结构联系起来的几何分析的一个方向。积分几何的哲学是存在比群不变性更一般的几何结构，它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、数学物理等方面具有重要的应用。积分几何是计算机断层扫描的理论基础。

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.