Probabilistic Approach in Geometric Functional Analysis

PI计划使用概率的方法来研究度量空间和凸体。对于凸几何中的许多问题，如求凸体的具有某些优良性质的截面或用另一个具有更好结构的体逼近凸体，其显式构造是未知的。在这些情况下，随机结构被证明是非常有效的。通常可以定义一个随机截面或近似值，并以高概率证明它具有所需的属性。这种方法结合了先进的概率工具，如测量浓度，导致了凸几何和泛函分析的重大发现，包括Dvoretzky定理和逆Santalo不等式。PI打算应用概率方法来研究不同度量空间的嵌入，例如群，图等，Banach空间。这些度量空间通常配备有概率测度，其具有强集中性。通过研究随机点的像的分布，可以得到关于将这种度量空间嵌入到Banach空间的重要信息。所提出的研究的另一个方向与凸体的研究有关，凸体不一定是对称的。包括PI在内的几位研究人员在过去3年中获得的结果表明，许多以前只知道凸对称体的性质在没有对称性假设的情况下仍然成立。PI计划继续他在这一领域的工作，目的是构建一个理论的一般凸体，这将是平行于现有的理论凸对称机构。拟议的研究将提供新的连接之间的功能分析，凸几何和概率。嵌入图到Banach空间的结果将有助于寻找图的小分离器。这可以导致在几个计算机科学问题，特别是在偏微分方程的数值解更有效的算法的建设。非对称凸集，这是所提出的研究的主要对象之一，自然出现在广泛的一类优化问题。因此，更好地了解这种机构的结构将导致构建更有效的优化算法。与凸体有关的随机过程在控制理论中也有相当大的意义。

The PI plans to study metric spaces and convex bodies using the methods of Probability. For many problems of Convex Geometry, like finding sections of a convex body with certain nice properties or approximating a convex body by another body, having a better structure, the explicit constructions are unknown. In these cases random constructions were proved to be very effective. It is often possible to define a random section or approximation and to show that it has the desired property with high probability. This approach combined with advanced probabilistic tools, like measure concentration, led to major discoveries in Convex geometry and Functional Analysis, including Dvoretzky's Theorem and inverse Santalo Inequality. The PI intends to apply the probabilistic method to study embeddings of different metric spaces, such as groups, graphs etc., into Banach spaces. These metric spaces are often equipped with a probability measure, which has strong concentration properties. Then it is possible to obtain significant information about the embedding of such metric space into a Banach space by studying the distribution of the image of a random point. Another direction of the proposed research is related to the study of convex bodies, which are not necessary symmetric. The results obtained in the last 3 years by several researchers, including the PI, show that many properties, previously known only for convex symmetric bodies, hold without the assumption of symmetry. The PI plans to continue his work in this area with the aim of constructing a theory of general convex bodies, which would be parallel to the existing theory of convex symmetric bodies.The proposed research will provide new connections between Functional Analysis, Convex Geometry and Probability. The results on embedding graphs into Banach spaces will be useful for finding small separators of graphs. This can lead to construction of more effective algorithms in several Computer Science problems, in particular in numerical solution of partial differential equations. The non-symmetric convex sets, which are one of the main objects of the proposed research, arise naturally in a broad class of optimization problems. So, better understanding of the structure of such bodies will result in constructing more effective optimization algorithms. The stochastic processes related to a convex body are also of considerable interest for the Control Theory.