Geometric analysis of high-dimensional convex bodies and random processes

拟议的研究是在泛函分析，凸几何，赋范空间的渐近理论和概率论的相关主题。从这些领域发展理论和工具，我打算研究有限维凸体之间的定量关系，凸体的各种参数和性质，随机矩阵和随机变量序列的行为，以及在纯数学和应用数学的各种相关领域提供应用。该项目的意义是扩大知识，加强数学的重要分支之间的联系，即凸几何，Banach空间的局部理论和概率论，以及在纯数学和应用数学的相关领域找到新的应用。计划解决几个开放的问题。在这些方向的调查将丰富新的思想，方法和技术的领域。研究结果也将在相关领域得到应用。 该项目涉及渐近理论及其非对称对应物的长期开放问题。特别是，我计划工作等著名的问题，如对偶熵和覆盖数，巴拿赫-马祖尔距离之间的任意凸体，照明问题，和其他人。概率方法在渐近理论中起着至关重要的作用。该项目还涉及研究随机矩阵的最小非平凡特征值和相应的偏差不等式，以及调查随机变量序列的顺序统计量的行为。在上个世纪，人们对顺序统计进行了深入的研究，但对建议中考虑的一般情况几乎一无所知。

The proposed research is on related topics in Functional Analysis, Convex Geometry, Asymptotic Theory of Normed Spaces, and Probability Theory. Developing theory and tools from these areas I intend to investigate quantitative relations among finite-dimensional convex bodies, various parameters and properties of convex bodies, behavior of random matrices and sequences of random variables, as well as to provide applications to various related areas in Pure and Applied Mathematics. The project's significance is to expand the knowledge and to strengthen the bonds between important branches of mathematics, namely, Convex Geometry, Local Theory of Banach Spaces, and Probability Theory, as well as to find new applications to related fields in Pure and Applied Mathematics. It is planed to solve several open problems. Investigations in these directions will enrich the areas with new ideas, methods, and techniques. The results will also have applications in related areas.    The project deals with longstanding open problems of Asymptotic Theory and its non-symmetric counterpart. In particular, I plan to work on such well-known problems as duality of entropy and covering numbers, Banach-Mazur distance between arbitrary convex bodies, Illumination problem, and others. Probabilistic technique plays a crucial role in Asymptotic Theory. The project also deals with the study of the smallest non-trivial eigenvalue of a random matrix and corresponding deviation inequalities, as well as with investigation of the behavior of order statistics of a sequence of random variables in terms of the sequence of their first moments. Order statistics have been intensively studied during last century but almost nothing is known about the general case considered in the proposal.