Geometric analysis of high-dimensional convex bodies and random processes

建议的研究是泛函分析、凸几何、赋范空间的渐近理论和概率论的相关课题。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性质，以及在纯数学和应用数学中的各个相关领域的应用。该项目的意义是扩大知识，加强重要数学分支之间的联系，即凸几何、Banach空间的局部理论和概率论，并在纯粹数学和应用数学的相关领域找到新的应用。计划解决几个悬而未决的问题。这些方向的研究将为这些领域提供新的思想、方法和技术。研究成果也将在相关领域得到应用。该项目解决了渐近理论及其非对称对应的长期悬而未决的问题。特别是，我计划研究一些众所周知的问题，如熵和覆盖数的对偶性、任意凸体之间的Banach-Mazur距离、照明问题等。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，以及研究随机变量序列的一阶矩序列的顺序统计量的行为。顺序统计量在上个世纪得到了深入的研究，但对该提议中所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.