Random matrices and geometric functional analysis

这个数学研究项目属于概率论、凸几何和泛函分析的界面。其中一个主要的研究领域是随机矩阵的非渐近理论，这是一个新兴的快速发展的研究领域，它分析一个大但固定大小的随机矩阵的谱特征，并努力获得具有高概率有效的界。这种边界的要求在凸几何和几何泛函分析中自然出现，其中使用随机矩阵来研究高维凸体的典型部分。本研究的另一个方向是非对称凸体的几何，其中概率方法也起着至关重要的作用。这个数学研究项目的几个组成部分是由计算机科学和工程中的问题所激发的。这些问题包括计算机断层扫描中出现的信号重建问题，以及在发布统计信息时保护隐私的相关问题。Rudelson和他的合作者将在概率和几何泛函分析之间建立新的联系，目的是研究随机矩阵的光谱特性。随机矩阵已经成为无线通信中信号重构的主要模型，并且可以更好地理解许多计算机科学算法的收敛性。

This mathematics research project belongs to the interface of probability, convex geometry, and functional analysis. One of the main areas of investigation is the non-asymptotic theory of random matrices, a new and rapidly developing area of research, which analyzes spectral characteristics of a random matrix of large, but fixed size, and strives to obtain bounds, which are valid with high probability. The requirements for such bounds arise naturally in convex geometry and geometric functional analysis, where random matrices are used to study typical sections of a high-dimensional convex body. Another direction of this research is the geometry of non-symmetric convex bodies, where the probabilistic methods also play a crucial role.Several components of this mathematics research project are motivated by problems in computer science and engineering. These include signal reconstruction problems arising in computer tomography, as well as questions related to protecting privacy while releasing statistical information. Rudelson and his collaborators will establish new connections between probability and geometric functional analysis with the objective to study the spectral properties of random matrices. Random matrices have become the main model for signal reconstruction in wireless communication and can lead to a better understanding of the convergence of many computer science algorithms.