Probabilistic aspects of asymptotic geometric analysis

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本课题属于渐近几何分析领域，主要研究凸几何和泛函分析中的高维现象，特别是概率论在该领域的应用。该项目有两个独立的重点领域，大型随机矩阵的光谱行为和高维凸体中的体积分布。提出的研究这些问题的技术包括传统的随机矩阵理论的概率工具和渐近几何分析，如矩量法和测度现象的集中法；以及来自理论概率论的新技术，比如斯坦的方法。在随机矩阵领域，研究了随机矩阵的范数波动和特征值波动。这里的一个主要目标是锐化和扩展由于提议者和其他人而产生的集中结果。作者还将继续研究大随机Toeplitz矩阵的行为，这是一类最近才在文献中被研究的随机矩阵，以及相关的随机矩阵集合。在凸几何领域，他将继续研究高维凸体截面体积的高斯近似定理，以及高维凸体中体积分布的相关问题。这里考虑的问题的一个重要特征是它们本质上的高维性质，这是该领域的典型特征。例如，许多定量几何问题在足够低的维数中是微不足道的，但当维数变得非常大时，就会出现深刻而意想不到的现象。在随机矩阵的情况下，主要关注的是对于大型有限矩阵的非平凡结果，而不是当大小变得无限时更传统的极限结果。这种高但有限维的方面对于几何、统计学或计算机科学等领域的潜在应用至关重要。每当研究涉及大量参数的定量问题时，就会出现高维现象。除了像凸几何和泛函分析这样的纯数学领域是这个项目的重点之外，这些问题自然会出现在统计学、计算机科学、数学生物学和物理学等不同领域。所谓的维数诅咒在许多定量领域都很常见，这表明处理这类问题的潜在困难。另一方面，渐近几何分析的目标是识别在依赖于大量参数的系统中出现的规律或模式，但对于少量参数则不明显。因此，高维在某些方面成为一种祝福而不是诅咒。概率论是这一领域的核心工具，因为许多问题以明确的方式涉及随机性，也因为许多先验的确定性问题可以通过引入概率观点来澄清。也就是说，当一个人以一种适当的随机方式看待事物时，高维中出现的模式就会变得明显。这个项目处理这两种类型的概率在高维现象中的应用。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project is in the field of asymptotic geometric analysis, which deals with high-dimensional phenomena in convex geometry and functional analysis, and more specifically in applications of probability theory in this field. The project has two separate areas of emphasis, the spectral behavior of large random matrices and the distribution of volume in high-dimensional convex bodies. The proposed techniques for studying these problems include both traditional probabilistic tools of random matrix theory and asymptotic geometric analysis, like the method of moments and the concentration of measure phenomenon; as well as novel techniques from theoretical probability, like Stein's method. In the area of random matrices the proposer studies the fluctuations of norms and eigenvalues of random matrices. One main goal here is to sharpen and extend concentration results due to the proposer and others. The proposer will also continue his study of the behavior of large random Toeplitz matrices, a class of random matrices which has only recently been investigated in the literature, and related random matrix ensembles. In the area of convex geometry the proposer will continue his work on Gaussian approximation theorems for volumes of sections of high-dimensional convex bodies, as well as related problems about how the volume is distributed in high-dimensional convex bodies. An important feature of the problems considered here, which is typical of the field, is their intrinsically high-dimensional nature. For example, many quantitative geometric questions are trivial in a sufficiently low number of dimensions, but deep and unexpected phenomena can arise when the dimension becomes very large. In the context of random matrices, the primary interest is in results which are nontrivial for large finite matrices, as opposed to more-traditional limit results as the size becomes infinite. This high- but finite-dimensional aspect is crucial for potential applications to fields like geometry, statistics, or computer science.High-dimensional phenomena arise whenever one studies quantitative problems involving a large number of parameters. Besides areas in pure mathematics like convex geometry and functional analysis which are the focus of this project, such problems naturally arise in fields as diverse as statistics, computer science, mathematical biology, and physics. The so-called curse of dimensionality is familiar in many quantitative fields, indicating the potential difficulty of dealing with such problems. The goal of asymptotic geometric analysis, on the other hand, is to identify regularity or patterns that arise in systems that depend on a very large number of parameters, but which are not apparent for a small number of parameters. Thus high-dimensionality in some ways becomes a blessing rather than a curse. Probability theory is a central tool in this field, both because many problems involve randomness in an explicit way, and because many a priori deterministic problems can be clarified by introducing a probabilistic viewpoint. That is, patterns that arise in high dimensions become apparent when one looks at things in an appropriately random way. This project deals with both these types of applications of probability to high-dimensional phenomena.