FRG: Collaborative Research: Fourier analytic and probabilistic methods in geometric functional analysis and convexity

这个项目的目的是将傅立叶分析、仿射凸几何、几何泛函分析、概率论和组合学的工具结合起来，解决几何、分析以及应用数学和计算机科学的各个领域中出现的问题。在技术层面上，重点是研究(通常是高维的)凸体、随机矩阵、高斯测量和过程的性质，以及逼近问题。计划研究的具体样本方向涉及切片问题、马勒猜想、高斯相关性猜想、函数类的组合维度、随机矩阵的奇数、信号重构(特别是压缩传感)以及与量子信息理论的链接。联合、集中的努力有望带来新的见解，以更好地了解参与者各自的研究领域，这些领域虽然相关，有时也有重叠，但并不完全相同，而且往往采用不同的视角。包括上述方法和问题的数学领域最近进入了快速增长期。这在很大程度上是由于与计算机科学和数学物理等其他领域的众多联系。简而言之，高维凸性和应用之间的丰富联系是由于人们想要分析的系统的复杂性(例如，物理、生物或经济)：这种系统中的大量自由参数可能反映在用作模型的数学对象的大维度中。此外，几何泛函分析中的许多结果可以表述为关于高维物体在存在凸性的情况下的复杂性的陈述；这解释了与计算机科学的联系。除了研究本身，该项目的一个主要组成部分是在综合研究环境中培训博士后和研究生。这包括组织暑期学校和会议。还计划在每个机构专门举办关于该项目的讲习班和研讨会。该领域的动态增长和应用程序的丰富使其成为研究生和年轻研究人员的理想学习主题，我们希望吸引他们。将特别注意招募在数学领域任职人数不足的群体的成员。

The aim of this project is to bring together tools from Fourier analysis, affine convex geometry, geometric functional analysis, probability theory, and combinatorics to attack problems arising in geometry, analysis, and in various areas of applied mathematics and computer science. On the technical level, the focus is on the study of properties of (generally high-dimensional) convex bodies, random matrices, Gaussian measures and processes, and of approximation problems. Specific sample directions of planned research are related to the slicing problem, the Mahler conjecture, the Gaussian correlation conjecture, combinatorial dimensions of classes of functions, singular numbers of random matrices, signal reconstruction (notably, compressed sensing), and links to quantum information theory. A combined, focused effort is expected to bring new insights toward a better understanding of the participants' respective fields of research, which - while related and occasionally overlapping - are not identical and often employ different perspectives.The area of mathematics encompassing the methods and the problems described above has recently entered a period of rapid growth. In large part this is due to numerous links to other fields such as computer science and mathematical physics. In a nutshell, the wealth of connections between high-dimensional convexity and applications is due to the complexity of the systems (e.g., physical, biological or economical) that one wants to analyze: the large number of free parameters in such systems may be reflected in the large dimension of the mathematical object that serves as a model. Additionally, many results in, say, geometric functional analysis, can be presented as statements about the complexity of high dimensional objects in presence of convexity; this explains the links to computer science. In addition to research per se, a major component of this project is the training of postdocs and graduate students in an integrated research environment. This includes organization of a summer school and of a conference. Workshops and seminars devoted to the project at each institution are also planned. The dynamic growth of the area and wealth of applications makes it an ideal topic of study for graduate students and young researchers, whom we expect to attract. Special attention will be paid to recruiting members of groups under-represented in the field of mathematics.