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

FRG：协作研究：几何泛函分析和凸性中的傅里叶分析和概率方法

• 批准号: 0652722
• 负责人: Stanislaw Szarek
• 金额: $ 30万
• 依托单位: Case Western Reserve University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652722&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Fourier; analytic

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.

该项目的目的是将工具从傅立叶分析，仿射凸几何形状，几何功能分析，概率理论和组合学中汇总起来，以攻击在几何，分析以及应用数学和计算机科学的各个领域中引起的问题。 在技​​术水平上，重点是研究（通常高维）凸体，随机矩阵，高斯测量和过程以及近似问题的性能。 计划研究的特定样本方向与切片问题，Mahler猜想，高斯相关性猜想，功能类别类别的组合维度，奇异数量的随机矩阵，信号重构（尤其是压缩感）以及与量子信息理论的联系。预计一项结合的，集中的努力将带来新的见解，以更好地理解参与者各自的研究领域，尽管相关和有时重叠的研究领域并不相同，并且经常采用不同的观点。涵盖方法的数学领域，并且上述问题最近已经进入了一段快速增长的时期。在很大程度上，这是由于与计算机科学和数学物理等其他领域的许多链接所致。简而言之，高维凸度和应用之间的丰富联系是由于要分析系统的复杂性（例如，物理，生物学或经济性）的复杂性：此类系统中的大量免费参数可能会反映在用于模型的数学对象的较大维度。此外，例如，几何功能分析中的许多结果可以作为有关在存在凸的情况下高维对物体复杂性的陈述。这解释了与计算机科学的链接。 除研究本身外，该项目的主要组成部分是在综合研究环境中对博士后和研究生进行培训。这包括组织暑期学校和会议。还计划了专门针对每个机构项目的研讨会和研讨会。该领域的动态增长和应用的财富使其成为研究生和年轻研究人员的理想研究主题，我们希望吸引他们。将特别关注数学领域中代表性不足的群体的招聘成员。