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.

该项目的目的是将傅立叶分析、仿射凸几何、几何泛函分析、概率论和组合数学的工具结合在一起，以解决几何、分析以及应用数学和计算机科学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）凸体、随机矩阵、高斯测度和过程以及逼近问题的性质。 计划研究的具体样本方向涉及切片问题、马勒猜想、高斯相关猜想、函数类的组合维数、随机矩阵的奇异数、信号重建（特别是压缩感知）以及与量子信息论的联系。联合、集中的努力有望为更好地理解参与者各自的研究领域带来新的见解，这些领域虽然相关且偶尔重叠，但并不完全相同，并且经常采用不同的观点。数学领域包括方法和上述问题近来已进入快速增长期。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，高维凸性与应用之间的丰富联系是由于要分析的系统（例如物理、生物或经济）的复杂性：此类系统中的大量自由参数可能反映出来在充当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示为关于存在凸性的高维对象的复杂性的陈述；这解释了与计算机科学的联系。 除了研究本身之外，该项目的一个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。还计划在每个机构举办专门针对该项目的讲习班和研讨会。该领域的动态增长和丰富的应用程序使其成为我们希望吸引的研究生和年轻研究人员的理想研究课题。将特别注意招募数学领域代表性不足的群体的成员。