CAREER: High-Dimensional Geometry and Its Applications

该项目是在高维几何分析领域，它包括一系列相对较新和非常活跃的数学领域，出现在调和分析和泛函分析、凸几何和概率学的交界处。这些领域的主要焦点是高维的研究，通常集中在几何对象，如凸体和超曲面，凸和凹函数，以及具有一定几何特征的随机向量。我们在低维方面的经验似乎表明，当维度变得非常大时，对象的几何性质变得越来越复杂和难以研究。然而，许多美好的、有时令人惊讶的属性出现在高维空间。这种性质被非正式地称为“高维现象”。对这一现象的研究对计算机科学中的许多应用至关重要，特别是在某些算法的速度问题上，以及在数据科学中。该项目的教育部分侧重于支持初级研究人员，特别强调鼓励女性数学家。首席调查员将为初级研究人员组织两次讲习班，在分配的时间内开展研究讨论，并举办该领域主要专家的简短讲座。这些工作坊旨在帮助初级数学家培养新的兴趣和创造新的合作。此外，首席研究员与肯纳索州立大学的Yulia Babenko共同举办了一场针对佐治亚州北部女性数学的研讨会。首席研究员一直在研究高维几何的几个方面。这个项目的一个重要方向是研究Brunn-Minkowski类型的不等式。更具体地说，耐人寻味的问题是，在某些对称性和凸性假设下，这些不平等是如何改善的。研究这类问题的技巧涉及到调和分析和凸几何的思想。此外，首席研究员将继续研究小球不等及其在信息论中的应用。过去首席研究者研究的一个重要对象是分布关于凸集的噪声敏感性，首席研究者将继续研究这个量及其与该领域中心问题的关系。最后，该项目的另一个方面涉及凸集的组合属性，如照度数。首席调查员过去曾研究过这一数字，并正在努力改进目前对这一数字的已知估计。

The project is in the area of High-dimensional Geometric Analysis, which comprises a family of relatively new and very active areas of mathematics, arising at the interface of Harmonic and Functional Analysis, Convex Geometry and Probability. The primary focus of these areas is the study of high-dimensionality; the consideration often focuses around geometric objects such as convex bodies and hypersurfaces, convex and concave functions, as well as random vectors with certain geometric characteristics. Our experience in low dimensions seems to suggest that when the dimension becomes very large, the geometric properties of objects become more and more complicated and difficult to study. However, many nice, and sometimes surprising, properties arise in high dimensions. Such properties are informally called ``high-dimensional phenomenon''. The study of this phenomenon has been crucial for many applications in computer science, in particular in questions regarding the speed of certain algorithms, as well as in data science. The educational component of this project focuses on supporting junior researchers, with the particular emphasis placed on encouraging female mathematicians. The principal investigator will organize two workshops for junior researchers, featuring research discussions during allocated time, and short lecture courses by leading experts in the field. These workshops are designed to help junior mathematicians to develop new interests and create new collaborations. In addition, a seminar for women in mathematics in Northern Georgia is run by the principal investigator jointly with Yulia Babenko from Kennesaw State University.The principal investigator has been working on several aspects of the geometry in high dimensions. An important direction of this project concerns the study of the inequalities of Brunn-Minkowski type. More specifically, the intriguing question is how those inequalities improve under certain symmetry and convexity assumptions. The techniques involved in studying such questions involve ideas from Harmonic Analysis and Convex Geometry. In addition, the principal investigator shall continue to study small-ball inequalities and their applications to Information theory. One of the important objects studied by the principal investigator in the past is the noise sensitivity of distributions with respect to convex sets, and the principal investigator shall continue to study this quantity and its relations to the central problems in the field. Finally, a different aspect of the project concerns combinatorial properties of convex sets, such as the illumination number. The principal investigator has studied this number in the past, and is working on improving current known estimates on this quantity.