CAREER: High-Dimensional Geometry and Its Applications

该项目是在高维几何分析领域，其中包括一个家庭的相对较新的和非常活跃的数学领域，在调和和泛函分析，凸几何和概率的接口出现。这些领域的主要焦点是高维的研究;考虑通常集中在几何对象，如凸体和超曲面，凸函数和凹函数，以及具有某些几何特征的随机向量。我们在低维的经验似乎表明，当维数变得非常大时，物体的几何性质变得越来越复杂，难以研究。然而，许多美好的，有时令人惊讶的性质出现在高维中。这种性质被非正式地称为“高维现象”。对这种现象的研究对于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以及数据科学方面。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。首席研究员将为初级研究人员组织两次研讨会，在分配的时间内进行研究讨论，并由该领域的主要专家进行简短的讲座。这些讲习班旨在帮助初级数学家发展新的兴趣，并创造新的合作。此外，在北方格鲁吉亚，首席研究员与肯尼索州立大学的尤利娅·巴班科共同举办了一个妇女数学研讨会，首席研究员一直在研究高维几何的几个方面。本项目的一个重要方向是Brunn-Minkowski型不等式的研究。更具体地说，有趣的问题是，这些不等式如何在某些对称性和凸性假设下得到改善。研究这些问题的技术涉及调和分析和凸几何的思想。此外，主要研究者应继续研究小球不等式及其在信息论中的应用。过去主要研究者研究的重要对象之一是关于凸集的分布的噪声敏感性，主要研究者将继续研究这个数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性质，例如照明数。首席研究员过去曾研究过这个数字，并正在努力改进目前已知的对这个数量的估计。

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.