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CAREER: High-Dimensional Geometry and Its Applications

职业：高维几何及其应用

基本信息

批准号：
1753260
负责人：
Galyna Livshyts
金额：
$ 42.5万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-02-01 至 2024-01-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1753260&HistoricalAwards=false
关键词：
CAREER Dimensional Geometry Its Applications

项目摘要

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.

该项目属于高维几何分析领域，该领域包括一系列相对较新且非常活跃的数学领域，这些领域出现在调和和泛函分析、凸几何和概率的界面上。这些领域的主要焦点是高维的研究；考虑的焦点通常集中在几何对象上，例如凸体和超曲面、凸函数和凹函数，以及具有某些几何特征的随机向量。我们在低维度的经验似乎表明，当维度变得非常大时，物体的几何特性变得越来越复杂且难以研究。然而，许多美好的、有时令人惊讶的特性出现在高维度中。这些特性被非正式地称为“高维现象”。这种现象的研究对于计算机科学的许多应用至关重要，特别是在有关某些算法的速度以及数据科学的问题上。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，并由该领域的顶尖专家举办短期讲座。这些研讨会旨在帮助初级数学家培养新的兴趣并建立新的合作。此外，首席研究员与肯尼索州立大学的尤利娅·巴本科 (Yulia Babenko) 联合举办了乔治亚州北部女性数学研讨会。首席研究员一直致力于高维几何的多个方面的研究。该项目的一个重要方向涉及 Brunn-Minkowski 型不等式的研究。更具体地说，有趣的问题是在某些对称性和凸性假设下这些不平等如何改善。研究此类问题所涉及的技术涉及调和分析和凸几何的思想。此外，首席研究员应继续研究小球不等式及其在信息论中的应用。过去负责人研究的重要对象之一是分布对于凸集的噪声敏感性，负责人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，例如照明数。首席研究员过去研究过这个数字，并正在努力改进目前已知的对该数量的估计。