Analytic and geometric aspects of convexity theory with applications

凸性理论的解析和几何方面及其应用

• 批准号: RGPIN-2018-05159
• 负责人: Ye, Deping
• 金额: $ 1.68万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=669281
• 关键词: Analytic; geometric; aspects; convexity; theory

The omnipresent convexity appears naturally when describing objects of interest in many mathematical related sciences. For instance, the set of quantum states in finite dimensional quantum systems and its subset containing all separable quantum states (i.e., not entangled quantum states) are convex compact sets. Hence, understanding the analytic and/or geometric aspects of convexity theory is in great demand and is prerequisite to understanding convex objects of interest. To this end, my proposed program of research aims to study properties of convexity theory, and to apply tools from convexity theory to attack problems arising in other areas such as mathematical physics, partial differential equations, probability theory and (quantum) information theory. ******One part of the proposed program of research deals with the modern geometric extensions of the Brunn-Minkowski theory and its dual. The emphasis is on understanding the properties of affine invariants (e.g., affine and geominimal surface areas), establishing new affine isoperimetric and isocapacitary inequalities, and solving Minkowski type problems (e.g., the Orlicz-Minkowski problem as well as its dual and/or polar analogues). Several projects are proposed to further explore the connections of the Brunn-Minkowski theory of convex bodies and its dual with partial differential equations, with particular attention paid to geometric inequalities, (polar or dual) Minkowski type problems, and the development of a dual Brunn-Minkowski theory for various variational functionals. ******Another part of the proposed program of research lies in the areas of geometrization of log-concave measures (or functions) and the information theory. The geometrization of log-concave measures can be viewed as the functional analogue of the Brunn-Minkowski theory. I aim to build a framework of the functional Lp and/or Orlicz Brunn-Minkowski theories for log-concave or quasi-concave functions, extend the entropy power inequality to their Lp and/or Orlicz analogues, and discover new geometric inequalities for quantum states. It is expected that these projects help further advance the connections between information theory and the Brunn-Minkowski theory, with particular attention paid to geometric inequalities for quantum states, and the generalizations of the entropy power inequality and Fisher information (in both classical and quantum settings). ******I will continue my commitment to the training of (undergraduate and graduate) students and postdocs. This program of research includes multiple diverse and interdisciplinary research topics, which makes it easier to attract Highly Qualified Personnel (HQP) and helps produce knowledgeable mathematicians of next generation.

在许多与数学相关的科学中，描述感兴趣的对象时，无所不在的凸性自然出现。例如，有限维量子系统中的量子态集合及其包含所有可分离量子态（即非纠缠量子态）的子集是凸紧集。因此，理解凸理论的解析和/或几何方面是非常必要的，也是理解感兴趣的凸对象的先决条件。为此，我提出的研究计划旨在研究凸性理论的性质，并应用凸性理论的工具来解决数学物理、偏微分方程、概率论和（量子）信息论等其他领域出现的问题。******研究计划的一部分涉及布伦-闵可夫斯基理论及其对偶的现代几何扩展。重点是理解仿射不变量的性质（例如，仿射和极小表面积），建立新的仿射等周和等容量不等式，并解决Minkowski型问题（例如，Orlicz-Minkowski问题及其对偶和/或极性类似问题）。为了进一步探讨凸体的Brunn-Minkowski理论及其对偶与偏微分方程的联系，提出了几个项目，特别关注几何不等式，（极或对偶）Minkowski型问题，以及各种变分泛函的对偶Brunn-Minkowski理论的发展。******提出的研究计划的另一部分在于对数凹测度（或函数）和信息论的几何化领域。对数凹测度的几何化可以看作是布伦-闵可夫斯基理论的泛函类比。我的目标是为对数凹或准凹函数建立一个泛函Lp和/或Orlicz Brunn-Minkowski理论的框架，将熵幂不等式扩展到它们的Lp和/或Orlicz类似物，并发现量子态的新的几何不等式。预计这些项目有助于进一步推进信息论和布伦-闵可夫斯基理论之间的联系，特别关注量子态的几何不等式，以及熵幂不等式和费舍尔信息的推广（在经典和量子设置中）。******我将继续致力于培养（本科生和研究生）学生和博士后。该计划的研究包括多个多元化和跨学科的研究课题，这使得它更容易吸引高素质人才（HQP），并帮助培养知识渊博的下一代数学家。