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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759178
• 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.

在描述许多数学相关科学中感兴趣的对象时，无处不在的凸度自然出现。例如，有限尺寸量子系统中的量子状态及其子集包含所有可分离量子状态（即，不是纠缠的量子状态）是凸形集合。因此，了解凸理论的分析和/或几何方面是极大的需求，并且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理论的属性，并应用凸理论的工具来攻击在其他领域（例如数学物理学，部分微分方程，概率理论和（量子）信息理论）中引起的问题。 拟议的研究计划的一部分涉及Brunn-Minkowski理论及其双重的现代几何扩展。重点在于了解仿生不变的特性（例如，仿生和土著表面积），建立了新的仿生等等等等和等距离的不平等，并解决了Minkowski类型问题（例如Orlicz-Minkowski问题，以及其双重和/或/或/或或或或或或或或或极性类似物）。提出了一些项目，以进一步探讨Brunn-Minkowski凸形机构及其双重差分方程的联系，并特别注意几何不平等，（Polar或Dual）Minkowski类型问题，以及对各种变异功能的双重Brunn-Minkowski理论的发展。拟议的研究计划的另一部分在于对数符号措施（或函数）和信息理论的几何化领域。 对数符号措施的几何化可以看作是布鲁恩·米科夫斯基理论的功能类似物。我的目标是建立一个功能性LP和/或Orlicz Brunn-Minkowski的框架，用于log-conconcave或Quasi-concave函数，将熵功率不等式扩展到其LP和/或Orlicz类似物，并发现量子状态的新的几何不平等。预计这些项目有助于进一步推进信息理论与布鲁恩·米科夫斯基理论之间的联系，并特别注意量子状态的几何不平等，以及熵能力不平等和渔民信息的概括（在古典和量子设置中）。我将继续致力于对（本科和研究生）学生和博士后的培训。该研究计划包括多种多样化和跨学科的研究主题，这使得吸引高素质的人员（HQP）变得更加容易，并帮助培养下一代知识渊博的数学家。