Analytic and geometric aspects of convexity theory with applications

在描述许多与数学相关的科学中感兴趣的对象时，无所不在的凸性自然地出现。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)的子集都是凸紧集。因此，理解凸性理论的解析和/或几何方面是非常必要的，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性理论的性质，并将凸性理论的工具应用于解决其他领域出现的问题，如数学物理、偏微分方程、概率论和(量子)信息论。 拟议的研究计划的一部分涉及Brunn-Minkowski理论及其对偶的现代几何扩展。重点是了解仿射不变量的性质(例如，仿射和几何最小表面积)，建立新的仿射等参等容不等式，以及解决Minkowski类型的问题(例如，Orlicz-Minkowski问题及其对偶和/或极类似问题)。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类型问题，以及各种变分泛函的对偶Brunn-Minkowski理论的发展。 拟议的研究计划的另一部分是对数凹度量(或函数)的几何化和信息论。对数凹测度的几何化可以看作是Brunn-Minkowski理论的泛函类比。我的目标是建立关于对数凹函数或准凹函数的泛函LP和/或Orlicz Brunn-Minkowski理论的框架，将熵功率不等式推广到它们的LP和/或Orlicz类似物，并发现新的量子态的几何不等式。预计这些项目有助于进一步推进信息论和Brunn-Minkowski理论之间的联系，特别关注量子态的几何不等式，以及(在经典和量子环境中)熵幂不等式和Fisher信息的推广。 我将继续致力于培养(本科生和研究生)学生和博士后。这一研究计划包括多个不同的跨学科研究主题，这使得吸引高素质人才(HQP)变得更容易，并有助于培养下一代知识渊博的数学家。

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.