Functional, geometric and matrix inequalities and applications

这项研究涉及两个主要方向：尖锐的泛函和几何不等式以及矩阵分析，以期将其应用于非线性偏微分方程、数学物理和量子信息理论。在泛函和几何不等式中，建议的研究项目将主要集中在Sobolev型和Hardy型不等式，这是分析中最常用的两个不等式。它们在变分、偏微分方程、几何等中出现的几个问题中发挥了关键作用。我和我的合作者一起，成功地开发了一些新的泛函和几何不等式的尖锐版本，以及它们在几个重要情况下的应用。尽管如此，仍有各种有趣而重要的问题是悬而未决的。这项提议的目的之一就是为这些问题做出贡献。具体地说，这项研究的一部分目的是开发新的技术和方法来研究各种几何环境下的Sobolev型不等式的尖锐形式、最佳常数和优化器。我还将致力于寻找精化版本的问题，这些精化版本将提供对文献中已知的Hardy型不等式的简单和直接的理解，以及它们的优化器的存在和不存在。这项研究提案还将集中于研究最优条件和发展双权Hardy不等式所适用的新概念，并利用它们来推导新类型的不等式。我还将研究这些概念之间的关系。还将研究Hardy型不等式和Sobolev型不等式在偏微分方程和谱理论中的应用。在矩阵分析中，我的主要兴趣是半正定矩阵集上的量子距离，这是衡量两个数据点(如量子力学中的量子态)之间的可区分性的主要工具。拟议的研究的一部分是建立矩阵分析中涉及迹、行列式、特征值等的新的或改进的不等式版本，并使用它们来推导合适的量子距离。还将研究量子距离的几个重要属性，如联合凸性、数据处理不等性、中间性、度量性，以验证新的量子距离是有意义的，并在从量子信息到机器学习和信号处理的许多实际应用中具有实验益处。这些量子距离在重心问题、几何问题、Karcher平均问题、量子信息论和矩阵优化中的应用也将被研究。本科生和研究生将在我的指导下接受研究培训，积极参与这个拟议的研究项目的两个方向。

This research proposal concerns topics in two main directions: sharp functional and geometric inequalities and matrix analysis, with a view toward applications to nonlinear partial differential equations, mathematical physics and quantum information theory. In functional and geometric inequalities, the proposed research project will mainly focus on the Sobolev type inequalities and the Hardy type inequalities, which are the two most frequently used inequalities in analysis. They play a crucial role in several problems arising in the calculus of variations, partial differential equations, geometry, etc. I, jointly with my collaborators, have successfully developed the sharp versions of some functional and geometric inequalities and their applications in several important cases. Still, there are various interesting and important problems that are open. One of the purposes of this proposal is to contribute toward these questions. In particular, part of this proposed research aims to develop new techniques and methods to study the sharp versions, best constants and the optimizers of the Sobolev type inequalities in various geometric settings. I will also work on the problems of finding the refined versions that will provide simple and direct understandings of known Hardy type inequalities in the literature, as well as the existence and nonexistence of their optimizers. This research proposal will also focus on investigating optimal conditions and developing new notions for which the two-weight Hardy inequalities hold, and using them to derive new types of inequalities. I will also study the relationships between these notions. Applications of the Hardy type inequalities and Sobolev type inequalities to partial differential equations and spectral theory will also be investigated. In matrix analysis, my main interests are in quantum distances on the set of positive semi-definite matrices, which are the main tools to measure the distinguishability between two data points (such as quantum states in quantum mechanics). Part of the proposed research is to establish new or refined versions of inequalities in matrix analysis involving trace, determinant, eigenvalues, etc, and use them to derive suitable quantum distances. Several important properties of a quantum distance such as joint convexity, data processing inequality, in-betweenness, metric property will also be studied to verify that the new quantum distances are meaningful and are experimentally beneficial in a number of applications varying from quantum information to machine learning and signal processing. The applications of these quantum distances in barycenter problems, geometry, Karcher mean problems, quantum information theory and matrix optimization will also be investigated. Undergraduate and graduate students will actively participate in both directions of this proposed research project by receiving research training under my supervision.