Functional, geometric and matrix inequalities and applications

本研究计划涉及两个主要方向：尖锐泛函和几何不等式以及矩阵分析，并着眼于非线性偏微分方程、数学物理和量子信息理论的应用。在函数不等式和几何不等式中，拟研究项目将主要关注Sobolev型不等式和Hardy型不等式，这是分析中最常用的两个不等式。它们在变分法、偏微分方程、几何等领域出现的一些问题中起着至关重要的作用。我和我的合作者一起，成功地开发了一些函数和几何不等式的尖锐版本，并在几个重要的情况下应用了它们。尽管如此，仍有各种有趣而重要的问题有待解决。本建议的目的之一就是为解决这些问题作出贡献。特别是，本研究的一部分旨在开发新的技术和方法来研究各种几何设置下Sobolev型不等式的尖锐版本，最佳常数和优化器。我还将致力于寻找精炼版本的问题，这些版本将提供对文献中已知的哈代型不等式的简单而直接的理解，以及它们的优化器的存在和不存在。本研究计划还将侧重于研究最优条件和发展双权Hardy不等式所适用的新概念，并利用它们推导出新的不等式类型。我还将研究这些概念之间的关系。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也将被研究。在矩阵分析中，我的主要兴趣是正半定矩阵集合上的量子距离，这是测量两个数据点之间可分辨性的主要工具（如量子力学中的量子态）。部分建议的研究是在矩阵分析中建立新的或改进版本的不等式，涉及迹，行列式，特征值等，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性质，如联合凸性、数据处理不等式、中间性、度量性质，以验证新的量子距离是有意义的，并且在从量子信息到机器学习和信号处理的许多应用中具有实验上的益处。这些量子距离在质心问题、几何、Karcher平均问题、量子信息理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训练，积极参与本拟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.