Functional, geometric and matrix inequalities and applications

函数、几何和矩阵不等式及其应用

• 批准号: RGPIN-2021-03584
• 负责人: Lam, Nguyen
• 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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=742624
• 关键词: Functional; geometric; matrix; inequalities; applications

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.

该研究计划涉及两个主要方向的主题：尖锐的功能和几何不等式和矩阵分析，以期应用于非线性偏微分方程，数学物理和量子信息理论。在函数和几何不等式中，拟议的研究项目将主要集中在Sobolev型不等式和哈代型不等式，这是分析中最常用的两个不等式。他们发挥了至关重要的作用，在一些问题中出现的变分法，偏微分方程，几何等，我，与我的合作者，已成功地开发了尖锐的版本的一些功能和几何不等式及其应用在几个重要的情况下。尽管如此，仍有各种有趣而重要的问题有待解决。本提案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尖锐的版本，最佳常数和优化的Sobolev型不等式在各种几何设置。我还将工作的问题，找到改进的版本，将提供简单和直接的理解已知的哈代型不等式在文献中，以及存在和不存在的优化。本研究计划也将集中于调查最佳条件和开发新的概念，其中两个重量哈代不等式举行，并使用它们来获得新类型的不等式。我也将研究这些概念之间的关系。哈代型不等式和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，我的主要兴趣是在半正定矩阵集合上的量子距离，这是测量两个数据点（如量子力学中的量子态）之间可互换性的主要工具。部分拟议的研究是在矩阵分析中建立新的或改进的不等式，涉及迹，行列式，特征值等，并使用它们来导出合适的量子距离。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优势。这些量子距离在重心问题，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监督下接受研究培训，积极参与这个拟议研究项目的两个方向。