Functional, geometric and matrix inequalities and applications

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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.