Matrix Analytics and Applications: Positivity, Graphs, and Stability

我的研究兴趣集中在线性代数和图论中的“正性”。沿着这条路线，我提出了各种研究项目，涉及完全正矩阵和与图及其相关组合参数和相应代数不变量相关的某些矩阵的性质。前者是一个被充分研究的基本矩阵类，在许多应用中出现，包括统计学、数学生物学和计算机辅助几何设计。我打算通过使用与某些矩阵分解相关的组合框架来研究与本课程相关的一些关键问题，并探索一些新的令人兴奋的方向，这些方向涉及到与潜在的基本代数结构的重要联系。我制定的计划将通过设定一系列具体目标，使人们对这类矩阵有更深的理解。关于从图中导出的矩阵的主要问题来自一个被称为反特征值问题的经典问题（即，提供了特征值并且需要相应的矩阵）。这个反特征值问题的一个重要部分是最小秩问题。这两者之间的联系如下，因为秩和null是密切相关的。这里的希望是吸引矩阵的代数性质，并将它们与图的组合特征结合起来，以产生关于这个特殊矩阵集合的有趣结果。总而言之，我将把现有的理论知识与新的尖端技术结合起来，探索一些重要应用的基本进展和联系，包括计算机科学中的通信复杂性，数学物理中的量子系统控制以及网络和图中某些搜索问题的引人注目的变化。这一前沿研究目前在全球范围内引起了人们的兴趣，因为它汇集了矩阵理论的材料，组合学的中心概念，并发展了一系列扩展的基本问题。

My research interests are centered on `Positivity' in Linear Algebra and Graph Theory. Along these lines, I have proposed various research projects involving totally positive matrices and properties of certain matrices associated with graphs and their related combinatorial parameters and corresponding algebraic invariants. The former is a well studied and fundamental class of matrices that arises in numerous applications, including statistics, mathematical biology, and computer aided geometric design. I intend to investigate a number of key problems associated with this class by making use of the combinatorial framework associated with certain matrix factorizations, and explore some new exciting directions involving important connections to an underlying fundamental algebraic structure. The plan that I have developed will lead to a deeper understanding of this class of matrices by setting out a sequence of concrete objectives. The main issue of interest concerning matrices that are derived from graphs comes from a classical problem known as an inverse eigenvalue problem (that is, the eigenvalues are provided and the corresponding matrix is desired). An important part of this inverse eigenvalue problem is the minimum rank problem. The connection between the two follows since rank and nullity are intimately related. The hope here is to appeal to the algebraic properties of matrices and combine them with the combinatorial characteristics of graphs to yield interesting results about this special collection of matrices. In summation, I will combine existing theoretical knowledge along with new cutting edge sophistication to explore a number of fundamental advances and connections to important applications including communication complexity in computer science, control of quantum systems in mathematical physics and compelling variations on certain searching problems in networks and graphs. This cutting edge research is of current interest across the globe, as it brings together material from the theory of matrices, central notions in combinatorics, and develops a sequence of expanding fundamental issues.