Matrix Theory with Applications to Positivity and Discrete Mathematics

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，我提出了令人兴奋的新的和持续的研究，涉及完全正矩阵和研究与图及其相关组合参数相关的某些矩阵的性质以及相应的代数不变量。这两个研究领域都根植于历史，具有当前的兴趣，并且具有丰富的应用（理论和实践），包括控制理论，有限自动机，矩阵刚性以及与纠缠和退相干相关的逆问题。总正性是一个被研究得很好的基本矩阵，它出现在许多应用中，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，并探索一些令人兴奋的新方向，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和前沿进展，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，而最小秩问题又等价于最大化与矩阵的零特征值或特征值相关的零或维数。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，并将它们与图的重要组合特征结合起来，以产生关于这一特殊矩阵集合的有趣结果和进展。总而言之，我将结合现有的理论知识和新的创新复杂性来探索一些基本的进步和重要应用的联系，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。这一当代重要的研究引起了全球的兴趣，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。我的研究植根于线性代数和离散数学的理论方面，但隐含地与许多数学分支和广泛的新兴和有趣的应用联系在一起。例如，我的一些工作是关于图传播（或感染）和某些图或网络搜索模型。

My research interests are centered on `Positivity' in Linear Algebra and Graph Theory. Along these lines, I have proposed exciting new and continuing research studies involving totally positive matrices and studies concerning the properties of certain matrices associated with graphs and their related combinatorial parameters along with corresponding algebraic invariants. Both of these research areas are rooted in history, of current interest, and rich in applications (both theoretical and practical), including control theory, finite automata, matrix rigidity, and inverse problems related to entanglement and decoherence. Total positivity is a well studied and fundamental class of matrices that appears 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 exciting new 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 and cutting-edge advances. 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, which, in turn, is equivalent to maximizing the nullity or dimension associated with the zero eigenvalue or characteristics value of a matrix. This 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 important combinatorial characteristics of graphs to yield interesting results and advances about this special collection of matrices. In summation, I will combine existing theoretical knowledge along with new innovative 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 contemporary and important 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. My research is rooted in theoretical aspects of linear algebra and discrete mathematics, but is implicitly connected to many branches of mathematics and with a wide range of emerging and interesting applications. For example, some of my work is concerned with graph propagation (or infection) and certain graph or network searching models.