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“积极性”。沿着这些路线，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有关的某些矩阵的性质与图形及其相关的组合参数沿着与相应的代数不变量。这两个研究领域都植根于历史，当前的兴趣，并在应用（理论和实践）丰富，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。全正性矩阵是一个研究得很好的基本矩阵类，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。我打算调查一些关键问题与这一类利用组合框架与某些矩阵分解，并探讨一些令人兴奋的新方向，涉及重要的连接到一个基本的代数结构。我制定的计划将通过设定一系列具体目标和前沿进展，加深对这类矩阵的理解。关于从图中导出的矩阵的主要问题来自一个被称为逆特征值问题的经典问题（即，提供特征值并期望相应的矩阵）。这个逆特征值问题的一个重要部分是最小秩问题，这反过来又相当于最大化与矩阵的零特征值或特征值相关联的零值或维数。两者之间的这种联系是因为等级和无效性密切相关。这里的希望是呼吁代数性质的矩阵和联合收割机他们的重要组合特征的图形产生有趣的结果和进展，这一特殊的收集矩阵。总之，我将联合收割机现有的理论知识沿着新的创新复杂性，探索一些基本的进步和连接到重要的应用，包括通信复杂性计算机科学，控制量子系统的数学物理和引人注目的变化，在某些搜索问题的网络和图形。这个当代和重要的研究是当前的兴趣在整个地球仪，因为它汇集了材料从理论的矩阵，在组合学的中心概念，并制定了一系列扩大的基本问题。我的研究植根于线性代数和离散数学的理论方面，但与数学的许多分支以及广泛的新兴和有趣的应用有着内在的联系。例如，我的一些工作涉及图传播（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.