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“积极性”。沿着这些路线，我提出了各种研究项目，涉及全正矩阵和某些矩阵的性质与图形及其相关的组合参数和相应的代数不变量。前者是一个很好的研究和基础类的矩阵，出现在许多应用，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，并探讨一些新的令人兴奋的方向，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，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。两者之间的联系如下，因为等级和无效是密切相关的。这里的希望是呼吁矩阵的代数性质和联合收割机结合起来，图形的组合特征，产生有趣的结果，这个特殊的矩阵集合。总之，我将结合联合收割机现有的理论知识沿着与新的尖端复杂性，探索一些基本的进步和连接的重要应用，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。这种前沿的研究是目前的兴趣在地球仪，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。

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.