Matrix Theory with Applications to Positivity and Discrete Mathematics

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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.