Matrix Analytics and Applications: Positivity, Graphs, and Stability

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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.