Mathematical Sciences: RUI: Extreme Points of Cones of Positive Semi-definite Matrices with Given Sparsity Pattern

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This research is concerned with two projects in matrix theory. For the first project, G is an undirected graph on n vertices and M(G) is an associated convex cone. The principal investigators will examine the structure of the extreme matrices, particularly when M(G) has extreme points of large rank. The second project is concerned with determining the extreme points of a collection of certain semidefinite matrices. They will investigate whether the rank and sparsity pattern alone determine if a matrix is an extreme point. A matrix is a rectangular array with possible entries from a wide variety of rings. This is a field which originated in the nineteenth century as part of algebra. It is currently very active in view of its many applications and connections with other parts of mathematics and other scientific and technical disciplines.