Total nonnegativity, quantum algebras and growth of algebras

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This is wide ranging project that involves the three areas of noncommutativealgebra, Poisson algebraic geometry and linear algebra. Also, the solutionsoften involve representation theory and combinatorics. In addition, theproject will consider problems concerning growth of algebras. The development of the theory of quantum algebras was motivated by problems inPhysics from the 1980s onwards. Totally nonnegative matrices have beeninvolved in problems in such diverse areas as mechanical systems, birth anddeath processes, planar resistor networks, computer aided geometric design,juggling, etc. Results concerning growth of algebras have been obtained fromthe 1960s onwards, but the subject was in a quiescent state until the 2000swhen significant advances have been made. In the past five years, surprising links between the three areas mentioned inthe first paragraph have been discovered and investigated. A partialunderstanding of these connections has been gained, especially in the particularcase of coordinate algebras of matrices. The present project aims to furtherthis understanding by deepening the knowledge of the matrix case and byexpanding the scope of the knowledge to include algebras such asgrassmannians, partial flag varieties and De Concini-Kac-Procesi algebras. The growth part of the project will concentrate on two specific types ofgrowth: quadratic growth/Gelfand-Kirillov dimension two, and intermediategrowth (super polynomial, but subexponential).