Tools for Positivity in Algebraic Combinatorics

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Nonnegative integer invariants provide an important way of understanding complicated algebraic or geometric objects. Examples include the degree of a polynomial and the number of holes of a surface. The goal of this project is develop general methods to obtain a detailed understanding of nonnegative integer invariants arising in several different areas of mathematics. This may form the foundation for developments in quantum information theory, knot theory, physics, and signal processing. In particular, this project offers potential new insights into the tensor decomposition problem, which is essentially the problem of recovering individual signals from a mixture of signals and has applications in medicine, computer vision, chemistry, and fast matrix multiplication.Positivity problems in algebraic combinatorics ask to find positive combinatorial formulae for nonnegative quantities arising in geometry and representation theory. The goal of this project is to develop tools to solve positivity problems arising in two areas of active research, Macdonald theory and geometric complexity theory. Macdonald polynomials are a two-parameter family of symmetric polynomials, which have ties to many areas including geometry, physics, and knot theory. A major breakthrough in this area came with the proof of the Macdonald positivity conjecture, which showed that important structure coefficients related to Macdonald polynomials are nonnegative. It remains a fundamental open question to give a positive combinatorial interpretation of these coefficients. Geometric complexity theory is an approach to P versus NP and related problems in complexity theory using algebraic geometry and representation theory. A fundamental problem in representation theory, believed to be important for this approach, is the Kronecker problem, which asks for a positive combinatorial formula for decomposing the tensor product of two irreducible representations of the symmetric group into irreducibles. This project will further develop the theory of noncommutative Schur functions, a powerful tool for solving positivity problems, particularly focusing on applications to Macdonald polynomials and the Kronecker problem.