This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
There are many examples in mathematics and science of polynomials which depend on several variables, and which have important applications. In this project the PI will investigate the combinatorics associated with some of the most useful of these, Macdonald polynomials, which are multi-variate symmetric functions which satisfy an orthogonality relation and play a central role in algebraic combinatorics with applications to special functions, algebraic geometry, and statistical mechanics. Their original definition was rather difficult and indirect, but in 2004 the PI found a nice combinatorial formula for them, which was proved in subsequent joint work between Haiman, Loehr and the PI. The PI and others have been finding that the new combinatorics of Macdonald polynomials often leads to new combinatorial formulas for related objects. For example, in recent joint work with Luoto, Mason, and Van Willigenburg, the PI has been investigating certain limiting cases of nonsymmetric Macdonald polynomials known as Demazure characters and Demazure atoms, which are connected to representation theory. The PI and his collaborators have shown that some of the fundamental relations satisfied by the important Schur function basis have refined versions involving Demazure characters and atoms, involving constructs in the new combinatorics of Macdonald polynomials. Another part of the project involves conjectures the PI has recently formed which are multi-variate versions of previous conjectures of the PI and others involving the zeros of rook and matching polynomials. The PI is using computers combined with current mathematical methods to make progress on these conjectures.
Polynomials in a single variable play a fundamental role in science and mathematics, for example when modeling discrete data by a continuous function. Certain families of polynomials known as orthogonal polynomials are especially useful. Macdonald polynomials are a master family of orthogonal polynomials in several variables which contain all sorts of previously studied families of orthogonal polynomials and other useful polynomials as special cases. Macdonald proved they exist, but gave no particularly simple description of them. The PI, in a previous collaboration with Haiman and Loehr, proved a direct combinatorial description of them. There are still many unsolved problems involving Macdonald polynomials, and in this projectthe PI will continue to develop the combinatorial side of the theory, with applications to a variety of polynomials from various branches of mathematics. Another aspect of this proposal involves values of the variables which make a given polynomial zero, which the PI is investigating using computers and experimental methods.
