Macdonald Polynomials, Diagonal Harmonics, and the Geometry of Hilbert Schemes

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Abstract: Macdonald polynomials, diagonal harmonics, and the geometryof Hilbert schemes.Professor Haiman is working to complete the proof of a series of conjectures involving Macdonald polynomials, the so-called "n-factorial" conjecture, and the character formula for diagonal harmonics. The methods involve a detailed algebraic geometrical study of the Hilbert scheme of points in the plane and related algebraic varieties. In earlier work Professor Haiman showed that certain hypotheses on the singularities and sheaf cohomology of these varieties would imply the desired algebraic results (and with Garsia, showed that the latter imply related combinatorial results). The current work is to establish these geometric hypotheses.Macdonald polynomials are a new family of symmetric functions. Their discovery by Macdonald in 1988 was a surprising development in the theory of symmetric functions, which is a fundamental and classical part of mathematics with roots in the work of Euler, Jacobi, and Cauchy over a century ago. Macdonald polynomials have since been found to have important applications in a wide range of areas including geometry, representation theory, and even theoretical physics. At the time of their discovery, Macdonald conjectured that certain coefficients associated with his polynomials should be positive integers, the proof of which remains the most important unsolved problem in this area. The successful completion of this project will solve this problem, proving the "Macdonald positivity conjecture," along with a related representation-theoretic conjecture of Garsia and the investigator known as the "n-factorial" conjecture, some related combinatorial conjectures, and strong new geometric properties of Hilbert schemes, which are likely to have further applications in geometry and representation theory.