Mathematical Sciences: Combinatorial Methods in Algebra and Geometry; Macdonald Polynomials, Diagonal Harmonics, and the Hilbert Scheme

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9400934 Haiman This project involves research into modules associated with the Macdonald polynomials and the module of diagonal harmonics. The work promises to establish new relationships between these objects and other parts of algebraic combinatorics and algebraic geometry. Preliminary work has already established the framework for the connections and has produced important new algebraic identities. This research falls in the broad category of combinatorics, which is one of the most active fields in today's mathematics. At its roots, combinatorics is the study of systematic counting. Counting can be incredibly difficult when the objects are difficult to list, and combinatorists look for general methods for overcoming these difficulties. Today's combinatorics makes use of a wide variety of the most advanced and modern mathematical techniques. Although its roots go back several centuries, the field has had an explosive development in the past few decades. This growth comes from its importance in communications and information technology and from the success of modern techniques to problems of counting.