Combinatorial aspects of geometry and representation theory

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Abstract for award of Haiman DMS-0301072Professor Haiman works on problems in combinatorics related toalgebra, geometry and representation theory. Using new discoveries onthe geometry of Hilbert schemes, he has recently proved the positivityconjecture for Macdonald polynomials and established a formula interms of these polynomials for the character of a "doubled" version ofthe classical spaces of harmonics for symmetric groups. Evidencesuggests that these results fit into a larger framework in whichNakajima's quiver varieties play the role of the Hilbert scheme. Inhis continuing research, Haiman hopes to establish the validity ofthis larger framework and further develop the connections betweencombinatorics, Macdonald polynomials and Hilbert schemes. He alsoplans to work on unsolved problems connected with q-analogs ofLittlewood-Richardson coefficients introduced by Shimozono and Weymanand the q-Schur functions of Lascoux, Lapointe and Morse. Finally, incollaboration with Grojnowski, he will return to a study he initiatedsome years ago of Hecke algebra characters and their connection withcombinatorial properties of immanants.Combinatorics is the mathematical study of things that can bedescribed, counted and manipulated in simple and explicit terms:things like trees, Young diagrams, or strings of symbols, that we canwrite on paper or encode in a computer. In my view, what makescombinatorics interesting is that deeply abstract mathematicalconcepts--such as geometric spaces in many dimensions, or algebras ofsymmetries--tend to have an underlying combinatorial structure. Mycurrent work connects the geometric properties of special spaces suchas Hilbert schemes and quiver varieties with the combinatorialproperties of Young diagrams and symmetric polynomials. Byunderstanding the connection, we can deal with these abstract spaceson a concrete level and get important new insights about both thegeometry of the spaces and the rules governing the combinatoricsassociated with them.