Combinatorial aspects of geometry and representation theory

海曼教授从事与代数、几何和表示理论有关的组合问题的研究。利用希尔伯特格式几何上的新发现，他最近证明了麦克唐纳多项式的正猜想，并建立了这些多项式的公式项，用于对称群的经典谐波空间的“双重”版本的特征。证据表明，这些结果符合一个更大的框架，在这个框架中，nakajima的箭囊变体扮演着希尔伯特方案的角色。在他继续的研究中，海曼希望建立这个更大框架的有效性，并进一步发展组合学、麦克唐纳多项式和希尔伯特格式之间的联系。他还计划研究由Shimozono和weymann引入的flittlewood - richardson系数的q-类似物以及Lascoux、Lapointe和Morse的q-Schur函数相关的未解决问题。最后，在与Grojnowski的合作下，他将回到他几年前开始的关于Hecke代数特征及其与内变量组合性质的联系的研究。组合学是一门研究可以用简单而明确的术语来描述、计算和操作的事物的数学，比如树、杨氏图或符号串，我们可以把它们写在纸上或用计算机编码。在我看来，组合学的有趣之处在于那些非常抽象的数学概念——比如多维几何空间，或者对称代数——往往有一个潜在的组合结构。我目前的工作将特殊空间的几何性质，如希尔伯特方案和颤振变体，与杨氏图和对称多项式的组合性质联系起来。通过理解这种联系，我们可以在具体层面上处理这些抽象空间，并获得关于空间几何和与之相关的组合规则的重要新见解。

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.