Mathematical Sciences: Hecke Algebras, MacDonald's Polynomials, and Conformal Field Theory

该奖项支持对仿射Hecke代数上的方程（微分和差分）的研究，该研究通过Hecke代数的表示理论为特殊函数理论和组合学提供了新的途径。这些方程是对共形场论及其差分对应物的Knizhnik-Zamolodchikov方程的推广。共形场论是描述凝聚态物理中的二维临界现象和弦理论中弦的经典运动的重要物理理论。除了在物理学上的重要性外，共形场论美丽而丰富的数学结构也引起了许多数学家的兴趣。在共形场论的研究中，揭示了无限维李代数与群的表示、黎曼曲面与代数曲线、怪物散群、模函数与模形式、椭圆属、结论等数学不同分支之间的新关系。人们相信，对共形场论的数学研究将导致新的数学结构，这在数学和理论物理中都是重要的。

This award supports the study of equations (both differential and difference) on affine Hecke algebras introduced by the principal investigator which leads to a new approach to the theory of special functions and combinatorics via the representation theory of Hecke algebras. These equations are a generalization of the Knizhnik-Zamolodchikov equation from conformal field theory and its difference counterpart. Conformal field theory is an important physical theory describing both two-dimensional critical phenomena in condensed matter physics and classical motions of strings in string theory. Besides its importance in physics, the beautiful and rich mathematical structure of conformal field theory also has interested many mathematicians. New relations between different branches of mathematics, such as representations of infinite-dimensional Lie algebras and groups, Riemann surfaces and algebraic curves, the Monster sporadic group, modular functions and modular forms, elliptic genera, and knot theory, is revealed in the study of conformal field theory. It is believed that the study of the mathematics involved in conformal field theory will lead to new mathematical structures which will be important both in mathematics and theoretical physics.