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

该奖项支持主要研究者介绍的关于仿射Hecke代数上的方程(包括微分和差分)的研究，这导致了通过Hecke代数的表示理论来研究特殊函数和组合学的新方法。这些方程是共形场理论中的Knizhnik-Zamolodchikov方程及其相应的差分场的推广。共形场理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。保形场理论除了在物理学中具有重要意义外，其优美而丰富的数学结构也引起了许多数学家的兴趣。在共形场理论的研究中，揭示了无穷维李代数和群的表示、Riemann曲面和代数曲线、Monster零星群、模函数和模形式、椭圆亏格和纽结理论等数学分支之间的新关系。人们相信，对保形场理论所涉及的数学的研究将导致新的数学结构，这在数学和理论物理中都将是重要的。

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.