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.
