9401186 Borcherds This award supports research on vertex algebras, generalized Kac-Moody algebras, quantum rings, and moonshine. Some generalized Kac-Moody algebras can be constructed using vertex algebras; in particular the monster Lie algebra is such an algebra and was used to prove Conway and Norton's moonshine conjectures. There is strong evidence that there are many similar algebras and this will be investigated. Quantum rings are a generalization of vertex algebras which include the space of field operators of a free field theory as another special case. This concept will be further explored by the principal investigator. 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.