Algebraic methods in modern mathematical physics have been influenced by efforts to understand the roles of symmetries in quantum field theory, and particularly by efforts to produce completely integrable systems (a notable example being the Seiberg-Witten equations). The symmetries behind such integrability tend to be hidden, and require sophisticated techniques for exposure. Members of our faculty are engaged in the study of such algebraic methods, including the representation theory of the Virasoro algebra and other infinite dimensional Lie algebras, which yield insights into modern mathematical physics, especially conformal field theory and string theory.