This research is concerned with representation theory of infinite-dimensional Lie algebras with applications to quantum field theory. A new approach to the study of integrals of motion in deformations of two-dimensional conformal field theories will be developed. The modular functor for affine Kac-Moody algebras at the rational level will be pursued, using geometrical properties of the corresponding representations. The Langlands-Drinfeld correspondence for complex algebraic curves will be investigated. Topological field theories, associated to affine algebras, will be studied from the cohomological point of view. The quantum deformation of the Lie algebra of differential operators on the circle will be studied in connection with the W-algebras. Finally, the formulas for singular vectors in Verma modules over the Virasoro algebra will be applied to the computation of the cohomologies of Lie algebras of vector fields. This research is concerned with a mathematical object called a Lie algebra. Lie algebras arise from another object called a Lie group. An example of a Lie group is the rotations of a sphere where one rotation is followed by another. Lie groups and Lie algebras are important in areas involving analysis of spherical motion.
