940841 Felder The main goal of the proposed activity is to give an explicit description of conformal blocks on Riemann surfaces, as a space of solutions of a system of differential equations, generalizing the Knizhnik-Zamolodchikov equations. Properties of the equations and their solutions will be investigated. In particular, integral representations of the solutions will be calculated, with applications to the topology of configuration spaces on surfaces, invariants of three-dimensional manifolds, and the theory of quantum groups and integrable systems. Conformal Field Theory finds its origin in the physics of surface at critical temperature, and string theory, a unified model of particle interactions. The object of this research is the mathematical structure of Conformal Field Theory. This theory explicitly associates a mathematical object called a complex vector space of "conformal blocks" with equations that model physical systems. ***
