Exact Results in Quantum Theory

This year remarkable relations have been discovered between four-dimensional, supersymmetric quantum gauge theories, andtwo-dimensional conformal field theories. As it turns out, various exact quantities computed for certain theories in both these classes are equal to each other. These quantities include four-dimensional partition functions on one hand, and two-dimensional correlators (three-point functions, conformal blocks) on the other hand. There is a growing computational and conceptual evidence that these relations are true in a very general context. I will discuss how this conjecture could be proved by relating both classes of theories in question to matrix models and non-commutative Riemann surfaces, and sketch the associated string theory interpretation.