One of the outstanding problems in present day theoretical physics is to bring together two central new ideas of the 20th century: quantum mechanics and general relativity. This is usually refered to as the problem of quantum gravity. In the last two decades string theory has emerged as the leading candidate for such a quantum gravity theory. It is based on the deceptively simple premise that at Planckian scales (of the order of cm), where the quantum effects of gravity are strong, particles are actually one-dimensional extended objects. It is at present our best hope to give concretely computable answers to fundamental questions such as the underlying symmetries of nature, the behaviour of black holes, and the quantum treatment of space-time singularities. It might also shed light upon larger issues such as the nature of quantum mechanics and space and time. In string theory all the forces and particles emerge in an elegant geometrical way, realizing Einstein's dream of building everything from the geometry of space-time.

Strings not only have applications in physics; the last two decades have seen important contributions to various mathematical disciplines, such as differential geometry, topology, Lie theory and algebraic geometry. Indeed the importance of physical ideas and intuition in modern mathematics has been demonstrated by the award of the 1990 Fields Medal to Edward Witten, the leading string theorist. The list of concrete mathematical applications of string theory is now long and growing rapidly. The natural occurrence of infinite-dimensional Lie algebras has shed light on such diverse topics as representation theory, modular properties of characters, and relations with knot theory and quantum groups. A ground-breaking idea in algebraic geometry has been the mirror symmetry phenomenon that relates the counting of holomorphic curves on Calabi-Yau three-folds to the variation of Hodge structures of a `mirror' manifold. This has led to spectacular predictions for the number of curves on certain three-folds. Also Kontsevich's proof of the Witten conjecture for the intersection calculus on the moduli space of complex curves should be mentioned.

Fri Mar 20 16:01:06 MET 1998