Number Theory and Theoretical Physics have common frontiers. The Riemann hypothesis can be cast as a Statistical Physics problem. In fact progress along this line was one result of the les Houches Winter Institute organised in 1989 during which 58 mathematicians and physicists met for two weeks for a combination of seminars and lecture series (published as two books by Springer-Verlag). Since then interactions have continued to blossom in Conformal invariant quantum field theories, Quantum chaos, Analytic number theory, String-M theory of Gravity. New geometries in particular Non-commutative Geometry and String or D-brane methods are becoming tools of the trade in Quantum field theory and in the quest for a quantized theory of Gravity. It is a serious challenge to physicists and actually to most mathematicians to grasp the algebraic concepts and abstract methods involved: K-theories, automorphic forms and arithmetic groups, categories.
Symmetrically it is much easier for mathematicians to talk to bilingual physicists than to penetrate the Physics literature.
This Winter Institute can be organised for clarity along three subjects (but personal interests and actual interrelations are more involved). Firstly Dynamical systems with their transcendence problems and chaotic/integrable dichotomy have seen the emergence of a minimal ("arithmetic") chaos controlled by symmetries, which is realised in gravity homogeneous models. Random matrix theory is related to quantum chaos, String models and Riemann's zeta function theory. A second topic is (poly- resp) di-logarithms which connects Quantum corrections, solutions of integrable models, hyperbolic geometry in higher dimensions than 2.
Note that three-logarithms appear in mirror symmetry (i.e. T-duality) of string models compactified to 4 dimensions. Algebraic K-theory is crucial there. The third main focus is going to be the analysis of perturbative divergences in quantum field theories and the issue of nonperturbative divergences in string theory. The tools to handle the problem will include Hopf algebras and arithmetic groups the so-called duality groups of supergravity-superstring theories. The recent emergence of striking nonperturbative results is bound to affect the development of Mathematics: a theory of general boundary conditions related to extended objects and K-theory, mirror symmetry and other dualities and an introduction to algebraic varieties important for Physics with Super symmetry will be presented. All three topics shall be studied sequentially.