The Moduli Space of Curves

Developments in theoretical physics, in particular in conformal field theory, have led to a surprising connection to algebraic geometry, and especially to the fundamental concept of the moduli space Mg of curves of genus g, which is the variety that parametrizes all curves of genus g. Experts in the field explore in this volume both the structure of the moduli space of curves and its relationship with physics through quantum cohomology. Witten's conjecture in 1990 describing the intersection behaviour of tautological classes in the cohomology of Mg arose directly from string theory. Shortly thereafter an interesting proof was provided by Kontsevich who, in this volume, describes his solution to the problem of counting rational curves on certain algebraic varieties and includes suggestions for further development. The same problem is given treatment in a paper by Manin. There follows a number of contributions to the geometry, cohomology and arithmetic of the moduli spaces of curves. In addition, several contributors address quantum cohomology and conformal field theory.