Singularity-theory and N=2 supersymmetry

N = 2 supersymmetry in two dimensions can be seen as a quantum realization of the geometry of singularity theory. We show this using non-perturbative methods. N = 2 susy is related to the Picard-Lefschetz theory much in the same way as N = 1 susy is related to Morse theory. All the concepts of singularity theory fit in the physics of the N = 2 Landau-Ginsburg models. The critical behaviour of the theory is encoded in a certain natural "gauge-connection" in coupling-constant space. It is flat for a quashihomogeneous superpotential, but not in general. We find an explicit formula relating it to the Gauss-Manin connection of the singularity associated to the superpotential. Our results are valid for both the quasihomogeneous and the non-quasihomogeneous case, but in the former our equations simplify dramatically. We discuss some preliminary applications.