Abstract:
Smooth riemannian manifolds enjoy beautiful properties. For example Poincare' duality. Or the fact that the classic geometric operators, such as the Gauss-Bonnet operator on differential forms, or the spin-Dirac operator on spinors, are essentially self-adjoint on $L^2$. Moreover, the analytic properties of these operators are related to the topology of the manifold through, for example, the Atiyah-Singer index theorem. What happens to these beautiful properties when we pass to singular spaces ? For example, when we consider singular projective varieties ? This question has been around for a long time and has generated, especially in the last 50 years, an intense research activity. In this talk I will survey old and new results concentrating, for the most recent contributions, on joint work with Pierre Albin and Markus Banagl.