Mathematical Conversations

The Ricci curvature of a Riemannian manifold is best viewed as the right replacement for the (nonlinear) laplacian of the metric g, which in particular explains why it so often appears in geometry and analysis.  Most commonly one studies either lower bounds on the Ricci curvature Rc>=-\lambda g , or Einstein manifolds Rc=\lambda g .  In recent years it has become a small business to understand how these conditions can be understood and even characterized through analysis; such characterizations have become the backbone of much of the recent structure theory.  We will discuss precise, but hopefully easy to understand, formalisms which show how lower bounds on Ricci curvature are 'equivalent' to analysis on M, and how being an Einstein manifold is 'equivalant' to analysis on the path space of M.