Title: Features of Ricci flow smoothing and singularities in four dimensions
Abstract: The Ricci flow, introduced by Hamilton in the 1980s, is a geometric-analytic tool that has since found many applications including Perelman's resolution of the Poincaré and Thurston's Geometrization conjectures, which classify three-dimensional compact manifolds and generalize the classical uniformization theorem for two-dimensional surfaces. The flow is a parabolic evolution equation which tends to smooth out a Riemannian metric on a manifold, but it may also encounter singularities; both are important features which can help us understand the topology of the manifold. When moving to four dimensions, conical singularities are an important new class of singularities that arise. I will discuss work on the existence of asymptotically conical expanding Ricci solitons in dimension four, joint with Richard Bamler, which may play a role in resolving these singularities and help in the ongoing construction of a theory of four-dimensional Ricci flow with surgery for the purpose of new applications.
