Some recent developments in Ricci flow

  • Richard H. Bamler

    Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA
Some recent developments in Ricci flow cover
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In this article we survey some of the recent developments in Ricci flow. We present a new theory of weak, 3-dimensional Ricci flows “through singularities,” which can be viewed as an improvement of Perelman’s Ricci flow with surgery. We point out two topological applications: the resolution of the Generalized Smale Conjecture regarding the diffeomorphism groups of 3-manifolds and the resolution of a conjecture regarding the space of positive scalar curvature metrics on 3-manifolds. We also describe ongoing research on the formation of singularities in higher dimensions, which may yield further interesting applications in the future.