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Thursday

Scalar curvature and smallness

  • Speaker: Davi Maximo
  • Time: 9:30-10:30
  • Abstract:

    In this talk, we will present new results where a positive lower bound on scalar curvature gives control over the geometry and topology of a manifold from above.

Existence of CMC hypersurfaces

  • Speaker: Xin Zhou
  • Time: 11:00-12:00
  • Abstract:

    In this talk, we will present some recent progress on the existence of constant mean curvature (CMC) hypersurfaces. We plan to talk about a recent work joint with Liam Mazurowski on the existence of infinitely many closed CMC hypersurfaces which enclose half of the ambient volume in a generic metric.

Mean curvature flow in R3 and the Multiplicity One Conjecture

  • Speaker: Bruce Kleiner
  • Time: 2:00-3:00
  • Abstract:

    An evolving surface is a mean curvature flow if the normal component of its velocity field is given by the mean curvature. First introduced in the physics literature in the 1950s, the mean curvature flow equation has been studied intensely by mathematicians since the 1970s with the aim of understanding singularity formation and developing a rigorous mathematical treatment of flow through singularities. I will discuss progress in the last few years which has led to the solution of several longstanding conjectures, including the Multiplicity One Conjecture. This is joint work with Richard Bamler.

Geometry of 4-dimensional Ricci solitons with (half) nonnegative isotropic curvature

Ricci solitons, introduced by R. Hamilton in the mid-80s, are self-similar solutions to the Ricci flow and natural extensions of Einstein manifolds. In this talk, I will first discuss the significant role they play in understanding singularity formation in the Ricci flow, especially in dimension 3, and then present some recent progress on the classification of 4-dimensional gradient Ricci solitons with nonnegative (or half nonnegative) isotropic curvature. It is based on my joint work with Junming Xie.

Dynamic instabilities of some Ricci solitons

  • Speaker: Dan Knopf
  • Time: 4:40-5:40
  • Abstract:

    I will describe work in progress with Sigurd Angenent in which we derive geometric criteria that imply that noncompact Ricci solitons in a certain class possess infinite-dimensional unstable manifold. Using a noncompact Einstein metric discovered by Christof Böhm as a motivating special case, we prove that the class is nonempty.