# Tuesday

- Speaker: Liam Mazurowski
- Time: 9:30-10:30
- Abstract: The solution to the isoperimetric problem shows that any closed manifold M contains a constant mean curvature hypersurface that encloses half the volume of M. This surface is obtained by minimizing area with a volume constraint. In this talk, we will investigate the existence of saddle type critical points for the area functional with half-volume constraint. First, we define the half-volume spectrum of a manifold. Then we explain how both the Allen-Cahn and Almgren-Pitts min-max theories can be used to construct surfaces achieving the half-volume spectrum. This is joint work with Xin Zhou.

- Speaker: Ao Sun
- Time: 11:00-12:00
- Abstract: Mean curvature flow is the fastest way to decrease the area of surfaces. It is the model in many disciplines such as material science, fluid mechanism, and computer graphics. The translators are a special type of mean curvature flow soliton. They are the singularity model of mean curvature flow and they play an important role in Ilmanen's elliptic regularization construction of weak mean curvature flow. I will present my recent work with Zhihan Wang (Princeton University) on the construction of translators with prescribed ends.

- Speaker: Lu Wang
- Time: 2:00-3:00
- Abstract: Minimal cones are models for singularities in minimal submanifolds, as well as stationary solutions to the mean curvature flow. In this talk, I will explain how to utilize mean curvature flow to yield near optimal estimates on density of topologically nontrivial minimal cones. This is joint with Jacob Bernstein.

- Speaker: Andre Neves
- Time: 3:30-4:30
- Abstract: We define the intersection between geodesic currents and conformal currents on closed hyperbolic 3-manifolds. As an application, we prove optimal inequalities between the Liouville entropy and the areas of minimal surfaces on negatively curved 3-manifolds. Joint with Fernando Marques.

- Speaker: Robert Haslhofer
- Time: 4:40-5:40
- Abstract: To capture singularities under mean curvature flow one wants to understand all ancient solutions. In addition to shrinkers and translators one also encounters ancient ovals, namely compact noncollapsed solutions that are not self-similar. In this talk, I will explain that any bubble-sheet oval for the mean curvature flow in R^4, up to scaling and rigid motion, either is the O(2)×O(2)-symmetric ancient oval constructed by White, or belongs to the one-parameter family of Z_2^2×O(2)-symmetric ancient ovals constructed by Du and myself. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar. This is joint work with Beomjun Choi, Toti Daskalopoulos, Wenkui Du and Natasa Sesum. I will also briefly mention the noncompact case, which is joint work with Kyeongsu Choi and Or Hershkovits.