Workshop

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In this talk we prove the uniqueness of pitchfork and helicoid translators of the mean curvature flow in $\mathbb{R}^3$.  This solves a conjecture by Hoffman, White and Martin.

The proof is based on   an arc-counting argument motivated by Morse-Rad\'o theory for translators and a rotational maximum principle. 

This is joint work with F. Martin and R. Tsiamis.

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We will talk about existence of Einstein metrics on manifolds with boundary, while prescribing the induced conformal metric and mean curvature of the boundary. In dimension 3, this becomes the existence of conformal embeddings of surfaces into constant sectional curvature space forms, with prescribed mean curvature. We will show existence of such conformal emebeddings near generic Einstein background. We will also discuss the existence question in higher dimensions, where things become more subtle and a non-degenerate boundary condition is used to construct metrics with nonpositive Einstein constant.

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The interplay between geometry and topology is always one of the central topics in Riemannian geometry. For open (noncompact and complete) manifolds with nonnegative Ricci curvature, it is known that the fundamental groups could be torsion-free nilpotent. This is distinct from open manifolds with nonnegative sectional curvature, whose fundamental groups are virtually abelian. This talk will cover how the virtual nilpotency/abelianness of the fundamental group is related to various geometric quantities and equivariant asymptotic geometry.

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Minimal surfaces in spheres, which are invariant by a group of symmetries, are closely related to group theory, hyperbolic geometry and geometric topology. In this talk, I will discuss a new connection with random matrices. As we will explain, from two permutations, one can associate a 2d minimal surface in a Euclidean sphere. The main property is a probabilistic rigidity phenomenon: if the two permutations are chosen uniformly at random, then with high probability, the minimal surface has a geometry close to that of the hyperbolic plane.

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We establish the conjectured mass-angular momentum inequality for multiple black holes, modulo the extreme black hole `no hair theorem'. More precisely it is shown that either there is a counterexample to black hole uniqueness, in the form of a regular axisymmetric stationary vacuum spacetime with an asymptotically flat end and multiple degenerate horizons which is `ADM stable', or the following statement holds. Complete, simply connected, maximal initial data sets for the Einstein equations with multiple ends that are either asymptotically flat or asymptotically cylindrical, admit an ADM mass lower bound given by the square root of total angular momentum, under the assumption of nonnegative energy density and axisymmetry. Moreover, equality is achieved in the mass lower bound only for a constant time slice of an extreme Kerr spacetime. The proof is based on a novel flow of singular harmonic maps with hyperbolic plane target, under which the renormalized harmonic energy is monotonically nonincreasing. Relevant properties of the flow are achieved through a refined asymptotic analysis of solutions to the harmonic map equations and their linearization. This is joint work with Qing Han, Gilbert Weinstein, and Jingang Xiong.

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Collapsed ancient solutions to the homogeneous Ricci flow on compact manifolds occur only on the total space of principal torus bundles. Under an algebraic assumption that guarantees flowing through diagonal metrics and a tameness assumption on the collapsing directions, we prove that such solutions have additional symmetries, i.e., they are invariant under the right action of their collapsing torus. As a byproduct of these additional torus symmetries, we prove that these solutions converge, backward in time, in the Gromov-Hausdorff topology to an Einstein metric on the base of a torus bundle.  (Joint work with F. Pediconi and S. Sbiti.)

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Siobhan Roberts will recount her trajectory from history major to math journalist, and give a behind-the-scenes look at writing about mathematics for the New York Times.

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The stability theory for minimal submanifolds in codimension one has been well studied. In this talk we will survey the state of affairs in higher codimension and discuss recent progress.