We demonstrate that any four-dimensional shrinking Ricci soliton (B×S², g), where B is any two-dimensional complete noncompact surface and g is a warped product metric over the base B, has to be isometric to the generalized cylinder R²×S² equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products — but not products — and provide rigorous examples of the formation of generalized cylinder singularity models R^k×S¹. This is joint work with Isenberg, Knopf and Ma.

In this talk we will work towards proving the following theorem: a contractible 3-manifold with positive scalar curvature and bounded geometry must be diffeomorphic to R³. The proof involves running an innermost weak inverse mean curvature flow on the manifold. This talk is based on joint work with Otis Chodosh and Yi Lai.

A well studied problem is the metric behavior of Calabi-Yau metrics on a fibration in an adiabatic family of Kähler classes. I will discuss recent progress showing that the Gromov-Hausdorff limit can be identified with the base of the fibration, generalizing results of Gross-Tosatti-Zhang, Song-Tian-Zhang and Li-Tosatti. A new ingredient is to exploit the RCD property of the Gromov-Hausdorff limit.

A lot of the focus of minimal submanifold theory has been for minimal submanifolds in codimension one. In the codimension one case there is a lot that is known, including Bernstein theorems, a regularity theory for minimizing hypersufaces, and even for stable minimal hypersurfaces. In higher codimension the situation is quite different, and is much less understood. There is a regularity theory for minimizers, but no Bernstein theorems in general, and the theory for stable minimal submanifolds in higher codimension is not well understood. In this talk I will describe some recent progress on stable minimal surfaces in higher codimension, in particular on Bernstein theorems, and applications in Riemannian geometry.

In this lecture, based on recent joint work with Yangyang Li (University of Chicago) and Zhihan Wang (Cornell University), I will present a generic regularity result for stationary integral n-varifolds with only strongly isolated singularities inside N-dimensional Riemannian manifolds, in absence of any restriction on the dimension (n≥2) and codimension. As a special case, we prove that for any n≥2 and any compact (n+1)-dimensional manifold M the following holds: for a generic choice of the background metric g all stationary integral n-varifolds in (M,g) will either be entirely smooth or have at least one singular point that is not strongly isolated. In other words, for a generic metric only "more complicated" singularities may possibly persist. This implies, for instance, a generic finiteness result for the class of all closed minimal hypersurfaces of area at most 4π²-ε (for any ε>0) in nearly round four-spheres: we can thus give precise answers, in the negative, to the well-known questions of persistence of the Clifford football and of Hsiang's hyperspheres in nearly round metrics. The aforementioned main regularity result is achieved as a consequence of the fine analysis of the Fredholm index of the Jacobi operator for such varifolds: we prove on the one hand an exact formula relating that number to the Morse indices of the conical links at the singular points, while on the other hand we show that the same number is non-negative for all such varifolds if the ambient metric is generic.

In 1989, Brian White conjectured that every Riemannian 3-sphere contains at least five embedded minimal tori. The number five is optimal, corresponding to the Lyusternik-Schnirelmann category of the space of Clifford tori. I will present recent joint work with Adrian Chu, where we confirm this conjecture for 3-spheres of positive Ricci curvature. Our proof is based on min-max theory, with heuristics largely inspired by mean curvature flow.

I will discuss a new geometric concentration phenomenon for equivariant harmonic maps from surfaces to Euclidean spheres. Specifically, for any unitary representation of the fundamental group of a Riemann surface S, one can associate a renormalized energy and an equivariant harmonic map from the universal cover of S to a unit sphere. The main result of this talk is that when the unitary representation is chosen at random, with high probability, the renormalized energy is close to an explicit constant and the shape of the harmonic map is close to a unique limit shape.

We show that any L^∞ Riemannian metric g on Rⁿ that is smooth with nonnegative scalar curvature away from a singular set of finite (n-α)-dimensional Minkowski content, for some α>2, admits an approximation by smooth Riemannian metrics with nonnegative scalar curvature, provided that g is sufficiently close in L^∞ to the Euclidean metric. The approximation is given by time slices of the Ricci-DeTurck flow, which converge locally in C^∞ to g away from the singular set. We also identify conditions under which a smooth Ricci-DeTurck flow starting from a L^∞ metric that is uniformly bilipschitz to Euclidean space and smooth with nonnegative scalar curvature away from a finite set of points must have nonnegative scalar curvature for positive times.

I will discuss some recent progress on the Donaldson-Segal programme, and in particular how calibrated cycles arise from the large mass limit of G₂ and Calabi-Yau monopoles.