Geometry
Organizers: Otis Chodosh, Filippo Gaia, & Rafe Mazzeo
Upcoming Events
Suppose we are given a sequence of smooth compact oriented Einstein 4-manifolds with fixed positive Einstein constant that Gromov-Hausdorff converges to a 4-dimensional Einstein orbifold. Suppose, moreover, that the limit metric is Hermitian with respect to some integrable complex structure on…
Past Events
Recent developments in scalar curvature geometry have revealed that positive scalar curvature (PSC) imposes quantitative restrictions on various geometric invariants. In this talk, I will present two results in this direction: an estimate for the stable two-dimensional systole of compact PSC…
The Ginzburg–Landau energy is often used to approximate the Dirichlet energy. As the perturbation parameter tends to zero, critical points of the Ginzburg–Landau energy converge, in an appropriate (bubbling) sense, to harmonic maps. In this talk I will first explain key analytical properties of…
This talk centers around non-smooth metric measure spaces with low regularity. We will exhibit a series of regularity results and structure theorems for such metric spaces whose Ricci curvature has a uniform lower bound in the synthetic sense. These…
It was shown recently by Chodosh-Choi-Mantoulidis-Schulze that for a generic initial surface in R^3, the mean curvature flow only has spherical and cylindrical singularities, confirming a well known conjecture of Ilmanen. I will discuss recent work showing that in fact for generic initial…
We prove the Riemannian positive mass theorem up to dimension 19, building on a combination of torical symmetrization and the singularity blow-up technique, together with the generic regularity theory for area-minimizing hypersurfaces developed by Chodosh, Mantoulidis, Schulze and Wang. This is…
Isoperimetric boundaries minimise area for fixed an enclosed volume, with sharp regularity theory ensuring smoothness away from a closed singular set of codimension seven. I will discuss recent work, with G. Niu, which constructs isoperimetric regions from hypersurfaces in closed manifolds. As a…
A classical theme in Riemannian geometry is that positive curvature imposes topological constraints on manifolds. In this talk, we investigate curvature conditions that distinguish Euclidean space among open contractible manifolds and the disk among compact contractible manifolds with boundary.…
Optimal bubble cluster problems concern the study of partitions of $\mathbb{R}^n$ into a finite collection of chambers, some with finite volume and some with infinite volume. One looks for local minimizers of interfacial area subject to volume constraints on the finite-volume chambers. The case…
Suppose we are given a globally hyperbolic spacetime (M,g) solving the Einstein vacuum equations, and a timelike geodesic in M. I will explain how to construct, on any compact subset of M, a solution g_\epsilon of the Einstein vacuum equations which is approximately equal to g far from the…
We prove existence for many examples of shrinkers by producing compact, smoothly embedded surfacesthat, under mean curvature flow, develop singularities at which the shrinkers occur as blowups. This is joint work with Paco Martin and Brian White.