Geometry
Organizers: Otis Chodosh, Filippo Gaia, Rafe Mazzeo, Paul Minter, & Doug Stryker
Upcoming Events
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…
Past Events
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.
We will introduce the half-volume spectrum, a new variational invariant inspired by the classical volume spectrum, but defined by considering only hypersurfaces that divide the ambient manifold into equal volumes. We will present a new min-max theory for constant mean curvature (CMC)…
The $\mathbb{Z}_{2}$-harmonic $1$-forms arises in various compactification problems in gauge theory, deformation problem in special holonomy and calibrated geometry, including those involving flat $PSL(2,\mathbb{C})$ connections, Hitchin equation, Fueter sections, branched deformations of…
In this talk, we will discuss solutions of the inverse mean curvature flow that satisfy the following boundary condition: each hypersurface stays tangential to the boundary of the ambient domain. We will also discuss some recently discovered connections between this class of solutions and scalar…
In recent years, the combined work of Guaraco, Hutchinson, Tonegawa, and Wickramasekera has established a min-max construction of minimal hypersurfaces in closed Riemannian manifolds, based on the analysis of singular limits of sequences of solutions of the Allen—Cahn equation, a semi-linear…
The bridge principle is the idea that you can join compact minimal submanifolds along their boundaries to produce an “approximately minimal” submanifold, called the approximate solution, and apply a small normal perturbation to make the new configuration minimal. It dates back to Lévy (1948…