Variational Methods in Geometry

Regularity of weakly stable codimension 1 CMC varifolds

Neshan Wickramasekera
University of Cambridge; Member, School of Mathematics
January 15, 2019
The lecture will discuss recent joint work with C. Bellettini and O. Chodosh. The work taken together establishes sharp regularity conclusions, compactness and curvature estimates for any family of codimension 1 integral $n$-varifolds satisfying: (i) locally uniform mass and $L^{p}$ mean curvature bounds for some $p > n;$ (ii) two structural conditions and (iii) two variational hypotheses on the orientable regular parts, namely, stationarity and (weak) stability with respect to the area functional for volume preserving deformations (supported on the regular parts).

Harmonic maps into singular spaces

Brian Freidin
Brown University; Visitor, School of Mathematics
December 11, 2018
In the 90's, Gromov and Schoen introduced the theory of
harmonic maps into singular spaces, in particular Euclidean buildings,
in order to understand p-adic superrigidity. The study was quickly
generalized in a number of directions by a number of authors. This
talk will focus on the work initiated by Korevaar and Schoen on
harmonic maps into metric spaces with curvature bounded above in the
sense of Alexandrov. I will describe the variational characterization

Global results related to scalar curvature and isoperimetry

Otis Chodosh
Princeton University; Veblen Research Instructor, School of Mathematics
December 4, 2018
I will first survey some recent progress on global problems related to scalar curvature and area/volume, focusing in particular on scale breaking phenomena in such problems. I will then discuss the role of the Hawking mass in the resolution of this scale-breaking issue for the stable CMC uniqueness problem in asymptotically Schwarzschild manifolds (joint work with M. Eichmair) and possibly mention some features of the isoperimetric problem in asymptotically Schwarzschild-anti-de Sitter manifolds (joint work with M. Eichmair, Y. Shi, J. Zhu).

Bubbling theory for minimal hypersurfaces

Ben Sharp
University of Warwick
November 27, 2018
We will discuss the bubbling and neck analysis for degenerating sequences of minimal hypersurfaces which, in particular, lead to qualitative relationships between the variational, topological and geometric properties of these objects. Our aim is to discuss the salient technical details appearing in both the closed and free-boundary setting, and to give an overview of the applications of such results. This will involve expositions of joint works with Lucas Ambrozio, Reto Buzano and Alessandro Carlotto.

Homotopical effects of k-dilation

Larry Guth
Massachusetts Institute of Technology
November 27, 2018
Back in the 70s, Gromov started to study the relationship between the Lipschitz constant of a map (also called the dilation) and its topology. The Lipschitz constant describes the local geometric features of the map, and the problem is to understand how it relates to the global geometric features of the map -- a bit like trying to understand the relationship between the curvature of a Riemannian manifold and its topology.

The min-max width of unit volume three-spheres

Lucas Ambrozio
University of Warwick; Member, School of Mathematics
November 20, 2018
The (Simon-Smith) min-max width of a Riemannian three dimensional sphere is a geometric invariant that measures the tightest way, in terms of area, of sweeping out the three-sphere by two-spheres. In this talk, we will explore the properties of this geometric invariant as a functional on the space of unit volume
metrics.

This is joint work with Rafael Montezuma.

Almgren's isomorphism theorem and parametric isoperimetric inequalities

Yevgeny Liokumovich
Massachusetts Institute of Technology; Member, School of Mathematics
November 20, 2018
In the 60's Almgren initiated a program for developing Morse theory on the space of flat cycles. I will discuss some simplifications, generalizations and quantitative versions of Almgren's results about the topology of the space of flat cycles and their applications to minimal surfaces.

I will talk about joint works with F. C. Marques and A. Neves, and L. Guth.

Morse-Theoretic Aspects of the Willmore Energy

Alexis Michelat
ETH Zurich
November 13, 2018
We will present the project of using the Willmore elastic energy as a quasi-Morse function to explore
the topology of immersions of the 2-sphere into Euclidean spaces and explain how this relates to the
classical theory of complete minimal surfaces with finite total curvature.

This is partially a joint work in collaboration with Tristan Rivière.