Student Geometry

I will introduce the Mean Curvature Flow in R^n and discuss its basic properties.

In this talk, we will introduce the Dirichlet-to-Neumann map and survey several important related results. In particular, we will we derive the spectral asymptotics for the Steklov problem on smooth Riemannian manifolds with boundary. We will then discuss a few open conjectures.

Following Chapter 6 of Levitin–Mangoubi–Polterovich, we will discuss the heat kernel and the asymptotics of its trace. We will leverage this theory to prove Weyl's law on a Riemannian manifold, albeit with a less-than-optimal remainder estimate. Time permitting, we will discuss isospectral…

Following Chapter 5 of Levitin–Mangoubi–Polterovich, we will survey three classical eigenvalue inequalities for the Laplacian: the Faber–Krahn, Cheeger, and Szegő–Weinberger inequalities. We will then focus on the Faber–Krahn inequality, giving a proof, discussing (quantitative) stability, and…

Following Chapter 4 in Levitin--Mangoubi--Polterovich’s book, we will investigate the nodal geometry of Laplace eigenfunctions on bounded domains and Riemannian manifolds. We will prove Courant’s nodal domain theorem, and then show that nodal sets are dense on the wavelength scale. Following a…

Following Chapter 3 in Levitin--Mangoubi--Polterovich's book, we will investigate the distribution of Laplace eigenvalues on bounded domains. We will discuss several proofs of Weyl's famous law for the asymptotic distribution of high-energy eigenvalues and Polya's conjecture that the leading…

Abstract

Abstract

Abstract

The authors study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a…

Abstract

The authors study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a…