Geometry & Analysis Seminar Fall 2018

Thursday October 11th, 2018

TBA

Thursday October 18th, 2018

Gabe Martins, MSRI

Different Aspects of Classical and Quantum Confinement

We survey results on the problem of confining a particle (either classical or quantum) to a bounded region in space using different types of force fields, which leads us to consider some subtleties in the canonical quantization procedure. We describe some earlier results for conservative forces and some recent progress and challenges on the problem for purely magnetic forces. Finally we suggest some extensions of this problem to Yang-Mills forces and its connections to the geodesic flow on pricipal bundles (through the so called Kaluza-Klein approach).

Thursday October 25th, 2018

Morgan Weiler, University of California Berkeley

Mean action of periodic orbits of area-preserving annulus diffeomorphisms

Given an area-preserving diffeomorphism ψ of an annulus, there is an “action function” f which measures how ψ distorts curves. The average value of f over the annulus is known as the Calabi invariant of ψ. We call the average value of f over a periodic orbit of ψ the mean action of the orbit. If ψ is rotation near the boundary of the annulus, and if the Calabi invariant is less than the maximum boundary value of f, then we show that the infimum of the mean action over all periodic orbits of ψ is less than or equal to the Calabi invariant.

***Special Day  Friday October 26th, 2018 Special Day***

Jingyi Chen, University of British Columbia

Some recent progress on Hamiltonian stationary Lagrangian submanifolds

In this talk, we will discuss recent results on regularity and compactness of Hamiltonian stationary Lagrangian submanifolds in $C^n$. These manifolds are critical points of the volume functional under Hamiltonian deformations.

Thursday November 1st, 2018

Thibault de Poyferre, University of California Berkeley

Gravity water waves and emerging bottom

To understand the behavior of waves at a fluid surface in configurations where the surface and the bottom meet (islands, beaches…), one encounters a difficulty: the presence in the bulk of the fluid of an edge, at the triple line. To solve the Cauchy problem, we need to study elliptic regularity in such domains, understand the linearized operator around an arbitrary solution, and construct an appropriate procedure to quasi-linearize the equations. Using those tools, I will present some a priori estimates, a first step to a local existence result.

Thursday November 8th, 2018

Selim Sukhtaiev, Rice University

The Maslov index and the spectra of second order elliptic operators
We describe relations between the Maslov index and the counting function for the spectrum of selfadjoint extensions of second order elliptic operators. Applications are given to the Schr\"odinger operators on domains in R^d.

Thursday November 15th, 2018

Marco Mazzucchelli, ENS Lyon & MSRI

Min-Max Characterizations of Zoll Riemannian manifolds

A closed Riemannian manifold is called Zoll when its unit-speed geodesics are all periodic with the same minimal period. This class of manifolds has been thoroughly studied since the seminal work of Zoll, Bott, Samelson, Berger, and many other authors. It is conjectured that, on certain closed manifolds, a Riemannian metric is Zoll if and only if its unit-speed periodic geodesics all have the same minimal period. In this talk, I will first discuss the proof of this conjecture for the 2-sphere, which builds on the work of Lusternik and Schnirelmann. I will then show an analogous result for certain higher dimensional closed manifolds, including spheres, complex and quaternionic projective spaces: a Riemannian manifold is Zoll if and only if two explicit min-max values in a suitable free loop space coincide. This is based on joint work with Stefan Suhr.

***Special Date & Time***

Tuesday November 20th, 2018

Bo Guan, Ohio State University

Fully nonlinear elliptic equations on Riemannian manifolds

Nonlinear partial differential equations play fundamental roles in many branches of mathematics, and especially in geometry. In this talk report recent progresses in our effort to search for general methods and techniques which enable us to solve fully nonlinear elliptic equations on real or complex manifolds under general conditions. We shall consider the Dirichlet and Neumann problems as well as equations on closed manifolds. For the Dirichlet problems, in particular, we are able to establish existence,results on Riemannian manifolds under conditions which are close to optimal. In the closed manifold case, we clarify relations between different notations of generalized subsolutions introduced by Szekelyhidi and myself independently.

Thursday November 29th, 2018

Pedro Morales, University of California Santa Cruz

Spectral Zeta Functions and their applications to the Casimir Effect.

In this talk, I will explore the Spectral Zeta Function associated with an elliptic (pseudo) differential operator on a compact Riemannian manifold. In this setting, the operator is self-adjoint and unbounded, preventing us for a well define trace and determinant. The Zeta Function formalism enables us to define a zeta-regularized trace for the differential operator, as well as a functional determinant. One direct application of this is in QFT effect, such as the vacuum energy and the Casimir Effect. These have to do with the fluctuations of quantum vacuum producing a non-zero vacuum energy that could produce an attractive or repulsive force depending on the boundary conditions.

Thursday December 6th, 2018

Or Hershkovits, Stanford University

Mean Curvature Flow of Surfaces.

In the last 35 years, geometric flows have proven to be a powerful tool in geometry and topology. The Mean Curvature Flow is, in many ways, the most natural flow for surfaces in Euclidean space.

In this talk, which will assume no prior knowledge, I will illustrate how mean curvature flow could be used to address geometric questions. I will then explain why the formation of singularities of the mean curvature flow poses difficulties for such applications, and how recent new discoveries about the structure of singularities (including a work joint with Kyeongsu Choi and Robert Haslhofer) may help overcome those difficulties.