Geometry & Analysis Seminar Fall 2019

Wednesday, October 9th, 2019

Thibault Lefeuvre, Paris-Sud University/MSRI

Geodesic stretch, pressure metric and the marked length spectrum rigidity conjecture

In 1985, Burns and Katok conjectured that the marked length spectrum of a negatively-curved Riemannian manifold (namely the collection of lengths of closed geodesics marked by the free homotopy of the manifold) should determine the metric up to isometries. This conjecture was independently proved for surfaces in 1990 by Croke and Otal but since then little progress has been accomplished in higher dimensions until our recent proof of the local version of the conjecture, obtained in collaboration with C. Guillarmou. Considering a geometric point of view in the moduli space of isometry classes, I will explain a new proof of this local version of the conjecture which relies on the notion of geodesic stretch. If time permits, I will show that this fits into a more general framework which generalizes Thurston's distance and the pressure metric (initially defined on Teichmuller space) to the setting of variable curvature and higher dimensions. Joint work with C. Guillarmou, G. Knieper.

Wednesday, October 16th, 2019 @ 2:45PM

Ruobing Zhang, Stony Brook University

Geometric analysis of collapsing Calabi-Yau spaces

This talk centers on the degenerations of Calabi-Yau manifolds, particularly on the interactions between algebraic degenerations and metric convergence in the collapsed contexts, which is a long standing question. As the complex structures degenerate, the collapsing Calabi-Yau metrics may exhibit various wild geometric properties and highly non-algebraic features. As motivating examples, we will also describe our recent results on the new collapsing mechanisms of K3 surfaces. The main part of the lecture is to explain the recent progress in higher dimensions.

Wednesday, October 16th, 2019 @ 4PM

Stephen Kleene, University of Rochester

Desingularizing Non-Degenerate Surfaces

I will discuss recent progress in singular perturbation from the point of view of the Cauchy data operator in the highly symmetric setting. In particular, we will show how to desingularize a generic highly symmetric compact surface satisfying a mean curvature type equation with a transversal self intersection.

Wednesday, October 23rd, 2019

Hadrian Quan, University of Illinois at Urbana-Champaign / MSRI

The Heat Kernel of a Contact Manifold in the Sub-Riemannian Limit

In this talk I will report on joint work with Pierre Albin in which we study the limits of heat kernels on contact manifolds corresponding to a Riemannian metric degenerating to a submetric of the contact structure. This question is approached using the tools of geometric microlocal analysis. Time permitting, I will discuss connections with the Rumin complex, and the associated limit of Analytic Torsion.

Wednesday, October 30th, 2019

TBA

Wednesday, November 6th, 2019

TBA

Wednesday, November 13th, 2019

TBA

Wednesday, November 20th, 2019

Zhen Huang, City University of New York

Blowing Up Closed Minimal Surfaces In Hyperbolic Three-Manifolds

In the late 70s, Uhlenbeck proposed a program to study the moduli space of minimal immersions of closed surfaces in hyperbolic 3-manifolds. This program addresses questions on existence, multiplicity and asymptotic behavior of such immersions. I will present several results in these directions, based on joint work with M. Lucia and G. Tarantello.

Wednesday, November 27th, 2019

TBA

Wednesday, December 4th, 2019

TBA 

Wednesday, December 11th, 2019

TBA