Geometry & Analysis Seminar Spring 2019

Wednesday April 3rd, 2019

Qing Han, Notre Dame University

Asymptotic Expansions of Solutions of the Yamabe Equation and the sigma-k Yamabe Equation near Isolated Singular Points

We study asymptotic behaviors of positive solutions to the Yamabe equation and the sigma-k Yamabe equation near isolated singular points and establish expansions up to arbitrary orders. Such results generalize an earlier pioneering work by Caffarelli, Gidas, and Spruck, and a work by Korevaar, Mazzeo, Pacard, and Schoen, on the Yamabe equation, and a work by Han, Li, and Teixeira on the sigma-k Yamabe equation. The proof is based on a study of the linearized operators at radial solutions, following an approach adopted by Korevaar et al.

Wednesday April 10th, 2019

TBA

Wednesday April 17th, 2019

TBA

Wednesday April 24th, 2019

TBA

Wednesday May 1st, 2019

TBA

Wednesday May 8th, 2019

TBA

Wednesday May 15th, 2019

TBA 

Wednesday May 22nd, 2019

Dave Smith, Yale-NUS College

Nonlocal problems for linear evolution equations

Linear evolution equations, such as the heat equation, are commonly studied on finite spatial domains via initial-boundary value problems. In place of the boundary conditions, we consider “multipoint conditions”, where one specifies some linear combination of the solution and its derivative evaluated at internal points of the spatial domain, and “nonlocal” specification of the integral over space of the solution against some continuous weight. There are physical models, including diffusion with practically measurable data, in which such problems are more realistic.

Wednesday May 29th, 2019

Moritz Reintjes, Regensburg University 

How to smooth wrinkles in spacetime

The problem whether singularities are removable by coordinate transformation is central to General Relativity. I will report on a recent breakthrough regarding the question whether there exists coordinates in which the gravitational metric exhibits optimal regularity (i.e., two derivatives above the Riemann curvature) or whether regularity singularities exist. This breakthrough came about by discovering a system of elliptic partial differential equations (the RT-equations) which determines whether coordinates exist in which the metric exhibits op timal regularity. By developing an existence theory for the nonlinear RT-equations, we prove that optimal metric regularity can always be achieved and that no regularity singularities exist above a threshold level of smoothness. Without resolving the problem of optimal metric regularity the initial value problem of General Relativity remains incomplete. For fluid dynamical shock wave solutions of the Einstein equations, optimal metric regularity is the threshold smoothness that guarantees causal structures, the Newtonian limit to Classical Mechanics, strong (point-wise) solutions and the Hawking-Penrose singularity theorems. Extending the existence theory for the RT-equations to the case of GR shock waves is work in progress. Since the RT-equations are elliptic regardless of metric signature, one may picture the geometry of the gravitational metric to be spanned over the curvature of spacetime like the skin of a drum.

Wednesday June 5th, 2019

TBA