A Quasi-Local Penrose Inequality and a Hoop Theorem

The Penrose Inequality is a conjecture in mathematical general relativity that relates the ADM mass of an asymptotically flat manifold with the area of Marginally Outer Trapped Surfaces in the manifold (MOTS). MOTS are Lorentzian analogs to minimal surfaces, and the conjecture may thought of as an ambitious refinement of the Positive Mass Theorem of Schoen Yau 79/81 and Witten 81, which is a fundamental result in the study of spaces with positive scalar curvature. The Hoop Conjecture aims to give necessary and sufficient conditions for the existence of MOTS within a given domain. We will present some recent progress in both directions. This is based on joint work with A.Alaee and S.T.Yau. Martin Lesourd

On the Cauchy problem for the Hall magnetohydrodynamics

In this talk, I will describe a recent series of work with I.-J. Jeong on the Hall MHD equation without resistivity. This PDE, first investigated by the applied mathematician M. J. Lighthill, is a one-fluid description of magnetized plasmas with a quadratic second-order correction term (Hall current term), which takes into account the motion of electrons relative to positive ions. Curiously, we demonstrate ill(!)posedness of the Cauchy problem near the trivial solution, despite the apparent linear stability and conservation of energy. On the other hand, we identify several regimes in which the Cauchy problem is well-posed, which not only includes the original setting that M. J. Lighthill investigated (namely, for initial data close to a uniform magnetic field) but also possibly large perturbations thereof. Central to our proofs is the viewpoint that the Hall current term imparts the magnetic field equation with a quasilinear dispersive character. Sung-Jin Oh

New Formulations of Relativistic and non-Relativistic compressible Euler flow: Miraculous Geo-Analytic Structures and Applications

We recently derived new formulations of the compressible Euler equations, in both the relativistic and non-relativistic cases. The new formulations exhibit miraculous geo-analytic structures, in- cluding I) A sharp decomposition of the flow into geometric “wave parts” and “transport-div-curl parts;” II) Null form source terms; and III) Structures that allow one to propagate one additional degree of differentiability (compared to standard estimates) for the vorticity and entropy. The structures allow one to import techniques and insights from mathematical general relativity and the theory of wave equations into the study of compressible fluid flow. We were inspired to search for such formulations by Christodoulou’s groundbreaking 2007 monograph on shock formation in irrotational and isentropic regions. I will then describe how the new formulations can be used to derive sharp results about the dynamics, including results on stable shock formation and the ex- istence of low-regularity solutions. I will emphasize the role that nonlinear geometric optics plays in the setup and highlight how the new formulation allows for its implementation. Finally, I will connect the new formulations to the broader goal of obtaining a rigorous mathematical theory that models the long-time behavior of solutions that can develop shocks. Various aspects of this work are joint with L. Abbrescia, J. Luk, M. Disconzi, C. Luo, and G. Mazzone. Jared Speck

The mass minimal extension of Bartnik's quasi-local mass

Given a compact Riemannian manifold $(\Omega, g)$ with boundary, it is of great interest to define an invariant determined by the boundary geometry, so-called the quasi-local mass. Among various proposals, R. Bartnik gave a definition of quasi-local mass in 1989 by considering admissible asymptotically flat extensions of $(\Omega, g)$ and minimizing the asymptotically defined masses. He proposed the conjecture that an extension that minimizes the mass (called a mass minimal extension), if exists, must be stationary. In this talk we will discuss partial progress toward this conjecture, based on joint work with Dan Lee. We identify a general obstruction to promote the dominant energy condition and then use it to show that an exterior region of the mass minimal extension must be vacuum and admit a Killing initial data. Lan-Hsuan Huang.

The Linear Stability of the Reissner-Nordstrom black hole in the full sub-extremal range.

The stability of black holes is one of the main open problems in General Relativity, and consists in understanding the long time behaviour of solutions to the Einstein equation. We give an overview of the problem of stability to the Einstein vacuum equation, and then focus on the case of the Einstein-Maxwell equation, i.e. gravitational and electromagnetic perturbations of a black hole. We describe the identification of gauge-invariant quantities and the symmetric coupled system of wave equations they satisfy in the case of Reissner-Nordstrom black hole. Boundedness of the energy and Morawetz estimates can be obtained in the full sub-extremal range by making use of a combined energy-momentum tensor for the system. Elena Giorgi

The Minkowski Inequality in de Sitter space

The classical Minkowski inequality in the Euclidean space provides a lower bound on the total mean curvature of a convex hypersurface in terms of the surface area, which is optimal on round spheres. In this talk, we present a curvature flow approach to prove a properly defined analogue in the Lorentzian de Sitter space. Julian Scheuer

Extension of the Mean Curvature Flow I

This is the first talk of a two-talk series. We will cover the first part (a compactness result) of a recent paper by Haozhao Li and Bing Wang that shows the mean curvature flow of embedded compact surfaces in $\mathbb{R}^3$ can be smoothly continued as long as the mean curvature remains bounded. Keaton Naff

Null Geometry and the Gibbons-Penrose Inequality

In this pair of talks, I will first spend some time introducing the basics of null geometry with a view toward discussing the Penrose Inequality for an imploding `null dust'. By way of the cosmic censorship hypothesis, this gives rise to an interesting geometric inequality for 2-spheres in Minkowski space that exhibit a certain convexity assumption. This so called Gibbons-Penrose inequality, although a simple case of the Penrose conjecture, remains unknown from Penrose's original 1973 paper. Henri Roesch

Geometric inequalities for quasi-local masses

In this talk, we establish lower bounds for quasi-local masses in terms of charge, angular momentum, and horizon area. In particular, we treat three quasi-local masses based on a Hamiltonian approach, namely the Brown-York, Liu-Yau, and Wang-Yau masses. The geometric inequalities are motivated by analogous results for the ADM mass. They may be interpreted as localized versions of these inequalities, and are also closely tied to the conjectured Bekenstein bounds for the entropy of macroscopic bodies. In addition, we give a new proof of the positivity property for the Wang-Yau mass which can be used to remove the spin condition in higher dimensions. Furthermore, we generalize a recent result of Lu and Miao to obtain a localized version of the Penrose inequality for the static Liu-Yau mass. This is joint work with Marcus Khuri and Shing-Tung Yau. Aghil Alaee/Marcus Khuri

Spacetimes with gravitational radiation

This is an elementary talk about the mathematical theory of gravitational radiation/waves. I will start with a brief introduction of General Relativity and the Einstein equation, and follow by a description of null infinity where gravitational radiation takes place. I will then move on to discuss several simplest dynamical spacetimes with radiation that include the Vaidya spacetime and the Robinson-Trautman spacetime. Mu-Tao Wang