When studying geometric stability questions it is common to obtain $L^p$ estimates for a sequence of Riemannian manifolds as a first step. As a next step one often would like to bootrstrap from $L^p$ estimates to Sormani-Wenger Intrinsic Flat (SWIF) and/or Gromov-Hausdorff (GH) convergence. We will discuss joint work with Raquel Perales and Christina Sormani which allows one to do just that. We will also present a Morrey type inequality which characterizes the $p$ for which one should expect to obtain both GH and SWIF convergence versus when one should only expect SWIF convergence. An example will be given which shows that $L^p$ convergence is strictly weaker than SWIF and GH convergence and we will give many examples which compare and contrast these notions that will provide intuition for all of the theorems discussed.

We consider the relativistic Euler equations with a physical vacuum boundary and an equation of state p(\varrho)=\varrho^\gamma$, $\gamma > 1$. We establish the following results. (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity only a half derivative above scaling; (iii) stability: our uniqueness in fact follows from a more general result, namely, we show that a certain nonlinear functional that tracks the distance between two solutions (in part by measuring the distance between their respective boundaries) is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions; (v) we establish a sharp continuation criterion, at the level of scaling, showing that solutions can be continued as long as the velocity is in $L^1_t Lip_x$ and a suitable weighted version of the density is at the same regularity level. This is joint work with Mihaela Ifrim and Daniel Tataru.

The classification of 3 + 1-static black hole solutions of the Einstein equations is central in General Relativity and important in geometry. In the realm of S1- symmetric spacetimes, a recent classification proved that, if the topology and the asymptotic are not constrained, S1-symmetric static vacuum black holes can be only of three kinds: (i) Schwarzschild black holes, (ii) Boost black holes, or (iii) Myers-Korotkin-Nicolai black holes, each one having its distinct asymptotic and topological type. In contrast to this, very little is known about the classifica- tion of S1-symmetric static electrovacuum solutions. In this talk we will discuss the remarkable feature that, while the Schwarzschild solution can be perturbed to a
charged static S1-symmetric solution, the Boosts and the Myers-Korotkin-Nicolai cannot: the addition of any electric charge would entirely destroy the asymptotic. We conjecture that the Boosts and the Myers-Korotkin-Nicolai topologies indeed do not accept an electrovacuum static black hole solution of any kind.

In this talk we introduce and solve the characteristic gluing problem for the Einstein vacuum equations. We show that obstructions to characteristic gluing come from an infinite-dimensional space of conservation laws along null hypersurfaces for the linearized equations at Minkowski. We prove that this space splits into an infinite-dimensional space of gauge-dependent charges and a 10-dimensional space of gauge-invariant charges. We identify the 10 gauge-invariant charges to be related to the energy, linear momentum, angular momentum and center-of-mass of the spacetime. Based on this identification, we explain how to characteristically glue a given spacetime to a suitably chosen Kerr spacetime. As corollary we get an alternative proof of the Corvino-Schoen spacelike gluing to Kerr. Moreover, we apply our characteristic gluing method to localise characteristic initial data along null hypersurfaces. In particular, this yields a new proof of the Carlotto-Schoen spacelike localization where our method yields no loss of decay, thus resolving an open problem. We also outline further applications. This is joint work with S. Aretakis (Toronto) and I. Rodnianski (Princeton).

The variational analysis of the Aviles-Giga functional is a classical problem in the calculus of variations. The energy functional was used to model smectic liquid crystals and thin film blisters. In this talk I will present some results on sharp regularity for unit vector fields in $\mathbb{R}^2$ satisfying certain entropy conditions. These conditions are related to the zero energy states of the Aviles-Giga functional. The problem can be reformulated as a differential inclusion into a non-elliptic set $K\subset\mathbb{R}^{2\times 2}$. As a consequence, we also get sharp regularity for solutions to this differential inclusion. This is joint work with Xavier Lamy and Andrew L.

A complete Riemannian manifold is called asymptotically hyperbolic if its ends are modeled on neighborhoods of infinity in hyperbolic space. There is a notion of mass for this class of manifolds defined as a coordinate invariant computed in a fixed asymptotically hyperbolic end and measuring the leading order deviation of the geometry from the background hyperbolic metric in the end. Asymptotically hyperbolic manifolds arize naturally in mathematical general relativity, in particular, as slices of asymptotically Minkowski spacetimes, in which case the mass of the slice coincides with the Bondi mass of the spacetime. Having reviewed these and related concepts, we will discuss our proof of the positive mass theorem in the asymptotically hyperbolic setting, which relies on the original ideas of Schoen and Yau and involves a blow-up analysis of the so-called Jang equation, a geometric PDE of mean curvature type

''Which manifolds admit a complete metric of positive scalar curvature R>0?'' has been the subject of a number of breakthrough works. The answer is known for certain classes of closed manifolds, but the story for noncompact manifolds is more subtle. A related question (emphasized by Gromov among others) is ``What is the geometry of spaces with R>0?''. Both of these questions turn out to be related to the Positive Mass Theorem for asymptotically flat manifolds. We describe this connection and some recent work with Ryan Unger and S-T Yau in this direction.

In any dimension, we classify which maximal globally hyperbolic flat spacetimes are foliated by constant mean curvature spacelike hypersurfaces. By a classic result of Cheng and Yau, such hypersurfaces are always complete. In dimension 2+1, we give a similar classification for foliations by surfaces of constant Gauss curvature. By contrast with the CMC case, there are many spacetimes whose Cauchy surfaces of constant Gauss curvature are not complete. I’ll talk about progress toward the still-open question of finding a criterion for completeness. An answer would give a characterization of all isometric embeddings of the hyperbolic plane into Minkowski 3-space. Everything is joint with Francesco Bonsante and Andrea Seppi.

Linkages, networks of nearly rigid bars meeting at freely rotating joints, serve as a simple mathematical model underpinning a variety of physical systems, from molecules to disordered spring networks to origami. Recent advances in fabrication has driven a resurgence of interest in designing mechanical systems based on linkages that can serve as “mechanical metamaterials”, materials whose properties can be controlled by cleverly designing their structure. I will discuss recent work in our group on understanding the mechanics of origami and other metamaterials from the point of view of designing and controlling “singularities,” places where the configuration space of the mechanism fail to be well-behaved. This allows us to understand the onset of rigidity in under-coordinated spring networks and, ultimately, develop mechanisms that can can perform simple — and robust — logical operations.