Southern California Analysis and Partial Differential Equations Conference
Titles and Abstracts

Manuel Tiglio (University of Maryland) "Dimensional Reduction and Einstein's Equations"
    Abstract: Modeling processes such as binary compact coalescences constitute a parametrized problem. An overview will be given of an effort to efficiently select which configurations to solve for, how to compactly represent them, and how to predict new ones based on an offline-online decomposition. Work done in collaboration with Harbir Antil (UMD), Sarah Caudill (LSU), Scott Field (UMD), Chad Galley (Caltech), Frank Herrmann (UMD), Jan Hesthaven (Brown), Ricardo Nochetto (UMD), Evan Ochsner (UWM), John Silberholz (MIT).

Greg Galloway (University of Miami) "Topological censorship from the initial data point of view."
    Abstract: The classical results on topological censorship, and more recent variations thereof, are spacetime results, that is, they are results involving assumptions essentially global in time. From the evolutionary point of view, there is the difficult question of determining whether a given initial data set will give rise to a spacetime satisfying these conditions. In order to separate out the principle of topological censorship from these difficult questions of global evolution, it would be useful to have a pure initial data version of topological censorship. In this talk we give a brief review of topological censorship, and we formulate and present such an initial data version. The approach taken here relies on recent developments in the existence theory for marginally outer trapped surfaces, and leads to a nontime-symmetric version of the purely Riemannian results of Meeks-Simon-Yau. This talk is based on joint work with Michael Eichmair and Dan Pollack.

Oscar Reula (Universidad Nacional de Cordoba) "Boundary conditions for coupled quasilinear wave equations with application to general relativity and electromagnetism"
    Abstract: In this talk we shall discuss the boundary value problem for second order quasilinear systems of equations. We shall introduce the notion of strongly stable boundary conditions and prove a theorem showing that a large set of boundary conditions fall into this class. In particular that set is appropriate to accomodate for the constraints in electromagnetism and general relativity in the harmonic gauge. We then relate this finding to the usual maximally dissipative boundary conditions. Finally we discuss the open problem of geometric uniqueness for these systems.

Spyros Alexakis (University of Toronto) "Bounds on the Bondi mass via curvature flux"
    Abstract: The Bondi mass is a positive quantity defined on null infinity of a given space-time, which measures the amount of gravitational energy that has escaped to infinity up to a given retarded time. In sufficiently regular space-times it is also the limit of the Hawking mass along infinite outgoing null surfaces. Given such a null surface in a vacuum space-time, we obtain bounds for the Bondi mass in terms purely of the curvature flux through the surface and the sphere from which it emanates. Joint work with Arick Shao.

Matt Choptuik (University of British Columbia) "Critical Phenomena in The Einstein-Vlasov System"
    Abstract: The Einstein-Vlasov system, in which collisionless matter is coupled to the general relativistic field, has been extensively studied for many years using both formal mathematical and computational techniques.
    In this talk I will discuss critical behaviour in the spherically symmetric version of the model. This involves investigation of the solutions that arise at the threshold of black hole formation (equivalently the threshold of blow-up) in evolutions of parametrized initial conditions. I will include a brief review of critical phenomena in general, as well as an overview of previous work for the spherically symmetric Einstein-Vlasov case. I will then describe ongoing efforts (with graduate students Roland Stevenson and Arman Akbarian) to improve the computational results using a direct solution of the Vlasov equation. Unresolved issues, including a possible mismatch between the numerics and results from perturbation theory will also be highlighted.

Jim Isenberg (University of Oregon) "AVTD Behavior in Smooth Solutions of Einstein's Equations"
    Abstract: One of the more useful approaches to studying the Strong Cosmic Censorship conjecture in a family of solutions of Einstein's equations is to first verify that generic solutions in that family exhibit AVTD (asymptotically velocity term dominated) behavior near their singular regions. It has been proven (by Ringstrom) that AVTD behavior occurs in generic Gowdy spacetimes, and it has also been shown that it occurs in at least some vacuum spacetimes with T^2 isometry, and in some with U(1) isometry. These T^2 and U(1) results have been proven using Fuchsian techniques, and have the unfortunate feature that, like many Fuchsian-based results, they require that the spacetimes be analytic. In work done with Florian Beyer, Philippe LeFoch, and Ellery Ames, we show that the analyticity condition can be removed, at least for the T^2 case. To prove this result, we have developed a variant of the Fuchsian technique which does not require analyticity. It is very likely that this variant can be applied to U(1) symmetric vacuum spacetimes as well as to those with T^2 symmetry.

Beverly Berger "Cosmological Spacetimes: Laboratories for the Mathematics of General Relativity"
    Abstract: Cosmological spacetimes with spatial symmetries provide simplified models to test theoretical, mathematical, and numerical approaches for both classical general relativity and quantum gravity. In this talk, I will focus on vacuum cosmological spacetimes with 3-torus spatial topology and two Killing fields, namely, the Gowdy and "galileo" models. Roughly speaking, these models consist of gravitational waves that drive the evolution of a background spacetime that originates in a big bang and expands forever. In the collapsing time-direction, the interesting question is the nature of the approach to the big bang singularity. I will describe how numerical simulations and heuristic methods suggest that the Gowdy models are velocity dominated while the galileo models are mixmaster-like in collapse. In the former case, proofs of this behavior exist. In the latter case, there is numerical evidence that the mixmaster behavior is not valid everywhere. These singularity studies are relevant for generic gravitational collapse. In contrast, in the expanding direction, each class of model spacetime has its own interesting features. Generic Gowdy models exhibit unexpected (but now understood) behavior found by Ringstrom. Numerical studies of the galileo spacetimes reveal a peculiar, attractor-like behavior that can be made plausible heuristically but is not yet understood.

Lydia Bieri (University of Michigan) "Null Asymptotic Analysis of Spacetimes"
    Abstract: This talk addresses the null asymptotic analysis of spacetimes in general relativity. We shall consider solutions of the Einstein equations describing isolated systems. In particular, I shall discuss new results where the metric of the solution spacetimes include non-isotropic mass terms. They naturally arise as global solutions to the initial value problem, when the initial data is suitably small. However, the null asymptotical results are largely independent of the smallness, which allows us to study corresponding large data scenarios. Whereas existence and uniqueness of such solutions (under smallness assumptions) has been implied by my former work, here, I am interested in more precise asymptotics. I will discuss the structures at null infinity, based on which we can derive results on gravitational radiation for the corresponding spacetimes.