Southern California
Analysis and Partial Differential Equations Conference

Titles and Abstracts

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.