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Mathematical Aspects of General Relativity
Organized by: Piotr T. Chruściel (1), James Isenberg (2) and Alan Rendall (3)
(1) Département de Mathématiques, Université François Rabelais, Parc de Grandmont, F-37200, TOURS, FRANCE(2) Department of Mathematics, University of Oregon, OR 97403-5203, EUGENE, UNITED STATES
(3) Albert-Einstein-Institut, Max-Planck-Institute for Gravitational Physics, Am Mühlenberg 1, D-14476, GOLM, GERMANY
The conference brought together people working in mathematical general
relativity, a field which lies at the interface of analysis, differential
geometry and physics. There was a mix of established workers in the
subject with participants at the start of their careers and researchers
from neighbouring fields.
The Einstein equations are at the heart of general relativity theory. They
form a system of evolution equations and their solutions are conveniently
parametrized by initial data. These initial data are required to satisfy
constraint equations and these were the topic of several lectures at the
conference. The talks of Corvino and Pollack were concerned with recent
progress in methods for constructing solutions of the constraints. An
important concept for solutions of the constraints is that of mass, including
quasilocal mass. This was a central theme of the talks of Degeratu, Huisken
and Shi. Huisken introduced a striking new approach to the definition of
mass based on isoperimetric inequalities. Dain presented a variational
characterization of the extreme Reissner-Nordstr\"om solution, thus relating
the study of the constraint equations to the theory of black holes. The
talk of Wohlfarth was rather outside the main area of the conference. He
described a new concept of geometry motivated by string theory which may
come to enrich the circle of ideas within mathematical relativity.
Many of the talks at the conference were on evolution equations related
to general relativity. Dafermos and Finster presented results related to
the dynamical stability of black holes. Dafermos described new results on
the rate of decay of solutions of the wave equation on the Schwarzschild
spacetime while Finster explained applications of methods of functional
analysis to the wave equation on the Kerr spacetime. Struwe's talk concerned
uniqueness for supercritical nonlinear wave equations which from the point
of view of the Einstein equations are an important example to compare
with. The Maxwell equations are another important comparison system and
Bauer showed how they can be used to understand more about radiation
formulae. Vel\'azquez gave an introduction to singularity formation
in the Keller-Segel model, a parabolic system coming from mathematical
biology. This is a possible source of insight for obtaining a rigorous
understanding of critical collapse in general relativity. The talks of
Andr\'easson, Choptuik and Lindblom dealt with various aspects of the
application of numerical techniques to the study of the Einstein evolution
equations. Choptuik showed impressive new simulations of coalescing black
holes due to Pretorius which could hardly have been imagined just a year ago.
Cosmology is at present a very active area of research in general
relativity. This is in part due to the challenge of understanding the
observed accelerated expansion of our universe. Mathematics is beginning to
make its mark in this subject and this was reflected by talks of Heinzle,
Rendall and Tod.
Mathematical relativity is a meeting point for many ideas and the abstracts
which follow give some idea of the variety of the subject. In fact the
spectrum of topics discussed by the participants at the conference was
much wider than those for which talks could be scheduled. By limiting
the number of presentations it was possible to leave plenty of time for
people to exchange insights. The lively interactions observed make us
hopeful that this conference has given a boost to the development of the
subject in the next few years.
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