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For the past forty years, Robert Bartnik has been one of the
leading mathematicians working on mathematical general relativity
and geometric analysis. Since his early dissertation work on the
existence of maximal hypersurfaces in general asymptotically flat
spacetimes, done under the guidance of S.T. Yau at the Institute
for Advanced Study at Princeton, Bartnik's work has had a major
impact on a number of different areas in mathematical relativity.
His careful definition of the ADM mass on asymptotically Euclidean
geometries, together with his analysis of the Laplace operator on
such geometries, has been highly influential in geometric analysis.
This work led in turn to his insightful definition of ""quasi-local
mass,"" a topic of intense interest to this day. Bartnik's
collaboration with his student John McKinnon resulted in their
iconic discovery of a globally regular static solution of the
Einstein-Yang-Mills equations. His proof that there exist globally
hyperbolic spacetime solutions of Einstein's equations, which
contain no constant mean curvature Cauchy surfaces, was very
surprising, and has led to a variety of further results of this
nature. The procedure he developed for generating solutions of the
Einstein constraint equations using a parabolic PDE system, has
already led to important applications and is likely to be very
useful in the future. With the publication of this volume, the
editors wish to honor Robert Bartnik's great contributions to their
field. Included in this collection are most of his published
papers, together with short essays by friends and colleagues who
have been strongly influenced by him. The editors dedicate this
collection to Robert, and to all those who will greatly benefit
from being introduced to his work.
During the 2015-2016 year at Harvard University's Center of
Mathematical Sciences and Applications (CMSA), several researchers
working in mathematical general relativity presented lectures on
modern topics of research in the field of "Non-linear Equations."
This volume presents articles-by those researchers and their
co-authors-drawn from their CMSA lectures. Specific topics include
the Cauchy problem for the Einstein equations in cosmological and
non-cosmological settings; investigation of stability as well as
singularities (black holes) of classes of spacetimes; initial data
engineering; gravitational radiation; and asymptotics of
spacetimes, quasi-local energies, and their limits. The content of
this volume reflects some of the activities at the Harvard CMSA
during the 2015-2016 program, and provides insights into active
areas of research in mathematical general relativity that can
benefit scholars working in PDEs, geometric analysis, and general
relativity.
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