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.