The efforts of geometers and physicists in exploiting the interconnections between differential geometry and topics in mathematical physics such as general relativity and string theory have led to significant progress in both domains. For example, one is reminded of the proofs of positive mass theorems via minimal surface or spinorial techniques, their connection to positive scalar curvature, and the subsequent use of concepts of mass in studying complete Riemannian manifolds satisfying appropriate asymptotic conditions such as flatness, hyperbolicity, or conformal compactness. As well, the recent proofs of the Riemannian Penrose inequality (for dimensions < 8) rely on understanding the inverse mean curvature flow and the conformal flow. What is common to these examples is that a problem in relativity is reformulated as a problem in Riemannian geometry, which is then solved by geometric analysis. By contrast, other more recent significant developments in general relativity have come about by studying the Einstein field equations directly as a hyperbolic partial differential system. Examples here include the proof of the the nonlinear stability of Minkowski space, and the numerious recent works motivated by the stability problem for black holes and singularities in general relativity. These works have underscored the importance of meeting head-on the challenges of applying geometric analysis in a Lorentzian background as well as resolving specific differential geometric issues about Lorentz manifolds. These challenges motivate the second subtheme of our proposal. Understanding the interaction between Lorentz geometry and Riemannian geometry is also essential to grand unification theories such as String Theory. For example, in studying the AdS/CFT correspondence, mathematicians have thus far largely switched to Euclidean signature and focussed on conformally compact Riemannian manifolds, or else turned to studying Sasaki-Einstein geometry separately. Again we hope to provide a forum for studying this fascinating relationship between Lorentzian and Riemannian geometries that physicists have suggested.