We first show that the intrinsic, geometrical structure of a dynamical horizon (DH) is unique. A number of physically interesting constraints are then established on the location of trapped and marginally trapped surfaces in the vicinity of any DH. These restrictions are used to prove several uniqueness theorems for DH. Ramifications of some of these results to numerical simulations of black hole spacetimes are discussed. Finally, several expectations on the interplay between isometries and DHs are shown to be borne out.

We study geometric transitions for topological strings on compact Calabi–Yau hypersurfaces in toric varieties. Large $N$ duality predicts an equivalence between topological open and closed string theories connected by an extremal transition. We develop new open string enumerative techniques and perform a high-precision genus zero test of this conjecture for a certain class of toric extremal transitions. Our approach is based on (a) an open string version of Gromov–Witten theory with convex obstruction bundle and (b) an extension of Chern–Simons theory treating the framing as a formal variable.

We extend the conformal gluing construction of Isenberg et al. [19] by establishing an analogous gluing result for field theories obtained by minimally coupling Einstein’s gravitational theory with matter fields. We treat classical fields such as perfect fluids and the Yang–Mills equations as well as the Einstein–Vlasov system, which is an important example coming from kinetic theory. In carrying out these extensions, we extend the conformal gluing technique to higher dimensions and codify it in such a way as to make more transparent where it can, and cannot, be applied. In particular, we show exactly what criteria need to be met in order to apply the construction, in its present form, to any other non-vacuum field theory.