On the topology and symmetry of black holes in five dimensions

The classification of higher dimensional stationary black hole solutions to the Einstein equations
is a major open problem in higher dimensional general relativity. Besides being of intrinsic
mathematical interest, this problem also has important applications in modern studies of high
energy physics, quantum gravity and string theory. This project will investigate the existence of
new asymptotically flat, stationary, black hole solutions to the five dimensional vacuum Einstein
equations which exhibit new types of topology or reduced symmetry. The first part of this project
will be to address a number of key open problems in the classification of solutions with biaxial
symmetry: the construction of black holes with lens-space horizon topology, and the existence of
black holes with non-trivial topology in the domain of outer communication (2- cycles). The second
part of the project will be to initiate the classification of solutions with a single axial
symmetry and in particular address the important conjecture that solutions of this type do actually
exist. Any definitive progress in these problems would be a valuable contribution to the black hole
classification program.