Colloquium

Black holes play a central role in astrophysics and in physics more generally. Candidate black holes are nearly ubiquitous in nature. They are found in the cores of nearly all galaxies, and appear to have resided there since the earliest cosmic times. They are also found throughout the galactic disk as companions to massive stars. Though these objects are almost certainly black holes, their properties are not very well constrained. We know their masses (often with errors that are factors of a few), and we know that they are dense.

This talk will discuss some surprising links which have emerged in the last few years between two at first sight distinct areas of mathematical physics: the spectral properties of certain simple schroedinger-like equations, and the Bethe ansatz techniques which are used to compute the energies of states in integrable quantum field theories. No knowledge of either area will be assumed.

An entirely new kind of band insulator was discovered recently. These new electronic states - called "topological insulators" - are fundamentally different from standard band insulators. They are distinguished by the fact that their edges (in the 2D case) or surfaces (in the 3D case) support gapless transport which is extremely robust. In the two dimensional case, topological insulators can be thought of as time reversal invariant analogues of integer quantum Hall states.

I will discuss distinctions between dark energy and modified gravity explanations of cosmic acceleration from the horizon scale to the deeply non-linear regime using the modified action f(R) and braneworld DGP models as worked toy examples.