Physics at the edge

In recent years, it has become clear that in certain physical situations the degrees of freedom in the bulk of a material or space-time can be repacked in terms of degrees of freedom confined to the edge of the material or space-time. In this dissertation, two physical instances of this effect were studied. In the first part of the dissertation we focus on the AdS/CFT correspondence; a duality between quantum gravity in asymptotically anti-de Sitter space-times and conformal field theories (CFTs). We formulate a set of necessary and sufficient conditions that the CFTs need to obey in order to reproduce Einstein gravity in the bulk. This gives insight into what CFTs can have a gravitational dual. Furthermore, in this part we also propose a deviation from the usual AdS/CFT correspondence for infinite volume to finite volume. This might be an essential first step in understanding holography in our own Universe.
The second part of the dissertation is devoted to topological insulators, which also allow for a description in terms of the physics at the edge. We focus on the classification of topological insulators and propose an efficient and intuitive algorithm to count all possible topological insulators with a given crystal symmetry and in the presence or absence of time-reversal symmetry. We check our results with known computations, but also perform the necessary computations in twisted equivariant K-theory ourselves in cases where no result was available and find perfect agreement.