Black Hole Final State Conspiracies

The principle that unitarity must be preserved in all processes, no matter how exotic, has led to deep insights into boundary conditions in cosmology and black hole theory. In the case of black hole evaporation, Horowitz and Maldacena were led to propose that unitarity preservation can be understood in terms of a restriction imposed on the wave function at the singularity. Gottesman and Preskill showed that this natural idea only works if one postulates the presence of "conspiracies" between systems just inside the event horizon and states at much later times, near the singularity. We argue that some AdS black holes have unusual internal thermodynamics, and that this may permit the required "conspiracies" if real black holes are described by some kind of sum over all AdS black holes having the same entropy.Comment: Various minor improvements, references added, 25 page

Pattern-Equivariant Homology of Finite Local Complexity Patterns

This thesis establishes a generalised setting with which to unify the study of finite local complexity (FLC) patterns. The abstract notion of a "pattern" is introduced, which may be seen as an analogue of the space group of isometries preserving a tiling but where, instead, one considers partial isometries preserving portions of it. These inverse semigroups of partial transformations are the suitable analogue of the space group for patterns with FLC but few global symmetries. In a similar vein we introduce the notion of a \emph{collage}, a system of equivalence relations on the ambient space of a pattern, which we show is capable of generalising many constructions applicable to the study of FLC tilings and Delone sets, such as the expression of the tiling space as an inverse limit of approximants. An invariant is constructed for our abstract patterns, the so called pattern-equivariant (PE) homology. These homology groups are defined using infinite singular chains on the ambient space of the pattern, although we show that one may define cellular versions which are isomorphic under suitable conditions. For FLC tilings these cellular PE chains are analogous to the PE cellular cochains \cite{Sadun1}. The PE homology and cohomology groups are shown to be related through Poincar\'{e} duality. An efficient and highly geometric method for the computation of the PE homology groups for hierarchical tilings is presented. The rotationally invariant PE homology groups are shown not to be a topological invariant for the associated tiling space and seem to retain extra information about global symmetries of tilings in the tiling space. We show how the PE homology groups may be incorporated into a spectral sequence converging to the \v{C}ech cohomology of the rigid hull of a tiling. These methods allow for a simple computation of the \v{C}ech cohomology of the rigid hull of the Penrose tilings.Comment: 159 pages, 8 figures, PhD thesi

A variational perspective on auxetic metamaterials of checkerboard-type

The main result of this work is a homogenization theorem via variational convergence for elastic materials with stiff checkerboard-type heterogeneities under the assumptions of physical growth and non-self-interpenetration. While the obtained energy estimates are rather standard, determining the effective deformation behavior, or in other words, characterizing the weak Sobolev limits of deformation maps whose gradients are locally close to rotations on the stiff components, is the challenging part. To this end, we establish an asymptotic rigidity result, showing that, under suitable scaling assumptions, the attainable macroscopic deformations are affine conformal contractions. This identifies the composite as a mechanical metamaterial with a negative Poisson's ratio. Our proof strategy is to tackle first an idealized model with full rigidity on the stiff tiles to acquire insight into the mechanics of the model and then transfer the findings and methodology to the model with diverging elastic constants. The latter requires, in particular, a new quantitative geometric rigidity estimate for non-connected squares touching each other at their vertices and a tailored Poincar\'e type inequality for checkerboard structures.Comment: 40 pages, 13 figure