3 + 1 dimensional Causal Dynamical Triangulations (CDT) describe a quantum theory of fluctuating geometries without the introduction of a background geometry. If the topology of space is constrained to be that of a three-dimensional torus we show that the system will fluctuate around a dynamically formed background geometry which can be understood from a simple minisuperspace action which conta...

This is a survey of various applications of analytical and geometric techniques to problems in manifold topology. The author has been involved in only some of these developments, but it seems more illuminating not to confine the discussion to these. We begin by recalling the notion of a manifold. Suppose we are provided with a large collection of small paper discs. Then we can construct a wide ...

Two-dimensional cell complexes form a remarkably rich class of objects; indeed, one can regard all of group theory as a sub-theory of 2-dimensional topology. One of the key invariants at our disposal is the second homotopy group, π2, and starting with the work of Whitehead and Reidemeister, a good theory of the structure of π2 has been developed. The aim of the talk is to describe this basic st...

We investigate Hamiltonian chaotic scattering in physically realistic three-dimensional potentials. We find that the basin topology of the scattering dynamics can undergo a metamorphosis from being totally disconnected to being connected as a system parameter, such as the particle energy, is varied through a critical value. The dynamical origin of the metamorphosis is investigated, and the topo...

The early work of Mostow, Margulis and Prasad on rigidity of arithmetic lattices has evolved into a broad use of quasi-isometry techniques in group theory and low dimensional topology. The word metric on a finitely generated group makes it into a metric space which is uniquely determined up to the geometric relation called quasi-isometry, despite the fact that the metric depends on the choice o...

We investigate topology changing processes in the WKB approximation of four dimensional quantum cosmology with a negative cosmological constant. As Riemannian manifolds which describe quantum tunnelings of spacetime we consider constant negative curvature solutions of the Einstein equation i.e. hyperbolic geometries. Using four dimensional polytopes, we can explicitly construct hyperbolic manif...

We investigate topology change in (1+1) dimensions by analyzing the scalarcurvature action 12 ∫ RdV at the points of metric-degeneration that (with minor exceptions) any nontrivial Lorentzian cobordism necessarily possesses. In two dimensions any cobordism can be built up as a combination of only two elementary types, the “yarmulke” and the “trousers.” For each of these elementary cobordisms, w...

The interplay between ferroelectricity and band topology can give rise to a wide range of both fundamental applied research. Here, we map out the emergence nontrivial corner states in two-dimensional ferroelectrics, remarkably demonstrate that are coupled together by crystallographic symmetry realize electric control higher-order topology. Implemented density functional theory, identify series ...