Limits of Geometries

Limits of Geometries

A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spaces, describing the basic process by which one homogeneous geometry may transf...

Casimir effect for curved geometries: proximity-force-approximation validity limits.

We compute Casimir interaction energies for the sphere-plate and cylinder-plate configuration induced by scalar-field fluctuations with Dirichlet boundary conditions. Based on a high-precision calculation using world-line numerics, we quantitatively determine the validity bounds of the proximity-force approximation (PFA) on which the comparison between all corresponding experiments and theory a...

Limits in modified categories of interest

‎We firstly prove the completeness of the category of crossed modules in a modified category of interest‎. ‎Afterwards‎, ‎we define pullback crossed modules and pullback cat objects that are both obtained by pullback diagrams with extra structures on certain arrows‎. ‎These constructions unify many corresponding results for the cases of groups‎, ‎commutative algebras and can also be adapted to ...

Generalized Covering Groups and Direct Limits

Homogeneous Geometries

For the purposes of this paper, a geometry will consist of a set together with a closure operation on that set, satisfying the exchange condition, and under which singletons are closed (for more precise definitions, see §2.1). The geometry is homogeneous if in the automorphism group of the geometry, the pointwise stabilizer of any finitedimensional closed subset of the geometry is transitive on...