‎For a given measure space $(X,{mathscr B},mu)$ we construct all measure spaces $(Y,{mathscr C},lambda)$ in which $(X,{mathscr B},mu)$ is embeddable‎. ‎The construction is modeled on the ultrafilter construction of the Stone--v{C}ech compactification of a completely regular topological space‎. ‎Under certain conditions the construction simplifies‎. ‎Examples are given when this simplification o...

We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton-MacPherson and Axelrod-Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embe...

We consider Kapranov’s Chow quotient compactification of the moduli space of ordered n-tuples of hyperplanes in P r−1 in linear general position. For r = 2, this is canonically identified with the Grothendieck-Knudsen compactification of M0,n which has, among others, the following nice properties: (1) modular meaning: stable pointed rational curves; (2) canonical description of limits of one-pa...

Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected components, due to spin and hyperelliptic structures. We prove that the spin p...

We present a non-supersymmetric compactification of M-theory down to seven dimensions which is close to a stable non-supersymmetric vacuum. This compactification is dual to a non-supersymmetric interacting conformal field theory in six dimensions. We argue that a given classical compactification of M-theory or string theory is close to a stable vacuum if it is dual to a conformal field theory w...

Vafa’s (11+1) F theory is extended by means of Bars’ 2T holographic theory to yield a 14d Multiverse theory that permeates the brane of a 12d Universe in which both the Universe and the Multiverse have (3+1) spacetimes. Given the 2d toroidal compactification of F theory, we conjecture that the Multiverse has a 4d Cartesian compactification that is filled with 3D+T spacetime via the standard 6d ...

We give an example of a horocycle in the Teichmüller space five-times-punctured sphere that does not converge Gardiner–Masur compactification, or equivalently horofunction compactification metric. As intermediate step, we exhibit simple closed curve whose extremal length is periodic but constant along horocycle. The lifts to any complex dimension greater than one via covering constructions.

We construct a normal projective rigid analytic compactification of an arbitrary Drinfeld modular variety whose boundary is stratified by varieties smaller dimensions. This generalizes work Kapranov. Using algebraic that Pink and Schieder's, we show the naturally isomorphic to analytification Pink's compactification. interpret forms as global sections natural ample invertible sheaves on deduce ...

We study the process of compactification as a topology change. It is shown how the mediating spacetime topology, or cobordism, may be simplified through surgery. Within the causal Lorentzian approach to quantum gravity, it is shown that any topology change in dimensions ≥ 5 may be achieved via a causally continuous cobordism. This extends the known result for 4 dimensions. Therefore, there is n...