Two manifolds are commensurable if they have diffeomorphic covers. We would like invariants that distinguish manifolds up to commensurability. A collection of such commensurability invariants is complete if it always distinguishes non-commensurable manifolds. Commensurability invariants of hyperbolic 3-manifolds are discussed in [NRe]. The two main ones are the invariant trace field and the inv...

asking about the place of religion in a democratic society refers straightforwardly to the kind of pluralism we adopt. given that intra-societal tensions mark out a democratic pluralistic society, then it seems that there is no doubt that there should be a place for religion and religious people in it. what is crucial for a democratic society is taking a suitable view on pluralism. there could ...

In the face of disuniacation and incommensurability, how can the scientiac community maintain itself and (re-)establish commensurability? According to Peter Galison’s investigations of twentieth-century microphysics, commensurability is achieved through local coordination even in the absence of global meaning: The “strength and coherence” of science is due to diverse, yet coordinated action in ...

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of Xa,b = (H ) × (H). A special case describes all Shimura subvarieties of type A1 Shimura varieties. We produce, for any n ≥ 1, examples of manifolds/Shimura varieties with precisely n commensurability classes of totally geodesic submanifolds/Shimura sub...

Tidal evolution has significantly affected the history of the Uranian satellite system. The presence of large chaotic zones at past mean-motion commensurabilities among the Uranian satellites has resulted in significant changes in the orbital elements of some of the satellites, while allowing them to escape from the commensurabilities and evolve to their present nonresonant configuration. Miran...

Fermionic atoms in two different hyperfine states confined in optical lattices show strong commensurability effects due to the interplay between the atomic density wave ordering and the lattice potential. We show that spatially separated regions of commensurable and incommensurable phases can coexist. The commensurability between the harmonic trap and the lattice sites can be used to control th...