One of the striking initial applications of the Seiberg-Witten invariants was to give new obstructions to the existence of Riemannian metrics of positive scalar curvature on 4– manifolds. The vanishing of the Seiberg–Witten invariants of a manifold admitting such a metric may be viewed as a non-linear generalization of the classic conditions [12, 11] derived from the Dirac operator. If a manifo...

We report on a proof-checked version of Stone’s result on the representability of Boolean algebras via the clopen sets of a totally disconnected compact Hausdor↵ space. Our experiment is based on a proof verifier based on set theory, whose usability can in its turn benefit from fully formalized proofs of representation theorems akin to the one discussed in this note.

We exhibit a closed, simply connected 4-manifold X carrying two symplectic structures whose rst Chern classes in H 2 (X; Z) lie in dis-joint orbits of the diieomorphism group of X. Consequently, the mod-uli space of symplectic forms on X is disconnected.

The Kaluza-Klein reduction of 4d conformally flat spacetimes is reconsidered. The corresponding 3d equations are shown to be equivalent to 2d gravitational kink equations augmented by a centrifugal term. For space-like gauge fields and nontrivial values of the centrifugal term the gravitational kink solutions describe a spacetime that is divided in two disconnected regions.

We show that any infinite compact Hausdorff space S is the continuous image of a totally disconnected compact Hausdorff space S', having the same topological weight as S, by a map 95 which admits a regular linear operator of averaging, i.e., a projection of norm one of C(S') onto <p°C(S), where <p°: C(S) -*■ C(S') is the isometric embedding which takes fe C(S) into/« ¡p. A corollary of this the...

In most financial markets, securities are traded in isolation. Such a disconnected market design can be inefficient if agents trade more than one security. I assess welfare effects of connecting markets by allowing orders for security to depend on prices other securities. show that everyone trades identical amounts under both structures and only the clearing perfectly correlated or all price-ta...

We investigate when thick 3D polygonal chains are locked, i.e., have a disconnected configuration space. In particular, we show that thick 4-chains are never locked, and we exhibit a class of locked thick 5-chains whose ratio of maximum edge length to minimum edge length is strictly less than 3 (the best known ratio for nonthick chains).