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Let (M-i, g(i))(i is an element of N) be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold (B, h) in the Gromov-Hausdorff topology. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator D-B on B. We give an explicit description of D-B and characterize the special case where D-B equals the Dirac operator on B.
When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory language, this means that the theory is twisted, which gives rise to an anomaly theory. In this paper, we give a detailed construction of this anomaly theory, as a functor that sends manifolds to infinite-dimensional Clifford algebras and bordisms to bimodules.
For a closed, connected direct product Riemannian manifold (M, g) = (M-1, g(1)) x ... x (M-l, g(l)), we define its multiconformal class [[g]] as the totality {integral(2)(1)g(1) circle plus center dot center dot center dot integral(2)(l)g(l)} of all Riemannian metrics obtained from multiplying the metric gi of each factor Mi by a positive function fi on the total space M. A multiconformal class [[ g]] contains not only all warped product type deformations of g but also the whole conformal class [(g) over tilde] of every (g) over tilde is an element of[[ g]]. In this article, we prove that [[g]] contains a metric of positive scalar curvature if and only if the conformal class of some factor (Mi, gi) does, under the technical assumption dim M-i = 2. We also show that, even in the case where every factor (M-i, g(i)) has positive scalar curvature, [[g]] contains a metric of scalar curvature constantly equal to -1 and with arbitrarily large volume, provided l = 2 and dim M = 3.
Was wird unter „nachhaltiger Prävention“ in der Präventionsforschung verstanden? Welche guten Beispiele für nachhaltige Prävention gibt es in der Praxis? Und v. a.: Wie lässt sich Prävention in den verschiedenen Bereichen wie Kriminalität, Gewalt und Rechtsextremismus nachhaltig gestalten? Diesen Fragen will der vorliegende Sammelband nachgehen und damit der Präventionsdebatte neue Impulse verleihen. Der Band will insbesondere die nationale sowie internationale Fachdebatte konstruktiv aufgreifen, Theorie und Praxis verbinden, „good practice“ Beispiele darstellen sowie Perspektiven nachhaltiger Prävention aufzeigen. Mit diesem Themenspektrum richtet er sich sowohl an die Wissenschaft als auch an die Praxis sowie insgesamt an eine interessierte Öffentlichkeit.