Characteristic Homomorphisms of Regular Lie Algebroids
Among all -algebras we characterize those which are algebras of -functions on second countable Hausdorff -manifolds.
We outline some of the tools C. Ehresmann introduced in Differential Geometry (fiber bundles, connections, jets, groupoids, pseudogroups). We emphasize two aspects of C. Ehresmann's works: use of Cartan notations for the theory of connections and semi-holonomic jets.
We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.
We present a direct analytic treatment of the Rokhlin congruence formula R2 by calculating the adiabatic limit of -invariants of Dirac operators on circle bundles. Extensions to higher dimensions are obtained.