Abstract: We discuss the definition and basic properties of the infinitesimal index as introduced be De Concini-Procesi-Vergne. Unlike the index of transversally elliptic operators, its domain is the equivariant cohomology with compact supports of the zeroes of the moment map. We also look at the special case of the cotangent bundle of a representation space, where connections to splines appear.

Abstract: It is classical that the topological K-theory KU(X) of a space X agrees with maps from X to KU, and that cohomology operations correspond to maps from KU to itself. Dual to this is the structure of "co-operations", i.e., the KU-homology of KU relative to the ring KU(point). This data has a structure, dubbed Hopf algebroid, which is related to combinatorics and numerical polynomials. In joint work with Pelaez, we determine the analogous structure for algebraic K-theory KGL, regarded as a motivic object. Applying the motivic slice filtration, we solve a problem of Voevodsky.