Abstract: Abstract: We construct an unbounded representative for the shriek class associated to the embeddings of spheres into Euclidean space. We equip this unbounded Kasparov cycle with a connection and compute the unbounded Kasparov product with the Dirac operator on R^n+1 . We find that the resulting spectral triple for the algebra C(S^n) differs from the Dirac operator on the round sphere by a so-called index cycle, whose class in KK_0(C, C) represents the multi- plicative unit. At all points we check that our construction involving the unbounded Kasparov product is compatible with the bounded Kasparov product using Kucerovsky’s criterion and we thus capture the composition law for the shriek map for these immersions at the unbounded KK-theoretical level. The necessary KK-theory will be introduced as a black box.

Abstract: A finite graph determines a Kirchhoff polynomial, which is a squarefree, homogeneous polynomial in a set of variables indexed by the edges. The Kirchhoff polynomial appears in an integrand in the study of particle interactions in high-energy physics, which provides some incentive to study the motives and periods arising from the projective hypersurface cut out by such a polynomial.

From this perspective, work of Bloch, Esnault and Kreimer (2006) suggested that the more natural object of study is, in fact, a polynomial determined by a hyperplane arrangement, which is closely related to the basis generating polynomial of the associated matroid. I will describe joint work with Mathias Schulze and Uli Walther on the singular loci of such polynomials.