Classical differential geometry can be encoded in spectral data, such as Connes’ spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes’ non-commutative spin geometry encompassing noncommutative Riemannian, symplectic, complex-Hermitian and (Hyper-) Kähler geometry...

This is a survey of the relationship between C *-algebraic deformation quan-tization and the tangent groupoid in noncommutative geometry, emphasizing the role of index theory. We first explain how C *-algebraic versions of deformation quantization are related to the bivariant E-theory of Connes and Higson. With this background, we review how Weyl–Moyal quantization may be described using the ta...

We are interested in making sense of an integral over a singular space, for example the quotient of the torus T 2 by the lines of irrational slope α / ∈ Q. More generally, we would like to study the quotient space of the global unstable manifolds for a hyperbolic dynamical system. In this chapter we present an “easy” version of Connes’ noncommutative integration theory. We start by studying int...

We give a natural and complete description of Ecalle’s mould–comould formalism within a Hopf–algebraic framework. The arborification transform thus appears as a factorization of characters, involving the shuffle or quasishuffle Hopf algebras, thanks to a universal property satisfied by Connes–Kreimer Hopf algebra. We give a straightforward characterization of the fundamental process of homogene...

In this paper, we use the KK-theory of Kasparov to prove exactness of sequences relating the K-theory of a real C∗ -algebra and of its complexification (generalizing results of Boersema). We use this to relate the real version of the Baum-Connes conjecture for a discrete group to its complex counterpart. In particular, the complex Baum-Connes assembly map is an isomorphism if and only if the re...