Index theory and noncommutative geometry: a survey

@article{Gorokhovsky2019IndexTA,
  title={Index theory and noncommutative geometry: a survey},
  author={Alexander Gorokhovsky and Erik van Erp},
  journal={Advances in Noncommutative Geometry},
  year={2019}
}
This chapter is an introductory survey of selected topics in index theory in the context of noncommutative geometry, focusing in particular on Alain Connes’ contributions. This survey has two parts. In the first part, we consider index theory in the setting of K-theory of C∗ algebras. The second part focuses on the local index formula of A. Connes and H. Moscovici in the context of noncommutative geometry. 
3 Citations
Noncommutative Geometry, the Spectral Standpoint
  • A. Connes
  • Mathematics
    New Spaces in Physics
  • 2019
We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for noncommutative
Spectral action and the electroweak θ-terms for the Standard Model without fermion doubling
Abstract We compute the leading terms of the spectral action for a noncommutative geometry model that has no fermion doubling. The spectral triple describing it, which is chiral and allows for
Local Index Formulae on Noncommutative Orbifolds and Equivariant Zeta Functions for the Affine Metaplectic Group.
We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$ generated by quantizations of isometric affine canonical transformations. The algebra $A$ includes as subalgebras all

References

SHOWING 1-10 OF 86 REFERENCES
The Local Index Formula in Noncommutative Geometry
In this chapter we present a proof of the Connes–Moscovici index formula, expressing the index of a (twisted) operator \(D\) in a spectral triple \((\mathcal {A},\mathcal {H},D)\) by a local formula.
The Local Index Formula in Noncommutative Geometry
In noncommutative geometry a geometric space is described from a spectral vantage point, as a tripleA, H, D consisting of a *-algebraA represented in a Hilbert spaceH together with an unbounded
TOPOLOGICAL INVARIANTS OF ELLIPTIC OPERATORS. I: K-HOMOLOGY
In this paper the homological K-functor is defined on the category of involutory Banach algebras, and Bott periodicity is proved, along with a series of theorems corresponding to the
K-homology and index theory on contact manifolds
This paper applies K-homology to solve the index problem for a class of hypoelliptic (but not elliptic) operators on contact manifolds. K-homology is the dual theory to K-theory. We explicitly
Noncommutative Geometry
Noncommutative Spaces It was noticed a long time ago that various properties of sets of points can be restated in terms of properties of certain commutative rings of functions over those sets. In
Classifying Space for Proper Actions and K-Theory of Group C*-algebras
We announce a reformulation of the conjecture in [8,9,10]. The advantage of the new version is that it is simpler and applies more generally than the earlier statement. A key point is to use the
K-homology and Fredholm operators I: Dirac operators
Quantum $K$-theory. I. The Chern character
We construct a cocycle on an infinite dimensional generalization of ap-summable Fredholm module. Our framework is related to Connes' cyclic cohomology and is motivated by our work on index theory on
...
...