arXiv research

We give a superconnection proof of Connes' index theorem for proper cocompact actions of etale groupoids. This includes Connes' general foliation index theorem for foliations with Hausdorff holonomy groupoid.

Survey on metrics and assembly maps in positive scalar curvature.

We use the symbol calculus for foliations developed in our previous paper to derive a cohomological formula for the Connes-Chern character of the semi-finite spectral triple. The same proof works for the Type I spectral triple of Connes-Moscovici. The cohomology classes of the two Connes-Chern characters induce the sam…

In this paper, we prove that infinitesimal equivariant Chern-Connes characters are well-defined. We decompose an equivariant index as a pairing of infinitesimal equivariant Chern-Connes characters with the Chern character of an idempotent matrix. We compute the limit of infinitesimal equivariant Chern- Connes character…

Paper introduces Lie algebroid index theory and a generalized Riemann-Roch theorem.

Extends coarse index theory to locally compact groups for Callias operators.

For a Lie groupoid G with a twisting (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. Fo…

Study of spinors and tangent groupoid for index theorem.

Defines and computes geometric pairings for discrete groups using Baum-Connes assembly map.

We construct certain spectral triples in the sense of A. ~Connes and H. Moscovici (``The local index formula in noncommutative geometry'' {\it Geom. Funct. Anal.}, 5(2):174--243, 1995) that is transversally elliptic but not necessarily elliptic. We prove that these spectral triples satisfie the conditions which ensure …

In this paper we consider a family of Dirac-type operators on fibration $P \to B$ equivariant with respect to an action of an etale groupoid. Such a family defines an element in the bivariant $K$ theory. We compute the action of the bivariant Chern character of this element on the image of Connes' map $Φ$ in the cyclic…

We present an index theorem for certain hypoelliptic differential operators on foliated manifolds. Our proof is a development of Alain Connes tangent groupoid proof of the Atiyah-Singer index theorem. The paper is largely self-contained.

Defines an equivariant index for proper actions by $H imes G$.

In his book (II.5), Connes gives a proof of the Atiyah-Singer index theorem for closed manifolds by using deformation groupoids and appropiate actions of these on R^N. Following these ideas, we prove an index theorem for manifolds with boundary.

Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of the index map of a twisted spectral triple in terms of $σ$-connections on finitely generated projective modules. This ma…

We prove a local index formula in conformal geometry by computing the Connes-Chern character for the conformal Dirac (twisted) spectral triple recently constructed by Connes-Moscovici. Following an observation of Moscovici, the computation reduces to the computation of the CM cocycle of an equivariant Dirac (ordinary) …

This paper is the second part of a series of papers on noncommutative geometry and conformal geometry. In this paper, we compute explicitly the Connes-Chern character of an equivariant Dirac spectral triple. The formula that we obtain for which was used in the first paper of the series. The computation has two main ste…

This paper is the continuation of Part I, expanding previous results of math.DG/9803051. This paper uses techniques in noncommutative geometry as developed by Alain Connes in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant u…

In this paper, we define the eta cochain form and prove its regularity when the kernel of a family of Dirac operators is a vector bundle. We decompose the eta form as a pairing of the eta cochain form with the Chern character of an idempotent matrix and we also decompose the Chern character of the index bundle for a fi…

The twisted Connes-Moscovici higher index theorem is generalized to the case of good orbifolds. The higher index is shown to be a rational number, and in fact non-integer in specific examples of 2-orbifolds. This results in a non-commutative geometry model that predicts the occurrence of fractional quantum numbers in t…

We study differential operators, whose coefficients define noncommutative algebras. As algebra of coefficients, we consider crossed products, corresponding to action of a discrete group on a smooth manifold. We give index formulas for Euler, signature and Dirac operators twisted by projections over the crossed product.…

The paper extends cluster validity indices for incremental analysis.

In this paper, the second of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs has girth tending to infinity, then the maximal coarse Baum-Connes assembly map is an isomorphism for the associated metric space $X$. As discussed in the first paper in this s…

This paper is part of a series of articles on noncommutative geometry and conformal geometry. In this paper, we reformulate the local index formula in conformal geometry in such a way to take into account of the action of conformal diffeomorphisms. We also construct and compute a whole new family of geometric conformal…

We give a new short proof of the index formula of Atiyah and Singer based on combining Getzler's rescaling with Greiner's approach of the heat kernel asymptotics. As application we can easily compute the Connes-Moscovici cyclic cocycle of even and odd Dirac spectral triples, and then recover the Atiyah-Singer index for…

In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum-Connes assembly map is injective, but not surjective, for the associated metric space $X$. Exp…

Researchers construct an index map for contact manifolds using K-theory.

We define an $L^2$-signature for proper actions on spaces of leaves of transversely oriented foliations with bounded geometry. This is achieved by using the Connes fibration to reduce the problem to the case of Riemannian bifoliations where we show that any transversely elliptic first order operator in an appropriate B…

We study primary and secondary invariants of leafwise Dirac operators on foliated bundles. Given such an operator, we begin by considering the associated regular self-adjoint operator $D_m$ on the maximal Connes-Skandalis Hilbert module and explain how the functional calculus of $D_m$ encodes both the leafwise calculus…

We study a two dimensional analogue of the Roe-Higson index theorem for a partitioned manifold. We prove that Connes' pairing of some invertible element with Roe's cyclic one-cocycle coincides to the Fredholm index of a Toeplitz operator. In the proof of this paper, we use some properties of a circle and use Higson's a…

Clarifies a trace for Heisenberg operators on contact manifolds.

The paper solves a general case of the cohomological relative index problem for foliations.

Researchers compute a residue cocycle for Dirac-type operators using modified Getzler calculus.

In the first part of this series, we defined an equivariant index without assuming the group acting or the orbit space of the action to be compact. This allowed us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In this paper, we investigate properties and applications of th…

We give a cohomological formula for the index of a fully elliptic pseudodifferential operator on a manifold with boundary. As in the classic case of Atiyah-Singer, we use an embedding into an euclidean space to express the index as the integral of a cohomology class depending in this case on a noncommutative symbol, th…

We prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity. We apply this theorem to study Riemannian metrics of positive scalar curvature on manifolds. For every two metrics of positive scalar curvature on a closed manifold and a Galois cover of the manifold…

Constructs a fusion product on spinor bundle over loop space.

Let $M$ be a complete Riemannian manifold and assume that $M$ is partitioned by a hypersurface $N$. In this paper we introduce a novel class of functions $C_{\mathrm{w}}(M)$ on noncompact manifolds, which is slightly larger than the algebra of Higson functions. Out of $φ$ that belongs to $C_{\mathrm{w}}(M)$ we construc…

Paper constructs a transfer map for codimension 2 submanifolds in higher index theory.

The Atiyah-Singer index theorem is a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their…

Survey of Baum-Connes conjecture history and methods.

The main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators, for proper actions by connected semisimple Lie groups on possibly noncompact manifolds, with compact quotients. For compact groups and manifolds, this reduces to the Atiyah-Segal-Singer fixed point formula…

Researchers compute Connes-Chamseddine cycle on 6D manifolds using noncommutative integral.

We discuss the behaviour of the signature index class of closed foliated bundles under the operation of cutting and pasting. Along the way we establish several index theoretic results: we define Atiyah-Patodi-Singer (APS) index classes for Dirac-type operators on foliated bundles with boundary; we prove a relative inde…

We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…

New index formula connects numerical and $K$-theoretic indices.

For any Lie groupoid we construct an analytic index morphism taking values in a modified $K-theory$ group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constru…

The aim of this note is to improve upon our earlier result which translates Weyl's (curvature) formulation of Chern character of a smooth vector bundle into the language of residues. The dualized Chern character is the functional on smooth differential forms on M. In our previous paper, this functional has been express…