Some available papers

The Atiyah covering index theorem for riemannian foliations (with J. Heitsch), submitted.

We use the symbol calculus for foliations developed in our previous paper to derive a cohomological formula
for the Connes-Chern character of the Type II spectral triple given associated with the transverse geometry of Galois coverings of bounded geometry foliations. The same proof works for the Type I spectral triple of Connes-Moscovici. The cohomology classes of the two Connes-Chern characters induce the same map on the image of the maximal Baum-Connes map in K-theory, thereby proving an Atiyah L2 covering index theorem.

Enlargeability, foliations and positive scalar curvature (with J. Heitsch), arXiv: 1703.02684, submitted.

We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature. In particular, we show that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold admits no metric of positive scalar curvature.

Gap labelling conjecture with nonzero magnetic field  (with V. Mathai), arXiv:1508.01064, to appear in Advances In Mathematics.

Abstract: Given a constant magnetic field on Euclidean space p determined by a skew-symmetric (p×p) matrix Θ, and a p-invariant probability measure μ on the disorder set Σ, we conjecture that the corresponding Integrated Density of States of any self-adjoint operator affiliated to the twisted crossed product algebra C(Σ)σp takes on values on spectral gaps in an explicit -module involving Pfaffians of Θ and its sub-matrices that we describe, where σ is the multiplier on p associated to Θ. We give a proof of the conjecture in 2D and 3D cases.

The higher twisted index theorem for foliations
(with A. Gorokhovsky and E. Leichtnam), arXiv:1607.04248,
J. Funct. Anal. 273 (2017), no. 2, 496–558.

Abstract:
Given a gerbe L, on the holonomy groupoid G of the foliation (M,F), whose pull-back to M is torsion, we construct a Connes Φ-map from the twisted Dupont-Sullivan bicomplex of G to the cyclic complex of the L-projective leafwise smoothing operators on (M,F). Our construction allows to couple the K-theory analytic indices of L-projective leafwise elliptic operators with the twisted cohomology of BG  producing scalar higher invariants. Finally by adapting the Bismut-Quillen superconnection approach, we compute these higher twisted indices as integrals over the ambiant manifold of the expected twisted characteristic classes.

A symbol calculus for foliations (with J. Heitsch), arXiv:1511.05697, J. Noncommut. Geom. 11 (2017), no. 3, 1141–1194.

Abstract: The classical Getzler rescaling theorem is extended to the transverse geometry of foliations. More precisely, a Getzler rescaling calculus, as well as a Block-Fox calculus of asymptotic operators, is constructed for all transversely spin foliations. This calculus applies to operators of degree
m globally times degree  in the leaf directions, and is thus an appropriate tool for a better understanding of the index theory of transversely elliptic operators on foliations. The main result is that the composition of AΨDOs is again an AΨDO, and includes a formula for the leading symbol.

# arXiv:1511.569

The Higson-Roe exact sequence and $\ell^2$ eta invariants (with I. Roy), J. Funct. Anal. 268 (2015), no 4, 974-1031.

Abstract: The goal of this paper is to solve the problem of existence of an $\ell^2$ eta morphism on the Higson-Roe structure group. Using the Cheeger-Gromov $\ell^2$ eta invariant, we construct a group morphism from the Higson-Roe maximal structure group to the reals. When we apply this morphism to the structure class associated with the spin Dirac operator for a metric of positive scalar curvature, we get the spin $\ell^2$ rho invariant. When we apply this morphism to the structure class associated with an oriented homotopy equivalence, we get the difference of the $\ell^2$ rho invariants of the corresponding signature operators. We thus get new proofs for the classical $\ell^2$ rigidity theorems of Keswani about Cheeger-Gromov rho invariants.

Spectral sections, twisted rho invariants and positive scalar curvature (with V. Mathai), J. Noncommut. Geom. 9 (2015), no. 3, 821-850.

Abstract: We had previously defined the rho invariant ρspin(Y,E,H,g)  for the twisted Dirac operator ̸EH on a closed odd dimensional Riemannian spin manifold (Y,g), acting on sections of a flat hermitian vector bundle E over Y, where H=ij+1H2j+1 is an odd-degree differential form on Y and H2j+1 is a real-valued differential form of degree 2j+1. Here we show that it is a conformal invariant of the pair (H,g). In this paper we express the defect integer ρspin(Y,E,H,g)ρspin(Y,E,g)   in terms of spectral flows and prove that ρspin(Y,E,H,g), whenever g is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum-Connes conjecture holds for π1(Y) (which is assumed to be torsion-free), then we show that ρspin(Y,E,H,rg)=0 for all r0, significantly generalizing our earlier results. These results are proved using the Bismut-Weitzenb\"ock formula, a scaling trick, the technique of noncommutative spectral sections, and the Higson-Roe approach.

Index type invariants for twisted signature complexes (with V. Mathai), Math. Proc. Cambridge Philos. Soc. 156 (2014), no 3, 473-503.

Abstract:  For a closed, oriented, odd dimensional manifold X, we define the rho invariant rho(X,E,H) for the twisted odd signature operator valued in a flat hermitian vector bundle E, where H = \sum i^{j+1} H_{2j+1} is an odd-degree closed differential form on X and H_{2j+1} is a real-valued differential form of degree {2j+1}. We show that the twisted rho invariant rho(X,E,H) is independent of the choice of metrics on X and E and of the representative H in the cohomology class [H]. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta invariant in terms of spectral flow and the usual eta invariant. In particular, we get a simple expression for it on closed oriented 3-dimensional manifolds with a degree three flux form. A core technique used is our analogue of the Atiyah-Patodi-Singer theorem, which we establish for the twisted signature operator on a compact, oriented manifold with boundary.

An interesting example for spectral invariants (with J. Heitsch & C. Wahl), J. of K-theory 13 (2014), no 2, 305-311.

Abstract: In "Illinois J. of Math. {\bf 38} (1994) 653--678", the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then showed that the associated heat operator converges to the Chern character of the index bundle of the operator. In "J. K-Theory {\bf 1} (2008) 305--356", we improved this result by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.

Leafwise homotopies and Hilbert-Poincaré complexes I. Regular HP complexes and leafwise pull-back maps. (with I. Roy), J. Noncommut. Geom. 8 (2014), no 3, 789-836.

Abstract: In this first part of our series of papers, we prove the leafwise homotopy invariance of $K$-theoretic signatures of foliations and laminations, using the formalism of Hilbert-Poincaré complexes as revisted by Higson and Roe. We use a generalization of their method to give a homotopy equivalence of de Rham-Hilbert-Poincaré complexes associated with leafwise homotopy equivalence for foliations. In particular, we obtain an explicit path connecting the signature classes in K-theory, up to isomorphism induced by Morita equivalence. Applications of this path on the stability properties of rho-invariants à la Keswani will be carried out in the later parts of this series.

Abstract:For a closed, spin, odd dimensional Riemannian manifold (Y,g), we define the rho invariant $\rho_{spin}(Y,\E,H, [g])$ for the twisted Dirac operator D^E_H on Y, acting on sections of a flat hermitian vector bundle E over Y, where H = \sum i^{j+1} H_{2j+1} is an odd-degree closed differential form on Y and H_{2j+1} is a real-valued differential form of degree {2j+1}. We prove that it only depends on the conformal class [g] of the metric g. In the special case when $H$ is a closed 3-form, we use a Lichnerowicz-Weitzenbock formula for the square of the twisted Dirac operator, to show that whenever $X$ is a closed spin manifold, then rho_{spin}(Y,E,H, [g])= rho_{spin}(Y,E, [g]) for all |H| small enough, whenever g is conformally equivalent to a Riemannian metric of positive scalar curvature. When H is a top-degree form on an oriented three dimensional manifold, we also compute rho_{spin}(Y,E,H).

Higher spectral flow and an entire bivariant JLO cocycle (with A. Carey), J. K-Theory 11 (2013), no. 1, 183–232.

Abstract: We prove that for any smooth fibration of closed manifolds, there is a well defined bivariant JLO cocycle which turns out to be analytic in the sense of Ralf Meyer's theory.

Homotopy invariance of the higher harmonic signature.(with J. Heitsch), J. Differential Geom. 87 (2011), no. 3, 389–467.

Abstract: We define the higher harmonic signature, twisted by a leafwise U(p,q)-flat bundle, of an even dimensional oriented Riemannian foliation of a compact Riemannian manifold, and prove that it is a leafwise homotopy invariant. In the process, we also prove that the projection onto the leafwise harmonic forms in the middle dimension for the associated foliation of the graph is transversely smooth in the definite case. Some consequences for the Novikov conjecture are also investigated.

Index, eta and rho invariants on foliated bundles. (with P. PiazzaAstérisque 327 (2009), 199-284.

Abstract: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 on the maximal Connes-Skandalis Hilbert module and explain how its functional calculus encodes both the leafwise calculus and the monodromy calculus in the corresponding von Neumann algebras. When the foliation is endowed with a holonomy invariant transverse measure, we explain the compatibility of various traces and determinants. We extend Atiyah's index theorem on Galois coverings to these foliations. We define a foliated rho-invariant and investigate its stability properties for the signature operator. Finally, we establish the foliated homotopy invariance of such a signature rho-invariant under a Baum-Connes assumption, thus extending to the foliated context results proved by Neumann, Mathai, Weinberger and Keswani on Galois coverings.

The higher fixed point theorem for foliations. I Holonomy invariant currents (with J. Heitsch)  J. of Funct. Analysis 259, (2010), 131-173.

Abstract: We prove a higher fixed point formula for foliations in the presence of a closed Haefilger current.To this end we associate with such current an equivariant cyclic cohomology class of Connes' C* algebra of the foliation, and compute its pairing with the localized equivariant K-theory in terms of local contributions near the fixed points.

Local index theorem for projective families (with A. Gorokhovsky). Perspectives on concommutative geometry, 1-27, Fields Inst. Commun. 61, Amer. Math. Soc., Providence, RI, 2011.

Abstract: We give a superconnection proof of the cohomological form of Mathai-Melrose-Singer index theorem for the family of twisted Dirac operators under relaxed conditions.