Some available papers      The higher fixed point theorem for foliations. Applications to rigidity and integrality (with J. Heitsch), arXiv: 2402.19283. Submitted. We give applications of the higher Lefschetz theorems for foliations that we proved in J. Funct. Anal. 259 (2010), no. 1, 131–173, primarily involving Haefliger cohomology. These results show that the transverse structures of foliations carry important topological and geometric information. This is in the spirit of the passage from the Atiyah-Singer index theorem for a single compact manifold to their families index theorem, involving a compact fiber bundle over a compact base. For foliations, Haefliger cohomology plays the role that the cohomology of the base space plays in the families index theorem. We obtain highly useful numerical invariants by paring with closed holonomy invariant currents. In particular, we prove that the non-triviality of the higher A-hat genus of the foliation in Haefliger cohomology can be an obstruction to the existence of non-trivial leaf-preserving compact connected group actions. We then construct a large collection of examples for which no such actions exist. Finally, we relate our results to Connes' spectral triples, and prove useful integrality results.

Higher relative index theorems for foliations (with J. Heitsch), arXiv:2402.10316. Submitted. In this paper we solve the general case of the cohomological relative index problem for foliations of non-compact manifolds. In particular, we significantly generalize the groundbreaking results of Gromov and Lawson, to Dirac operators defined along the leaves of foliations of non-compact complete Riemannian manifolds, by involving all the terms of the Connes-Chern character, especially the higher order terms in Haefliger cohomology. The zero-th order term corresponding to holonomy invariant measures was carried out previously (in J. Funct. Anal. 284 (2023), no. 2) and becomes a special case of our main results here. In particular, for two leafwise Dirac operators on two foliated manifolds which agree near infinity, we define a relative topological index and the Connes-Chern character of a relative analytic index, both being in relative Haefliger cohomology. We show that these are equal. This invariant can be paired with closed holonomy invariant currents (which agree near infinity) to produce higher relative scalar invariants. When we relate these invariants to the leafwise index bundles, we restrict to Riemannian foliations on manifolds of sub-exponential growth. This allows us to prove a higher relative index bundle theorem, extending the classical index bundle theorem that we obtained in 2008. Finally, we construct examples of foliations and use these invariants to prove that their spaces of leafwise positive scalar curvature metrics have infinitely many path-connected components, completely new results which are not available from J. Funct. Anal. 284 (2023), no. 2. In particular, these results confirm the well-known idea that important geometric information of foliations is embodied in the higher terms of the A-hat genus.

Dirac operators on foliations with invariant transverse measures (with J. Heitsch), arXiv:2109.09806 J. Funct. Anal. 284 (2023), no. 2, Paper No. 109742, 52 pp. We extend the groundbreaking results of Gromov and Lawson on positive scalar curvature and the Dirac operator on complete Riemannian manifolds to Dirac operators defined along the leaves of foliations of non-compact complete Riemannian manifolds which admit invariant transverse measures. We prove a relative measured index theorem for pairs of such manifolds, foliations and operators, which are identified off compact subsets of the manifolds. We assume that the spectral projections of the leafwise operators for some interval [0,ϵ], ϵ>0, have finite dimensional images when paired with the invariant transverse measures. As a prime example, we show that if the zeroth order operators in the associated Bochner Identities are uniformly positive off compact subsets of the manifolds, then they satisfies the hypotheses of our relative measured index theorem. Using these results, we show that for a large collection of spin foliations, the space of positive scalar curvature metrics on each foliation has infinitely many path connected components. Chern-Connes-Karoubi character isomorphisms and algebras of symbols of pseudodifferential operators, Cyclic cohomology at 40: achievements and future prospects (with A. Baldare and V. Nistor), Proc. Sympos. Pure Math., 105 American Mathematical Society, Providence, RI, 2023, 1-27. We introduce a class of algebras for which the Chern-Connes-Karoubi character is an isomorphism after tensoring with C. We provide several examples of such algebras, such as invariant sections of Azumaya bundles, crossed products with  nite groups and certain algebras of pseudodifferential operators.

The Gromov-Lawson index and the Baum-Connes assembly map, arXiv:2103.03495 World Scientifi Publishing, 2023, 835-882. This is a survey paper which gathers some results related with the study of Positive Scalar Curvature metrics in connection with the Baum-Connes assembly map.

An equivariant PPV theorem and Paschke-Higson duality (with I. Roy), arXiv:2001.09811. Annals of K-theory 7, (2022) no2, 237-278. We prove an equivariant localized and norm-controlled version of the Pimsner-Popa-Voiculescu theorem. As an application, we deduce a proof of the Paschke-Higson duality for transformation groupoids.

The index of leafwise G-transversally elliptic operators on foliations (with A. Baldare), arXiv:2001.02428, J. of Geom. and Physics 163 (2021), 34pp. We introduce and study the index morphism for G-invariant leafwise G-transversally elliptic operators on smooth closed foliated manifolds which are endowed with leafwise actions of the compact group G. We prove the usual axioms of excision, multiplicativity and induction for closed subgroups. In the case of free actions, we relate our index class with the Connes-Skandalis index class of the corresponding leafwise elliptic operator on the quotient foliation. Finally we prove the compatibility of our index morphism with the Gysin Thom isomorphism and reduce its computation to the case of tori actions. We also construct a topological candidate for an index theorem using the Kasparov Dirac element for euclidean G-representations.

The relative L^2-index theorem for Galois coverings, arXiv:2009.10011, Annals of K-theory 6, (2021) no3, 503-541. Given a Galois covering of complete spin manifolds where the base metric has PSC near infinity, we prove that for small enough epsilon > 0, the epsilon spectral projection of the Dirac operator has finite trace in the Atiyah von Neumann algebra. This allows us to define the L2 index in the even case and we prove its compatibility with the Xie-Yu higher index. We also deduce L2 versions of the classical Gromov-Lawson relative index theorems. Finally, we briefly discuss some Gromov-Lawson L2 invariants.

Geometric noncommutative geometry (with J. Heitsch), arXiv:1909.00063, Expositiones Mathematicae 39 (2021), 454-479. In a recent paper, the authors proved that no spin foliation on a compact enlargeable manifold with Hausdorff homotopy graph admits a metric of positive scalar curvature on its leaves. This result extends groundbreaking results of Lichnerowicz, Gromov and Lawson, and Connes on the non-existence of metrics of positive scalar curvature. In this paper we review in more detail the material needed for the proof of our theorem and we extend our non-existence results to non-compact manifolds of bounded geometry. In addition, we give an interpretation of the Gromov-Lawson relative index theorem, using our extension of Haefliger cohomology to non-compact manifolds.

The Higson-Roe sequence for étale groupoids II. The universal sequence for equivariant families (with I. Roy), arXiv:1812.04371, J. Noncomm. Geometry 15 (2021) no1, 1-39. This is the second part of our series about the Higson-Roe sequence for étale groupoids. We devote this part to the proof of the universal K-theory surgery exact sequence which extends the seminal results of N. Higson and J. Roe to the case of transformation groupoids. In the process, we prove the expected functoriality properties as well as the Paschke-Higson duality theorem.

Enlargeability, foliations and positive scalar curvature (with J. Heitsch), arXiv: 1703.02684, Invent. Math.  215 (2019), no. 1, 367–382. We extend the deep 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.

The Higson-Roe sequence for étale groupoids. I. Dual algebras and compatibility with the BC map (with I. Roy), arXiv:1801.06040, J. Noncommut. Geom. 14 (2020), no. 1, 25–71. We introduce the dual Roe algebras for proper étale groupoid actions and deduce the expected Higson-Roe short exact sequence. When the action is cocompact, we show that the Roe C∗-ideal of locally compact operators is Morita equivalent to the reduced C∗-algebra of our groupoid, and we further identify the boundary map of the associated periodic six-term exact sequence with the Baum-Connes map, via a Paschke-Higson map for groupoids. For proper actions on continuous families of manifolds of bounded geometry, we associate with any G-equivariant Dirac-type family, a coarse index class which generalizes the Paterson index class and also the Moore-Schochet Connes' index class for laminations.

Proof of the magnetic gap-labelling conjecture for principal solenoidal tori (with V. Mathai), arXiv:1806.06302, J. Funct. Anal. 278 (2020), no. 3, 108323, 9 pp.   In this note, we prove the magnetic spectral gap-labelling conjecture as stated in [arXiv:1508.01064], in all dimensions, for principal solenoidal tori.

Gap labelling conjecture with nonzero magnetic field  (with V. Mathai), arXiv:1508.01064, Advances In Mathematics 325 (2018), 116-164. Given a constant magnetic field on Euclidean space ℝp determined by a skew-symmetric (p×p) matrix Θ, and a ℤp-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)⋊σℤp takes on values on spectral gaps in an explicit ℤ-module involving Pfaffians of Θ and its sub-matrices that we describe, where σ is the multiplier on ℤp associated to Θ. We give a proof of the conjecture in 2D and 3D cases.

Transverse noncommutative geometry of foliations (with J. Heitsch), arXiv:1804.06837, J. Geom. Phys. 134 (2018), 161–194. Dedicated to Alain Connes on the occasion of his 70-th birthday. 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 Beals-Greiner calculus, satisfying the usual axioms, gives rise to a semi-finite spectral triple over the crossed product algebra of the foliation by the action, and hence a periodic cyclic cohomology class through the Connes-Chern character. The Connes-Moscovici hypoelliptic signature operator yields an example of such a triple and gives the dierential denition of our ''L^2-signature". For Galois coverings of bounded geometry foliations, we also define an Atiyah-Connes semi-finite spectral triple which generalizes to Riemannian bifoliations the Atiyah approach to the L^2-index theorem. The compatibility of the two spectral triples with respect to Morita equivalence is proven, and by using an Atiyah-type theorem proven in [BH17], we deduce some integrality results for Riemannian foliations with torsion-free monodromy groupoids.

The Atiyah covering index theorem for riemannian foliations (with J. Heitsch), arXiv:1804.07033, Trans. Amer. Math. Soc. 371 (2019), no. 8, 5875–5897. 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.

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. 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.   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 ℓ 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)∈ℚ, 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 r≫0, 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.

Conformal invariants of twisted Dirac operators and positive scalar curvature (with V. Mathai),  J. Geom. & Phys. 70 (2013), 39-47. 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. Piazza)  Astérisque 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.