arXiv research

The paper proves a Connes trace theorem for curved noncommutative tori.

We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus $\mathbb{T}_θ^2$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed L…

The central result here is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyc…

In the present paper we prove a statement closely related to the cyclic formality conjecture. In particular, we prove that for a divergence-free Poisson bivector field on R^d, the Kontsevich star-product with the harmonic angle function is cyclic. We also prove a globalization of this theorem in the case of arbitrary P…

In this paper we study the curved geometry of noncommutative 4-tori $\mathbb{T}_θ^4$. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time hea…

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…

Clarifies a trace for Heisenberg operators on contact manifolds.

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…

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.

Paper derives sub-Riemannian versions of Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki theorems for twisted BCV spaces.

We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry). The result is also a generalization of Conn's linearization theorem from one-point leaves to arbitrary symplectic leaves (however, we do not make use of Conn's theorem).

The cyclotomic trace of Bökstedt-Hsiang-Madsen, the subject of Bökstedt's lecture at the congress in Kyoto, is a map of pro-abelian groups K_*(A) -> TR_*^.(A;p) from Quillen's algebraic K-theory to a topological refinement of Connes' cyclic homology. Over the last decade, our understanding of the target and its relatio…

In Alain Connes noncommutative geometry, the question of the existence of a non-trivial integral can be described in terms of the singular traceability of the compact operator |D|^(-d), D being the Dirac operator, namely of the existence of a finite non-trivial singular trace on the ideal generated by |D|^(-d). A condi…

Researchers create a Fredholm module on fractal shapes like the Cantor set.

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

We give a soft geometric proof of the classical result due to Conn stating that a Poisson structure is linearizable around a singular point (zero) at which the isotropy Lie algebra is compact and semisimple.

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…

Strong bolicity helps prove Baum-Connes conjecture for certain hyperbolic groups.

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash-Moser fast convergence method. In the case of one-point submanifolds (fixed points), this immediately implies a stronger version of Conn's linearization theorem, also proving that Conn's theorem is, indeed, just a manife…

This thesis studies normal forms for Poisson structures around symplectic leaves using several techniques: geometric, formal and analytic ones. One of the main results (Theorem 2) is a normal form theorem in Poisson geometry, which is the Poisson-geometric version of the Local Reeb Stability (from foliation theory) and…

A trace on the C^*-algebra A of quasi-local operators on an open manifold is described, based on the results in \cite{RoeOpen}. It allows a description `a la Novikov-Shubin \cite{NS2} of the low frequency behavior of the Laplace-Beltrami operator. The 0-th Novikov-Shubin invariant defined in terms of such a trace is pr…

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

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…

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

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…

Anti-self-dual (ASD) connections for a compact smooth four manifold arise as critical values for the Yang-Mills action functional. Nahm transform is a nice correspondence between a vector bundle with ASD connections and a vector bundle with ASD connections over Picard torus associated to X. In this talk we propose a no…

We discuss analogies between number theory and the theory of dynamical systems on spaces with a one-codimensional foliation. The emphasis is on comparing the "explicit formulas" of analytic number theory with certain dynamical Lefschetz trace formulas. We also point out a possible relation between an Arakelov-Euler cha…

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…

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 (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold…

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.

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…

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.

The present paper is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan' theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin^C-structures, Dirac operators, exterior algebra bundles and Connes' differential algebras in the commutative case…

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…

The Novikov-Shubin numbers are defined for open manifolds with bounded geometry, the Gamma-trace of Atiyah being replaced by a semicontinuous semifinite trace on the C*-algebra of almost local operators. It is proved that they are invariant under quasi-isometries and, making use of the theory of singular traces for C*-…

We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using …

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…

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) …

The abstract theorem is extended to higher genus surfaces.

We give a local classification of generalized complex structures. About a point, a generalized complex structure is equivalent to a product of a symplectic manifold with a holomorphic Poisson manifold. We use a Nash-Moser type argument in the style of Conn's linearization theorem.

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

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

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…

We use adiabatic limits to study foliated manifolds. The Bott connection naturally shows up as the adiabatic limit of Levi-Civita connections. As an application, we then construct certain natural elliptic operators associated to the foliation and present a direct geometric proof of a vanshing theorem of Connes[Co], whi…

For an orbifold X and $α\in H^3(X, Z)$, we introduce the twisted cohomology $H^*_c(X, α)$ and prove that the Connes-Chern character establishes an isomorphism between the twisted K-groups $K_α^* (X) \otimes C$ and twisted cohomology $H^*_c(X, α)$. This theorem, on the one hand, generalizes a classical result of Baum-Co…

New rigidity result for Steklov eigenvalues on manifolds.

We present an alternative approach to the result of Guentner, Higson, and Weinberger concerning the Baum-Connes conjecture for finitely generated subgroups of SL(2,C). Using finite-dimensional methods, we show that the Baum-Connes assembly map for such groups is an isomorphism.

Survey on metrics and assembly maps in positive scalar curvature.