arXiv research

New calculus for pseudodifferential operators on manifolds with cylindrical ends.

This paper provides a new proof of the Lefschetz fixed point formula using groupoids.

We give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural $\mathbb{R}^\times_+$-action. Specifically, we show that a properly supported semiregular distribution on $M\times M$ is the Schwartz kernel of a classical …

In this paper we define a new convergence called "asymptotically conic convergence" in which a smooth family of Riemannian metrics on a fixed compact manifold degenerate to a metric with isolated conic singularity. Our results are: convergence of the spectrum of the geometric Laplacians and uniform convergence of the c…

We consider two calculi of pseudodifferential operators on manifolds with fibered boundary: Mazzeo's edge calculus, which has as local model the operators associated to products of closed manifolds with asymptotically hyperbolic spaces, and the phi calculus of Mazzeo and the second author, which is similarly modeled on…

Characterizes polyhomogeneous symbols and applies to Heisenberg calculus.

Develops calculus for QFB metrics, proving Fredholm properties and decay of harmonic forms.

The paper studies formal geometry of dg manifolds and proves isomorphism of their calculi.

A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a one-to-one correspondence, between the module structure of the 1-forms and the …

We recall an extension of Kirby's Calculus on non-simply connected 3-manifolds given in [FR], and the surgery calculus of bridged links from [Ke], which involves only local moves. We give a short combinatorial proof that the two calculi are equivalent, and thus describe the same classes of 3-manifolds. This makes the p…

Study geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differ…

New Fredholm criteria for pseudodifferential operators on manifolds.

In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in…

Researchers study spectral asymmetry using pseudodifferential projections on the massless Dirac operator.

Holomorphic structures on quantum flag manifolds uniquely defined.

A well known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a pseudodifferential projection vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in a filtered algebra the value of a proje…

We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $Γ\subset\R^p$, we construct a uniq…

We discuss the construction of finite noncommutative geometries on Hopf algebras and finite groups in the `quantum groups approach'. We apply the author's previous classification theorem, implying that calculi in the factorisable case correspond to blocks in the dual, to classify differential calculi on the quantum cod…

We present the theory of pseudodifferential operators acting on a vector orbibundle over an orbifold, construct the zeta function of an elliptic pseudodifferential operator and show the existence of a meromorphic extension to the complex plane with at most simple poles. We give formulas for generalized densities on the…

For an arbitrary Riemannian manifold $X$ and Hermitian vector bundles $E$ and $F$ over $X$ we define the notion of the normal symbol of a pseudodifferential operator $P$ from $E$ to $F$. The normal symbol of $P$ is a certain smooth function from the cotangent bundle $T^*X$ to the homomorphism bundle $Hom (E,F)$ and dep…

We provide evidence for the conjecture that the Wodzicki-Chern classes vanish for all bundles with the group Z of invertible zeroth order pseudodifferential operators as structure group. In particular, we prove this vanishing if the structure group reduces to pseudodifferential operators with leading order symbol the i…

Extends Lie bialgebroids for string and M theories with new calculus framework.

We consider the crossed product $G_{ga}$ by $\R_+^*$ of the adiabatic groupoid associated with any Lie groupoid $G$. We construct an explicit Morita equivalence between the exact sequence of order 0 pseudodifferential operators on $G$ and (a restriction of) the natural exact sequence associated with $G_{ga}$. As an imp…

Researchers extend pseudodifferential calculus on filtered manifolds using fixed point algebras.

We show that an elliptic uniform pseudodifferential operator over a manifold of bounded geometry defines a class in uniform K-homology, and that this class only depends on the principal symbol of the operator.

Study resolvents of Bochner Laplacians on compact manifolds.

We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion $a\sim \sum_{j=0}^\infty a_{m-j}, a_{m-j}(x,ξ)=\sum_{l=0}^k a_{m-j,l}(x,ξ) \log^l|ξ|,$ where $a_{m-j,l}$ is homogeneous in $ξ$ of degree $m-j$. We will explain why this algebra of pseudo…

Study travel time tomography for transversely isotropic media using modified pseudodifferential calculus.

Researchers create a parametrix for resolvents on manifolds with ends.

Introduces generalized products for pseudodifferential operators on manifolds with corners.

This paper is concerned with pseudodifferential calculus on manifolds with fibred corners. Following work of Connes, Monthubert, Skandalis and Androulidakis, we associate to every manifold with fibred corners a longitudinally smooth groupoid which algebraic and differential structure is explicitely described. This grou…

We compute $K$-theory invariants of algebras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem for operators invariant with respect to an action of $\R^k.$ We discuss the relation between our results and the $η$-invariant.

We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a gen…

We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. The generalization will follow from a local index theorem that is valid on any manifold of bounded geometry. This local formula incorporates the uniform estimates …

We define the equivariant family index of a family of elliptic operators invariant with respect to the free action of a bundle $\GR$ of Lie groups. If the fibers of $\GR \to B$ are simply-connected solvable, we then compute the Chern character of the (equivariant family) index, the result being given by an Atiyah-Singe…

Study connects spectral and algebraic torsion in geometric contexts.

The Hochschild and cyclic homology groups are computed for the algebra of `cusp' pseudodifferential operators on any compact manifold with boundary. The index functional for this algebra is interpreted as a Hochschild 1-cocycle and evaluated in terms of extensions of the trace functionals on the two natural ideals, cor…

Researchers find spectral gaps in quantum flag manifolds using twisted operators.

The paper studies how geometric transformations affect semi-classical operators on specific Lie groups.

In a previous paper ([1]), we associated a holonomy groupoid and a C*-algebra to any singular foliation (M,F). Using these, we construct the associated pseudodifferential calculus. This calculus gives meaning to a Laplace operator of any singular foliation F on a compact manifold M, and we show that it can be naturally…

The purpose of this note is to extend the results of V. Guillemin on elliptic self-adjoint pseudodifferential operators of order one, from operators defined on smooth functions on a closed manifold to operators defined on smooth sections in a vector bundle of Hilbert modules of finite type over a finite von Neumann alg…

Isomorphism found between filtered calculus and crossed products.

In previous papers (arxiv:math/0612370 and arxiv:0909.1342) we defined the C*-algebra and the longitudinal pseudodifferential calculus of any singular foliation (M,F). Here we construct the analytic index of an elliptic operator as a KK-theory element, and prove that the same element can be obtained from an "adiabatic …

A notion of an algebroid - a generalization of a Lie algebroid structure is introduced. We show that many objects of the differential calculus on a manifold M associated with the canonical Lie algebroid structure on T^M can be obtained in the framework of a general algebroid. Also a compatibility condition which leads,…

An analytic index is defined for a family of cusp pseudodifferential operators, $P_b,$ on a fibration with fibres which are compact manifolds with boundaries, provided the family is elliptic and has invertible indicial family at the boundary. In fact there is always a perturbation $Q_b$ by a family of cusp operators of…

The extended Heisenberg algebra for a contact manifold has a symbolic calculus that accommodates both Heisenberg pseudodifferential operators as well as classical pseudodifferential operators. We derive here a formula for the index of Fredholm operators in this extended calculus. This formula incorporates in a single e…

Characterizes Fredholm conditions for group-invariant pseudodifferential operators.