Raphael Ponge

Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds

Introduction Heisenberg manifolds and their main differential operators Intrinsic approach to the Heisenberg calculus Holomorphic families of

A New Short Proof of the Local Index Formula and Some of Its Applications

\mathbf{\Psi_{H}}

SPECTRAL ASYMMETRY, ZETA FUNCTIONS, AND THE NONCOMMUTATIVE RESIDUE

DOs Heat equation and complex powers of hypoelliptic operators Spectral asymptotics for hypoelliptic operators Appendix A. Proof of Proposition 3.1.18 Appendix B. Proof of Proposition 3.1.21 Bibliography.

Noncommutative Geometry and Lower Dimensional Volumes in Riemannian Geometry

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 an application we can easily compute the CM cyclic cocycle of even and odd Dirac spectral triples, and then recover the Atiyah-Singer index formula (even case) and the Atiyah-Patodi-Singer spectral flow formula (odd case).

Noncommutative geometry and conformal geometry, II. Connes–Chern character and the local equivariant index theorem

In this paper we study the spectral asymmetry of (possibly nonselfadjoint) elliptic ΨDOs in terms of the difference of zeta functions coming from different cuttings. Refining previous formulas of Wodzicki in the case of odd class elliptic ΨDOs, our main results have several consequence concerning the local independence with respect to the cutting, the regularity at integer points of eta functions and a geometric expression for the spectral asymmetry of Dirac operators which, in particular, yields a new spectral interpretation of the Einstein–Hilbert action in gravity.

Noncommutative residue invariants for CR and contact manifolds

In this paper we explain how to define “lower dimensional” volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential way noncommutative geometry and the analysis of Dirac operators on spin manifolds. However, the ultimate definitions of the lower dimensional volumes do not involve noncommutative geometry or spin structures at all.

ON THE SINGULARITIES OF THE ZETA AND ETA FUNCTIONS OF AN ELLIPTIC OPERATOR

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 steps. The first step is the justification that the CM cocycle represents the Connes-Chern character. The second step is the computation of the CM cocycle as a byproduct of a new proof of the local equivariant index theorem of Donnelly-Patodi, Gilkey and Kawasaki. The proof combines the rescaling method of Getzler with an equivariant version of the Greiner-Hadamard approach to the heat kernel asymptotics. Finally, as a further application of this approach, we computate the short-time limit of the JLO cocycle of an equivariant Dirac spectral triple.

ON THE ASYMPTOTIC COMPLETENESS OF THE VOLTERRA CALCULUS

Abstract In this paper we produce several new invariants for CR and contact manifolds by looking at the noncommutative residue traces of various geometric Ψ H DO projections. In the CR setting these operators arise from the -complex and include the Szegö projections acting on forms. In the contact setting they stem from the generalized Szegö projections at arbitrary integer levels of Epstein-Melrose and from the contact complex of Rumin. In particular, we recover and extend recent results of Hirachi and Boutet de Monvel and we answer a question of Fefferman.

Calcul fonctionnel sous-elliptique et résidu non commutatif sur les variétés de Heisenberg

Let P be a self-adjoint elliptic operator of order m > 0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form , where k ranges over all nonzero integers ≤ n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions become singular at all points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, self-adjoint first-order differential operators and self-adjoint elliptic pseudodifferential operators. As consequences, we obtain genericity results for the singularities of the zeta and eta functions in those settings. In particular, in the setting of Dirac-type operators we obtain a purely analytical proof of a well-known result of Branson–Gilkey [Residues of the eta function for an operator of Dirac type, J. Funct. Anal. 108(1) (1992) 47–87], which was obtained by invoking Riemannian invariant theory. As it turns out, the results of this paper contradict Theorem 6.3 of [R. Ponge, Spectral asymmetry, zeta functions and the noncommutative residue, Int. J. Math. 17 (2006) 1065–1090]. Corrections to that statement are given in this paper.

Noncommutative geometry and conformal geometry. III. Vafa–Witten inequality and Poincaré duality

The Volterra calculus is a simple and powerful pseudodifferential tool for inverting parabolic equations which has found many applications in geometric analysis. An important property in the theory of pseudodifferential operators is asymptotic completeness, which allows the construction of parametrices modulo smoothing operators. In this paper, we present new and fairly elementary proofs of the asymptotic completeness of the Volterra calculus.