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The Novikov conjecture for linear groups

Erik Guentner, Nigel Higson, Shmuel Weinberger (2005)

Publications Mathématiques de l'IHÉS

Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C*-algebra theory.

The positive mass theorem for ALE manifolds

Mattias Dahl (1997)

Banach Center Publications

We show what extra condition is necessary to be able to use the positive mass argument of Witten [12] on an asymptotically locally euclidean manifold. Specifically we show that the 'generalized positive action conjecture' holds if one assumes that the signature of the manifold has the correct value.

The proof of the Nirenberg-Treves conjecture

Nils Dencker (2003)

Journées équations aux dérivées partielles

We prove the Nirenberg-Treves conjecture : that for principal type pseudo-differential operators local solvability is equivalent to condition ( Ψ ). This condition rules out certain sign changes of the imaginary part of the principal symbol along the bicharacteristics of the real part. We obtain local solvability by proving a localizable estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman)...

The Quantum Birkhoff Normal Form and Spectral Asymptotics

San Vũ Ngọc (2006)

Journées Équations aux dérivées partielles

In this talk we explain a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy E . This permits a detailed study of the spectrum in various asymptotic regions of the parameters ( E , ℏ ) , and gives improvements and new proofs for many of the results in the field. In the completely resonant...

The radiation field is a Fourier integral operator

Antônio Sá Barreto, Jared Wunsch (2005)

Annales de l’institut Fourier

We show that the ``radiation field'' introduced by F.G. Friedlander, mapping Cauchy data for the wave equation to the rescaled asymptotic behavior of the wave, is a Fourier integral operator on any non-trapping asymptotically hyperbolic or asymptotically conic manifold. The underlying canonical relation is associated to a ``sojourn time'' or ``Busemann function'' for geodesics. As a consequence we obtain some information about the high frequency behavior of the scattering...

The resolvent for Laplace-type operators on asymptotically conic spaces

Andrew Hassell, András Vasy (2001)

Annales de l’institut Fourier

Let X be a compact manifold with boundary, and g a scattering metric on X , which may be either of short range or “gravitational” long range type. Thus, g gives X the geometric structure of a complete manifold with an asymptotically conic end. Let H be an operator of the form H = Δ + P , where Δ is the Laplacian with respect to g and P is a self-adjoint first order scattering differential operator with coefficients vanishing at ∂ X and satisfying a “gravitational” condition. We define a symbol calculus for...

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