10w5106 Geometric Scattering Theory and Applications

Arriving Sunday, March 14 and departing Friday, March 19, 2010

Organizers: Peter Hislop (University of Kentucky), Rafe Mazzeo (Stanford University), Peter Perry (University of Kentucky), Antonio Sa Barreto (Purdue University).

Confirmed Participants

Press Release: Geometric Scattering Theory and Applications

Information for Participants

Objectives

The goal of this workshop is to bring together researchers in geometric scattering theory, conformal and CR geometry, and dynamics to make further progress in building this dictionary between dynamics, scattering and fine aspects of the geometry and analysis at infinity of this class of spaces. Areas of focus will include:

begin{enumerate} item Conformal and CR-Geometries: elucidating connections between scattering theory, or other measurements of asymptotics of textquotedblleft bulktextquotedblright quantities in the AdS/CFT correspondence, with conformal and CR invariants.

item Trace formulae , spectral asymptotics, and resonances: extending non-perturbative trace formulas to asymptotically hyperbolic manifolds, complex hyperbolic manifolds, and other geometric settings

item Chaotic quantum scattering: elucidating further connections between classical and quantum chaos in the setting of geometric scattering;

item Scattering theory in various spacetimes: scattering in Kerr and Schwarzschild geometries, using techniques from geometric scattering to obtain the space-time asymptotics of solutions to the wave equation on Schwarzschild and Kerr models of black holes, and connections with other areas of geometric scattering theory end{enumerate}

The workshop program will be designed to highlight the recent work of young researchers in the field, which will be accomplished by asking certain of them to give 50 minute talks on their research. Of the remaining talks, three will be designated as 50-minute survey talks, and the rest, by senior researchers, will be 30 minutes. The daily program will consist of one 50-minute and one 30-minute talk in the morning, followed by a long midday break, and one 50-minute talk and two 30-minute talks in the afternoon. The long break from 11:15 until 4:00 is designed to allow participants to work and interact informally. We will also hold two moderated open problem sessions during the week to make sure that everyone, particularly the young researchers, have a good sense of some of the most important open directions in the field.

Our proposed list of participants includes experts in microlocal analysis, dynamics, CR-geometry, and scattering theory; we hope to provide an environment in which broader connections can be established and in which younger researchers can benefit from the perspective and experience of accomplished senior researchers in these respective fields.

We now discuss the scientific rationale for our areas of focus in greater detail.

A central role in geometric scattering theory and its applications is played by Poincar'{e}-Einstein spaces, which are asymptotically real hyperbolic, in the AdS/CFT correspondence in string theory. The asymptotic geometry on the boundary at infinity in this case is conformal geometry. There is now an extensive dictionary between scattering theory for the Laplace operator on the interior and various natural conformal quantities. In physics this is the beginning of the Maldacena correspondence, but in mathematics, it has led to some very surprising new ways to approach old problems in conformal geometry, e.g. the Fefferman-Graham ambient metric construction to analyze scalar conformal invariants, or the Graham-Zworski theory to relate the conformally covariant GJMS operators on the boundary (which are higher order analogues of the conformal Laplacian and Paneitz operator) to residues of the scattering matrix. This setting also includes the various trace formulas on convex cocompact hyperbolic manifolds which involve Selberg zeta functions to spectral or scattering information. This is also interesting from a microlocal point of view since these manifolds provide examples of chaotic quantum scattering which can be analyzed in detail using techniques from dynamics. One goal is to extend these relationships between dynamics of the geodesic flow and scattering to more general spaces which do not have constant curvature.

There is an emerging parallel development for the complex case. Natural examples of asymptotically complex hyperbolic spaces include the canonical Bergman and K"ahler-Einstein metrics on strictly pseudoconvex domains. The asymptotic geometry at infinity is now CR geometry. This was one of the original motivations of the Fefferman-Graham program, but the connections with scattering theory are quite recent. There should be similar stories for spaces modeled on higher rank symmetric spaces of noncompact type, and these are also now just beginning to be developed. One important task is for the geometric scattering community to absorb the content of the more difficult trace formulae in the higher rank setting, e.g. the Arthur trace formula, so as to gain insight into how to find appropriate generalizations.

On the analytic side, efficient calculi of pseudodifferential operators adapted to each of these geometries have made it possible to analyze the resolvent of the Laplacian and the associated scattering operator in detail. These theories have led to the definition of new renormalized traces and determinants for resolvents, heat kernels, scattering operators, etc.; this lies at the heart of some important new non-perturbative trace formulae , as developed by Guillarmou and others. Conversely, there are many techniques from dynamics to study geodesic flow and the associated (Ruelle-type) zeta functions, and these have had an impact on geometric scattering by leading to results new results concerning distribution of resonances, and new trace formulae .

As mentioned above, there is also a more algebraic side to this subject. Many old questions in conformal and CR geometry have been resolved using ideas from the Fefferman-Graham program, i.e. extending a complete asymptotically symmetric Einstein metric. One recent example is Alexakis' tour de force resolution of the Deser-Schwimmer conjecture about the structure of conformally invariant scalar invariants. More broadly, and drawing significant inspiration from representation theory, Juhl has developed a far-reaching generalization of the Fefferman-Graham theory which has introduced a huge panoply of new and interesting conformally invariant operators.

Complementing all of this are the many deep questions in nonlinear geometric analysis about the existence and nature of these asymptotically symmetric Einstein metrics and of the special submanifolds associated to these geometries. This part of the subject has taken on a broad life of its own, and for reasons of focus, should not be one of the main themes of this workshop. Nonetheless, there are several important researchers working on the interface of these two aspects of the subject who should be included.

Finally, as indicated earlier, another set of spaces amenable to treatment by the techniques of geometric scattering theory are some interesting examples of cosmological spacetimes such as the Schwarzschild, Kerr, de Sitter and anti de Sitter spaces, and once again, other spaces which have the same asymptotic structure as one of these at infinity. One of the key challenges in mathematical relativity is to study the stability properties of these spacetimes (as solutions of the Einstein equation), which motivates the study of spaces which are perturbations of the exact solutions, and also gives great urgency to the big problem of studying the asymptotic properties of solutions of the wave equation on these spacetimes. There are already some important results in this direction using techniques from geometric scattering, and this is a very important direction of research.