Oberwolfach Seminar: Recent Developments in Conformal Differential Geometry

Date:

November 18th - November 24th, 2007

Organizers:

Helga Baum, Berlin

Andreas Juhl, Uppsala

Programme:

Of central interest in conformal differential geometry are conformal invariants, for example, conformally covariant differential operators, conformal curvature tensors, conformal holonomy groups or groups of conformal diffeomorphisms. Conformally covariant operators arise often in physics. For example, the classical Maxwell equation on 4-dimensional Minkowski space is conformally covariant. Further conformally covariant operators are the Dirac operator, the Yamabe operator, the Paneitz operator and the twistor operator. In recent years the AdS/CFTcorrespondence in quantum gravity motivated new studies in conformal differerential geometry. The aim of the seminar is to present some of these ideas and developments.

The seminar is organized like a summer school and adressed to graduate students and post doc's. There will be 2 lecture series, one on Q-curvature, its origin and relevance in geometry, spectral theory and physics and one on holonomy theory of conformal manifolds and its relation to Einstein metrics and to conformally invariant twistor equations. In the two courses we intend to cover the following subjects:

Andreas Juhl: Q-curvature

• The flat model of conformal geometry. Conformal group. Actions and Representations.
• The Fefferman-Graham construction and conformal invariants. Poincare-Einstein metrics. Conformally covariant powers of the Laplacian (GJMS-operators). GJMS-operators and scattering theory of asymptotically hyperbolic spaces.
• Origins and various routes to Q-curvature. Q-curvature in spectral theory and geometry. Q-curvature in dimension 4, Paneitz-operator and Paneitz-curvature.
• Holography and Q-curvature. The holographic formula and its consequences.
• Families of conformally covariant differential operators. Residue families and Q-polynomials. Recursive structures.
• Eastwoods' curved translation principle: from Verma modules to tractor calculus. Applications to families of conformally covariant operators. Tractor construction of Q-curvature. Extrinsic Q-curvature.
  

Helga Baum: Holonomy theory of conformal structures

• Cartan connections, curvature and holonomy groups. Tractor bundles and tractor connections.
• Conformal structures and the normal conformal Cartan connection. The conformal tractor connection and its curvature. Holonomy groups of conformal structures.
• Conformal holonomy groups and Einstein metrics. Splitting theorem in conformal holonomy theory. Cone and ambient metric constructions and relation to metric holonomy groups. Classification results for Riemannian and Lorentzian conformal holonomy groups.
• Twistor equation on spinors, conformal Killing spinors and related geometries. Link to conformal holonomy.
• CR-geometry and Fefferman spaces.
• Conformal structures with unitary and special unitary holonomy groups. Results for other holonomy groups.
  

Prerequisites:

Basic aquaintance with Riemannian Geometry, fibre bundle methods, Lie groups and Lie algebras.

Deadline for applications:

October 1, 2007

The seminars take place at the Mathematisches Forschungsinstitut Oberwolfach. The number of participants is restricted to 24. The Institute covers accommodation and food. Travel expenses cannot be reimbursed. Applications including

• full name and address, including e-mail address
• present position, university
• name of supervisor of Ph.D. thesis
• a short summary of previous work and interest

should be sent as hard copy or by e-mail (.ps or .pdf file) to:

Prof. Dr. Gert-Martin Greuel
Universität Kaiserslautern
Fachbereich Mathematik
Erwin Schrödingerstr.
67663 Kaiserslautern, Germany
.