Analysis and index theory on singular spaces (a groupoid approach)

Victor Nistor, Mathematics Department, Pennsylvania State University.

Abstract: I will give an introduction to the analysis on singular and non-compact spaces, concentrating on an approach developed by Ammann, Connes, Lauter, Mazzeo, Melrose, Monthubert, Moroianu, Schrohe, Schulze, myself, and others using groupoids. This approach requires first a compactification of the non-compact manifold that is studied to a compact manifold with corners, such that the geometry is encoded in a Lie algebra of vector fields. I call a manifold with these properties a Lie manifold.

The class of Lie manifolds is convenient because it is large enough to be able to describe a large class of interesting examples, yet small enough to allow for nontrivial results. Examples of Lie manifolds include: compact (smooth) manifolds, manifolds with cylindrical ends, manifolds that are euclidean at infinity, conformally compact manifolds, and many others. Many singular spaces can be studied using Lie manifolds by conformally changing the natural metric on the smooth part. Examples include polyhedral domains and conical manifolds. Manifolds with cusps and locally symmetric spaces can also be studied by conformally changing the metric to a manifold with cylindrical ends or to more general Lie manifolds.

Here are the topics of the course and the tentative subject of the lectures (each lecture is two hours long and may contain a computer presentation):

  1. 11:05--13, Friday, February 9, room 314:

    Introduction, main definition, and motivations. (The Laplace operator in cylindrical and spherical coordinates, definition of Lie manifolds, the basic examples, motivation: Computational challenges in three dimensions.)

  2. 14--16, Wednesday, February 14, room 314:

    Lie manifolds and analysis on these spaces (geometric properties, differential operators, Sobolev spaces, presentation for lectures 2-4).

  3. 10--12, Thursday, February 15, Amphi Darboux:

    Pseudodifferential operators on Lie manifolds and groupoids (the constructions due to Connes, Melrose, Monthubert, and myself of natural algebras of pseudodifferential operators on Lie manifolds using groupoids; the properties of the pseudodifferential operators in these calculi of pseudodifferential operators, including mapping properties.)

  4. 16:10--18, Monday, February 19, room ??? :

    Fredholm and Index theory on Lie manifolds.

  5. 14--15:50, Wednesday, February 21, Amphi Darboux:

    Applications to analysis on polyhedral domains and numerical analysis (regularity results, quasi-optimal rates of convergence, presentation).

The plan and the number of lectures may slightly change depending on the interests of the audience.

(Last revised 01/30/07.)