CMA Workshop on Pseudo-differential operators, Lie groupoids and index theory

1 - 3 September 2009

This workshop is part of the CMA's Special Year on Spectral Theory and Operator Theory within the Mathematical Sciences Institute.

Speakers:

Claire Debord
Jean-Marie Lescure
Sylvie Paycha
Bryan Wang
DOP

Schedule:

Tuesday, September 1

9:30 - 10:50 (JD LG15)

Claire Debord: Introduction to Lie groupoids

10:50- 11:20 Morning Tea (Common Room)

11:20 - 12:40 (JD LG15)

Jean-Marie Lescure: Pseudo-differential operators on Lie groupoids

2:00 - 3:00 (JD LG15)

Bryan Wang: Longitudinal index theorem and longitudinal index formulae for foliations

Wednesday, September 2

9:30 - 10:50 (JD G35)

Claire Debord: K-duality for pseudomanifolds

10:50- 11:20 Morning Tea (Common Room)

11:20 - 12:40 (JD G35)

Jean-Marie Lescure: An index theorem on pseudomanifolds with isolated conical singularities.

3:00 - 4:30 (JD LG15)

Sylvie Paycha: From defect formulae for integrals of pseudodifferential symbols to a local formula for the index of an operator, Part I

Thursday:

9:30 - 10:50 (JD G35)

Sylvie Paycha: From defect formulae for integrals of pseudodifferential symbols to a local formula for the index of an operator, Part II

10:50- 11:20 Morning Tea (Common Room)

11:20 - 12:40 (JD G35)

Bryan Wang: Twisted K-theory and twisted Riemann-Roch theorem

2:30 - 3:30 (JD LG15)

Discussion and Open problems Session

Speaker/Title/Abstract:

Claire Debord: (1) Introduction to Lie groupoids

In this talk, we will define the notion of Lie groupoid, we will describe several famous examples of groupoids involved in Noncommutative geometry (e.g. : goup action, foliation, tangent groupoid ...) . Finally we will explain the construction of the $C^*$-algebras associated to a Lie groupoid.

It will be a survey talk with no more prerequisite than usual differential geometry.

Claire Debord: (2) K-duality for pseudomanifolds

Index theory on a smooth manifold involves the tangent space of the manifold. Thus, when one wants to make index theory on singular spaces, one first need to understand what should be the tangent space in this situation. Here we will be interested in pseudomanifolds. We will first recall their definition. Then we will see how one can construct a tangent space $T^{S} X$ associated to such a singular space $X$. It will no longer be a fibre bundle, but a Lie groupoid. We will then see that, as in the case of smooth manifolds, there is a K-duality between the $C^*$-algebra of $T^{S} X$ and the algebra $C(X)$ of continuous functions on $X$.

Jean-Marie Lescure: (1) Pseudo-differential operators on Lie groupoids.

We will briefly review basic definitions and properties of pseudo-differential operators on manifolds. We will then explain the notion of pseudodifferential operators on Lie groupoids. We will see through several examples that this notion covers many different situations (ordinary calculus, symbols, families, operators on stratified spaces).

Jean-Marie Lescure: (2) An index theorem on pseudomanifolds with isolated conical singularities.

In this talk, we will see how the two descriptions of the index map on a manifold given by Atiyah and Singer can be generalized to pseudomanifolds with isolated conical singularities. The heart of the construction is the use of the K-theory of the noncommutative tangent space. We will describe both the analytical and topological index map in this situation and we will prove their equality. We will explain how the groupoids technics allow an unified presentation of index theory for manifolds and pseudomanifolds with isolated conical singularities.

Sylvie Paycha: From defect formulae for integrals of pseudodifferential symbols to a local formula for the index of an operator

Translation invariant linear forms on the whole algebra of pseudodifferential symbols on R^d are proportional to the noncommutative residue. As a result, the Lebesgue integral on L^1 functions unfortunately does not extend to a translation invariant linear form on the whole algebra of pseudodifferential symbols on R^d, forcing to work with ordinary linear extensions which fail to be translation invariant.

Defect formulae which express the difference between various linear extensions of the ordinary integral, show that these differ by local terms involving the noncommutative residue. Defects given by these local terms are responsible for discrepancies/anomalies arising from the lack of properties such as Stokes' theorem, covariance which one expects of an integral but which fail to hold for these linear extensions.

When extended to pseudodifferential operators on closed manifolds, such defect formulae lead to expressions of zeta regularised traces of a differential operator in terms of a residue of its logarithm. In particular, one can express the index of a Dirac type operator on a closed manifold in terms of the residue of a logarithm of a generalized Laplacian, thus giving an a priori local description of the index.

This point of view on the index, seen as a local defect or discrepancy, will be discussed in these lectures.

Bryan Wang: Longitudinal index theorem and longitudinal index formulae for foliations

I will review the longitudinal index theroem for foliations due to Connes-Skandalis. In the presence of an invariant transverse measure, I will explain how to get Connes index formulae using longitudinal cohomology and longitudinal characteristic classes of foliations.

Bryan Wang: Twisted K-theory and twisted Riemann-Roch theorem

I will review some latest developments in twisted K-theory, twisted Riemann-Roch theorem and twisted index formulae.

Bryan Wang: Geometric cycles and D-branes on foliated spaces

I will discuss the notion of twisting on Lie groupoids and twisted longitudinal index theorem using geometric cycles (also called D-branes) on foliated spaces.

Further information

For further information about the program please contact the organiser: Bryan Wang
For administrative matters, please contact Annette Hughes.