Superconnection heat kernels, microlocal analysis and the geometry of certain non-compact manifolds

Información del proyecto

Identificador del acuerdo de subvención: FMBI961573

Proyecto cerrado

Fecha de inicio 1 Febrero 1997

Fecha de finalización 31 Julio 1997

Financiado con arreglo a

• Specific research and technological development programme in the field of the training and mobility of researchers, 1994-1998

Coste total

Sin datos

Aportación de la UE

Sin datos

Coordinado por

UNIVERSITE DE PARIS-SUD XI
France

Objetivo

Research objectives and content
Among the interesting invariants of a differentiable compact closed manifold are those defined by elliptic operators and in particular the so called generalized Dirac operators. One of the central results in this direction is the Atiyah-Singer index theorem for families of Dirac operators. The superconnection heat-kernel proof of the family index theorem, given by Jean-Michel Bismut almost 10 years ago, has allowed for interesting extensions of the theory, as for example the family index theorem for Dirac operators on manifolds with boundary by Bismut-Cheeger and Melrose-Piazza. The treatment given by Melrose-Piazza is also based on the use of a pseudo differential calculus on manifolds with boundary (the b-calculus).
The family index theorem is also the starting point of two fundamental topics in non-commutative geometry: the foliation index theorem and the higher index theorem on Galois coverings (by Connes and Connes-Moscovici respectively). Galois coverings and the leaves of a foliations are fundamental examples of non-compact manifolds.
The super connection approach to the family index theorem has been recently extended to these two non-commutative contexts by Heitch and Lott respectively.
The objectives of this research project are the followings. 1) prove a general higher index theorem on Galois coverings with boundary ( maybe assuming the covering group to be of polynomial growth with respect to a word metric). 2) apply the first theorem in order to investigate the homotopy invariance of higher signatures on manifolds with boundary. 3) prove a higher foliation index theorem on foliations with boundary. 4) investigate surgery results for higher index classes on Galois coverings which are the union along a hypersurface of two Galois coverings with boundary.
We have partial results on the first project (in collaboration with Eric Leichtnam). We expect the superconnection heat kernel approach and the use of techniques from microlocal analysis (as in the work of Melrose-Piazza) to be very relevant for this project.
Training content (objective, benefit and expected impact)
The research project involves a clear understanding of the analytic properties of the (superconnection) heat-kernel on Galois coverings and on foliations. The responsible scientist of the host institution, Jean-Michel Bismut, is one of the world experts on heat kernels. I hope to benefit very much from the interaction with him and in particular to deepen my present knowledge of the field. The techniques that I can learn in this way will be influential in much of my future research activity. I also expect the interaction with other members of the faculty at Orsay and with mathematicians of the nearby "Institut des Hautes Etudes Scientifiques" to be very important both for the research project and the training aspect of it.
I should also mention that despite their importance the topics I intend to research and be trained on, are very little represented in Italy. Links with industry / industrial relevance (22)
As for the majority of the research results of pure Mathematics, we do not expect immediate applications to industry.

Ámbito científico (EuroSciVoc)

• ciencias médicas y de la salud › medicina clínica › cirugía
• ciencias naturales › matemáticas › matemáticas puras › geometría

Programa(s)

• FP4-TMR - Specific research and technological development programme in the field of the training and mobility of researchers, 1994-1998

Tema(s)

• 0302 - Post-doctoral research training grants
• TM22 - Geometry and Topology

Convocatoria de propuestas

Datos no disponibles

Régimen de financiación

RGI - Research grants (individual fellowships)

Coordinador

UNIVERSITE DE PARIS-SUD XI

• Aportación de la UE: Sin datos
• Dirección: Université de Paris 11, Bètiment 425 ; 91405 GOMETZ LA VILLE ; Francia
• Coste total: Sin datos

Participantes (1)

Not available

Italia

• Aportación de la UE: Sin datos
• Coste total: Sin datos