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INdAM Fellowship Programs in Mathematics and/or Applications cofunded by Marie Skłodowska-Curie Actions

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Sara Azzali

INdAM-COFUND Outgoing fellow between 2012-01-31 and 2014-01-30

Project information


Higher Primary and Secondary Invariants for Foliations in Noncommutative Geometry

Outgoing host organisation                                  Return host organisation

Institut de Mathématiques                                             Dipartimento di Matematica
Université Paris Diderot                                                  Università  degli Studi di Roma “La Sapienza”

175, rue du Chevaleret                                                     Piazzale Aldo Moro, 5
75013 Paris, France                                                          00185 Roma, Italy                     


In global analysis, the spectral properties of elliptic operators are an essential tool to detect the geometric characteristics of spaces. In particular, index theory provides fundamental results relating the analysis of a manifold to the topology.

In this project we investigate index theory for foliations, i.e. spaces which are partitioned into noncompact manifolds, with a very singular parameter space. Foliations can be successfully modeled by the techniques of noncommutative geometry.

Our main goal is to construct higher invariants for foliations and to find new applications to the geometry of such spaces. An important tool of our project is the heat-equation method, which is applied to produce geometric refinements of index formulae, and so called secondary invariants “eta” and “torsion”. We plan also to use different points of view and construct secondary invariants from the comparison of different representatives of the index class.

One of the main objectives is to construct the “noncommutative eta form” for the family of tangential signature operators on foliated bundles, and prove a higher local index theorem.

Fellow information

Research interests

Global analysis, index theory, heat-kernel techniques, secondary invariants eta and torsion. Noncommutative geometry.

Previous positions, awards

  • 2008-2011: Post doc, Mathematisches Institut, Georg-August-Universität Göttingen
  • 2007-2008: Post doc, LMAM, University of Metz, Groupe de travail Géométrie non commutative

Publications, preprints, other works

  1. Sara Azzali, Charlotte Wahl, Spectral flow, index and the signature operator, Journal of Topology and Analysis 3 (2011) 37-67
  2. Sara Azzali, L2 Rho form for normal coverings of fibre bundles, International Journal of Mathematics 22 (2011) 1139-1161
  3. Sara Azzali, Large time limit and the L2 local index theorem, Oberwolfach Rep. 28 (2010) 10.4171.
  4. Sara Azzali, Two spectral invariants of type Rho, Oberwolfach Rep. 4 (2007) 41/2007

Conferences, schools, other events

  • Torsion analytique et ses applications
    Département de Mathématiques d’Orsay, Paris, France, 18-22 June 2012
  • INdAM meeting “Index Theory, noncommutative geometry and applications”
    Cortona, Italy, 11-16 June 2012
  • Analysis and geometric singularities
    MFO, Oberwolfach, Germany, 7-12 May 2012
  • Invited talk “Invariants eta et courbure scalaire positive”
    at Séminaire Géométrie Dynamique
    Université Lille 1, France, 3 February 2012

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