2004/05, 2nd semester

Course Materials
Foliations, C* algebras and index theory

Part I of this course describes the classical approach to characteristic classes for foliations. This include the Chern-Weil type construction, Bott’s vanishing theorem, the Godbillon-Vey class, and the Gelfand-Fuks cohomological realization.

Part II is devoted to the non-commutative approach to characteristic classes of foliations, via transverse Hopf symmetry and Hopf-cyclic cohomology. The main application discussed is the index theorem for transversely hypoelliptic operators on foliations.

In part III Bott periodicity in its classical form and in its Banach algebra are discussed. An outline of characteristic classes leading to the Atiyah-Singer formula are presented next. The Atiyah-Singer formula is proved as a corollary of Bott periodicity. At the end, the index theorem for families of elliptic operators is developed.

Part I, II
Henri Moscovici
Notes by P. Witkowski
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Table of contents
  1. Foliations
    1. What is a foliation and why is it interesting
    2. Equivalent definitions
    3. Holonomy grupoid
    4. How to handle M/F
    5. Characeristic classes
  2. Characteristic classes
    1. Preamble: Chern-Weil construction of Pontryagin ring
    2. Adapted connection and Bott's theorem
    3. The Godbillon-Vey class
    4. Nontriviality of Godbillon-Vey class
    5. Naturality under transversality
    6. Transgressed classes
  3. Weil algebras
    1. The truncated Weil algebras
    2. W_q and framed foliations
  4. Gelfand-Fuks cohomology
    1. Cohomology of Lie algebras
    2. Gelfand-Fuks cohomology
    3. Some "soft" results
    4. Spectral sequences
  5. Characteristic classes and Gelfand-Fuks cohomology
    1. Jet groups
    2. Jet bundles
    3. Characteristic map for foliation
  6. Index theory and noncommutative geometry
    1. Classical index theorems
    2. General formulation and proto-index formula
    3. Multilinear formulation: cyclic cohomology (Connes)
    4. Connes cyclic cohomology
    5. An alternate route via the families index theorem
    6. Index theory for foliations
  7. Hopf cyclic cohomology
    1. Preliminaries
    2. The Hopf algebra H_n
Clarifications for the last lectures
Henri Moscovici
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Table of contents
  1. Geometric background
    1. Diff-invariant setting
    2. Hypoelliptic signature operator
    3. Noncommutative local index formula
    4. Hopf algebraic symmetries
  2. Cohomological background
    1. Hopf cyclic cohomology
    2. Godbillon-Vey cocycle
    3. Schwarzian cocycle
    4. Transverse fundamental cocycle
    5. Isomorphism with Gelfand-Fuks cohomology
Part III
Paul Baum
Notes by P. Witkowski
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Table of contents
  1. Bott periodicity and index theorem
    1. Bott periodicity
    2. Elliptic operators
    3. Topological formula of Atiyah-Singer
    4. Index theorem for families of operators
Dirac operators and Spin structures
Paul Baum
Notes by P. Witkowski
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Table of contents
  1. The Dirac operator of R^n
    1. Dirac operator
    2. Bott generator vector bundle
  2. Spin representation and Spin^c
    1. Clifford algebras and spinor systems
Exam Exam questions

The exam was on 12th June 2005. It consisted of the written part (six exercises) and oral part. In the oral part each student had to answer two questions: easy one and difficult one (chosen from the two difficult questions).

Three students (on the graduate and undergraduate level) passed the exam.

Written part

Oral part

Set 1

Designed by: Pawel Witkowski