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Abstract
Understanding the basic concepts of singular foliation, holonomy, foliation C*-algebra, and some aspects of the associated representation theory.
Objective
Understanding the basic concepts of singular foliation, holonomy, foliation C*-algebra, and some aspects of the associated representation theory.
Content
1. Regular foliations: The Frobenius theorem and foliation charts. 2. The holonomy groupoid of a regular foliation. Construction and examples. Explanation why the holonomy groupoid replaces the leaf space. 3. The C*-algebra of a regular foliation. Construction and examples. 4. Singular foliations: Definition and examples. Correspondence of projective modules of vector fields with almost regular foliations. 5. Bisubmersions and bisections: Definition and examples. 6. The holonomy groupoid of a singular foliation: Construction and examples. Proof that the holonomy groupoid of an almost regular foliation is always a Lie groupoid. 7. Applications in Poisson geometry: The almost regular case and log- symplectic Poisson structures. (Computation of the holonomy groupoid in this case and proof that it is a Poisson groupoid.) 8. The C*-algebra of a singular foliation: Construction and computations (exact sequence of foliation C*-algebra) 9. Some representation theory: The desintegration theorem. If me allows it, a choice of the following topics can be presented, depending on the interest of the audience: - The hierarchy of singularities. - Deformation to normal cone and the analytic index map. Also strict quantisation. - Laplacians of singular foliations as unbounded multipliers of the foliation C*-algebra, and their spectrum. - More generally, longitudinal pseudodifferential operators.
Resources
Literature
- Lecture Notes by the Instructor Further material: - Alberto Candel and Lawrence Conlon. Foliations I. AMS Graduate Studies in Mathema cs, vol. 23. - Kirill C. H. Mackenzie. General theory of Lie groupoids and Lie algebroids. LMS Lecture Notes Series 2013. https://doi.org/10.1017/CBO9781107325883 - Jean Renault. A groupoid approach to C*-algebras. Lecture Notes in Mathematics vol. 798, Springer https://doi.org/10.1007/BFb0091072 - Alan L. T. Paterson. Groupoids, Inverse Semigroups and their Operator Algebras. Progress in Mathematics vol. 170, Springer https://doi.org/ 10.1007/978-1-4612-1774-9 - Ieke Moerdijk. Introduction to Foliations and Lie groupoids. Cambridge University Press. https://doi.org/10.1017/CBO9780511615450
General Information
- Language
- English
- Levels
- MSC
Examination
- Type
- graded semester performance
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture |
Singular Foliations (University of Zurich)
**Course at University of Zurich**
|
No time listed | 4 h weekly |
| exercise |
Singular Foliations (University of Zurich)
**Course at University of Zurich**
|
No time listed | 2 h weekly |
Offered In
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Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
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