Steven Hurder
Department of Mathematics
University of Illinois at Chicago
hurder@uic.edu, foliations.org

A foliation is like an onion: you just peel space back layer by layer to see what's there. The word foliation is related to the french term feuilletage, as in the very tasty pastry mille feuilles.

Foliations arise natually in many areas of geometry and dynamical systems. They are an essential tool in the study of smooth dynamical systems, can be used to study the topology of 3-manifolds, and are the core geometry underlying non-commutative geometry and index theory.

The most current view on the subject, however, is that a foliation is a "Stack" modeled on an étale differentiable Lie groupoid - a continuous field of groupoids over the foliated manifold M.

• Lecture 1 - "Foliations and groupoids" (Wednesday, January 31: 10:00-12:00, Room 314, IHP)

  We discuss foliations as groupoids and their classifying spaces.



• Lecture 2 - "Stackie dynamics" (Friday, January 31: 9:00-10:55, Room 314, IHP)

  We discuss dynamical properties of a foliation which are well-defined up to Morita equivalence of the associated groupoid. These are the invariants of the coarse dynamics of a foliation.



• Lecture 3 - "Analysis of leafwise operators" (Wednesday, February 7: 13:30-15:30, Amphi Darboux, IHP)

  We discuss the basic properties of analysis along the leaves of a foliation, considered as a complete Riemannian immersed submanifold.



• Lecture 4 - "Convex hulls & spectral flow" (Friday, February 9: 9:00-10:55, Room 314, IHP)

  We discuss a generalization of the notion of the algebraic hull of a quasi-periodic function, and how the odd foliation index theory is applied to study the spectral flow of self-adjoint leafwise elliptic operators.



• Lecture 5 - "Spectrum & asymptotic invariants" (Thursday, February 22: 9:00-11:00, Amphi Darboux, IHP)

  We discuss the construction of the coarse index of a foliation, which constructs a families index problem over M and, via descent, to yield index classes in K^*(M). When applied to self-adjoint leafwise elliptic operators paired with almost flat vector bundles, this yields a new method to estimate the spectral density of leafwise operators.



• Lecture 6 - "Coarse invariants of chaotic foliations" (Friday, February 23: 9:00-10:55, Room 314, IHP)

  In this last lecture, we discuss the "coarse cohomology theory" of a foliation. This yields invariants of foliations based on categorical methods applied to coarse geometry. It will be calculated in a "typical" example of a chaotic foliation, the generalized Hirsch foliations.

• Books on foliations & Flow diagram
• Some pictures of foliations
• References for talks

• "Foliations and foliated vector bundles" by John Milnor (1970)
• "Dynamics and the Godbillon-Vey Classes: a History and Survey" by Steve Hurder (2000)
• "Classifying spaces for groupoid structures" by Takahi Tsuboi (2001)
• "Naissance des feuileatges, d'Ehresmann-Reeb a Novikov" by André Haefliger (2003)
• "Characteristic classes of Riemannian foliations" by Steve Hurder (2003)

There are now several good introductory books to the geometric theory of foliations:

• "Global analysis on foliated spaces", by Calvin C.Moore and Claude Schochet, with Appendices by S. Hurder and Robert J. Zimmer, sec. ed. (2006)
• "Dynamics of foliations, groups and pseudogroups", by Pawel Walczak (2004)
• "Foliations 1 & 2", by Alberto Candell and Lawrence Conlon, (2000 & 2003)
• "Introduction to Foliations and Lie Groupoids", by Ieke Moerdijk and Janez Mrcun (2003)
• "Confoliations", by Yakov Eliashberg and William Thurston (1998)
• "Geometry of foliations", by Philippe Tondeur (1997)
• "Feuilletages", by Claude Godbillon (1991)
• "Foliations on Riemannian manifolds", by Philippe Tondeur (1988)
• "Introduction to the geometry of foliations. Parts A & B", by Gilbert Hector and Ulrich Hirsch (1986 & 1987)
• "Differential geometry of foliations", by Bruce Reinhart (1983)
• "Geometric Theory of Foliations", by Caesar Camacho and Alcides Lins Neto (1979) (transl. 1985)
• "Topology of foliations: an introduction", by Itiro Tamura (1976) (transl. 1992)
• "The Quantitative Theory of Foliations", by H. Blaine Lawson (1975)

• Ergodic theory and Weil measures for foliations (1987)
• The ∂-operator (1988)
• Eta invariants and the odd index theorem for coverings (1990)
• Cyclic cocycles, renormalization and eta-invariants (1991)
• The longitudinal cocycle and the index of Toeplitz operators (1991)
• Topology of covers and the spectral theory of geometric operators (1993)
• Exotic theory for foliated spaces (1993)
• Spectral theory of foliation geometric operators (1994)
• Pure-point spectrum for foliation geometric operators (1994)
• Coarse geometry of foliations (1994)
• Exotic index theory and the Novikov conjecture (1995)
• Coarse cohomology for families (2001)
• Tangential LS category and cohomology for foliations (2002)
• Hirsch foliations in codimension greater than one (2005)
• Transverse LS-category for Riemannian foliations (2006)
• Lecture notes for talks (to be posted)