**Time and Place:**MWF from 11:00-11:50am in TBA. First half of semester only, ends Oct 15.**Section:**FTM**CRN:**57739**Instructor:**Nathan Dunfield**E-mail:**nmd@illinois.edu**Office:**378 Altgeld.**Office Phone:**(217) 244-3892**Office Hours:**TBA.

**Web page:**http://dunfield.info/595A**Lecture notes**

As with many areas of topology and geometry, the study of
3-dimensional manifolds begins with its codimension-1 submanifolds,
namely embedded surfaces. From there, it is very natural to consider
foliations of 3-manifolds by surfaces, that is, locally trivial
decompositions into (typically noncompact) surface leaves. While
every closed orientable 3-manifold has a foliation, the seemingly mild
condition of requiring the foliation to be taut restricts both the
topology of the ambient manifold (e.g. the universal cover must be
**R**^{3}) and of the leaves, and forces the existences of interesting
transverse dynamics.

I will start with an overview of foliations, focusing on examples, and then discuss the basic properties of taut foliations on 3-manifolds. Then I will then turn to the foundational work of Gabai, specifically his tool of sutured manifold hierarchies for constructing foliations. Next, I will discuss connections to contact geometry, namely the work of Eliashberg-Thurston which builds from a taut foliation a pair of tight contact structures. Finally, I will explain how taut foliations fit into the various Floer homology theories of 3-manifolds (the Heegaard, instanton, and monopole Floer homologies).

**Prerequisites:** Basic knowledge of smooth manifolds and
algebraic topology, e.g. Math 518 and Math 525. No prior knowledge of
foliations or 3-manifolds will be assumed.

I will not follow any particular text, but recommend the following:

- A. Candel and L. Conlon,
*Foliations I*, Graduate Studies in Mathematics**23**, AMS, 1999. - A. Candel and L. Conlon,
*Foliations II*, Graduate Studies in Mathematics**60**, AMS, 2003. - D. Calegari,
*Foliations and the geometry of 3-manifolds*, Oxford Univ. Press, 2007.

Students registered for the course will need to write a short (2-4 page) paper which will be due on Friday, October 15th. This paper is largely free-form, and can be about any subject related to the content of this course. For instance, it could be a brief account of a result not covered in class, a review of the some related results explaining why they are interesting, a detailed work-out of a proof only sketched in class, or careful solutions to problems from class or taken from one of the texts or other readings.

Notes from each lecture will be posted here.

- Aug 23. First day of class!