Diffeomorphism groups: algebra, topology, homology is a week-long summer school for graduate students on classical diffeomorphism groups and their connections with problems in topology, geometry, and dynamics.

Open to and appropriate for graduate students at all levels, the summer school features two self-contained minicourses, guided problem sessions, and guest lectures.

A very short intro to diffeomorphism groups

### Minicourses

1. The algebra of diffeomorphism groups:   Kathryn Mann (UC Berkeley)

Lecture notes, including problem sets (will be updated throughout the week)
Proof of perfectness for Diff_0(M) from Tuesday's lecture

This course introduces classical and new results on the algebraic structure of diffeomorphism groups. These groups are algebraically simple (no nontrivial normal subgroups) -- for deep topological reasons due to Epstein, Mather, Thurston... but nevertheless have a very rich algebraic structure. We'll see that:

1. The algebraic structure of Diff(M) determines M
If Diff(M) is isomorphic to Diff(N), then M and N are the same smooth manifold (Filipkiewicz)
2. The algebraic structure of Diff(M) "captures the topology" of Diff(M)
Any group homomorphism from Diff(M) to Diff(N) is necessarily continuous. Any homomorphism from Homeo(M) to any separable group is necessarily continuos (Hurtado, Mann)

We'll explore consequences of these theorems and related results, as well as other fascinating algebraic properties of diffeomorphism groups of manifolds (for instance, distorted elements, left-invariant orders, circular orders...). In the last lecture, we'll touch on recent work on the geometry and metric structure of diffeomorphism groups.

Beginner -- assumend background knowledge.

1. Basic knowledge on smooth manifolds, Lie groups, and group actions, as in Lee's Introduction to smooth manifolds. The chapter on Lie groups and group actions is particularly recommended.

Intermediate level:

1. Ghys, Groups acting on the circle (survey paper)
2. Banyaga, The structure of classical diffeomorphism groups (book), chapter 1.

Advanced Level -- not seriously recommended until after the summer school!

2. Topological aspects of diffeomorphism groups:   Bena Tshishiku (Chicago)

Lecture notes, including problem sets (will be updated throughout the week)

The cohomology of the diffeomorphism group Diff(M) of a manifold M and its classifying space BDiff(M) are important to the study of fiber bundles with fiber M. In particular, we can learn a lot about M bundles by
(1) finding nonzero elements of H*(BDiff(M)) and
(2) relating these classes to the topology/geometry of individual bundles.
A good start to (1) is to understand the topology of Diff(M), and this has been done in low dimensions (by Smale, Hatcher, Earle-Eells, Gabai, and others). An example of (2) is the study of fiber bundles admitting a flat connection (as pioneered by Milnor and Morita). This course will discuss (1) and (2) through a few rich examples and in connection to major areas of current research. Our discussion will include

1. the homotopy type of Diff(M) when dim(M) < 4;
2. circle bundles, the Euler class, and the Milnor-Wood inequality; and
3. surface bundles, the Miller-Morita-Mumford classes, and Nielsen realization problems.

Beginner -- good background knowledge to have.

1. From Hatcher, Algebraic topology: basic knowledge of homotopy groups, fibrations, long exact sequence of homotopy groups
2. Hatcher, Vector bundles and K-theory Chapters 1 and 3.

Intermediate level:

1. Stasheff, Continuous cohomology of groups and classifying spaces (survey paper)
2. Hatcher, A 50 -Year View of Diffeomorphism Groups (talk notes)

1. Morita, Geometry of characteristic classes (book)
2. Calegari, The Euler class of planar groups . This short research paper gives some nice connections between the two minicourses.

Participants are strongly encouraged to attend both minicourses, as each will use material taught in the other.

### Guest lectures

We are please to have a series of short research talks by senior graduate students:

Ying Hu (Louisiana State)
Sander Kupers (Stanford)   [Notes]
Sam Nariman (Stanford)
Nathan Perlmutter (Oregon)
Wouter van Limbeek (Chicago)

### Questions list

Click here for the list of problems and discussion questions submitted by participants.

### Registration

Registration is now closed.

### Financial Support

We have a limited amount of financial support for U.S. citizens and permanent residents. To apply, fill out the registration form, which asks also for a short letter of reference from your advisor or other faculty member.

### Schedule

Available here!
Registration begins at 8:45 on Monday morning.
All talks will take place in Evans hall.

Suggestions for activities on Wednesday afternoon (free time):

### Contact info

Questions? Contact Kathryn Mann or e-mail diffgroups2015@gmail.com.