Instructor: Peter Teichner

Lectures: Tu 3:30-5:00, 51 Evans

Course Control Number: 54973

Recommended Reading: Simons-Sullivan.pdf, Freed.pdf, Hopkins-Singer.pdf, Freed-Hopkins.pdf, Bunke-Schick.pdf, Freed-Moore-Segal.pdf

Syllabus: We’ll study Smooth Cohomology, a combination of (generalized) cohomology and differential forms. In the first meeting we’ll give a survey of the subject, the later lectures will be given by participants, with the material and speaker determined two weeks in advance.

Sept. 2  Peter Teichner, Introduction to Smooth Cohomology

Sept. 16 Nate Watson, Differential Characters, Part 1

This talk is going to explain the definition of a differential character, which is a "smooth" map from the (singular) cycles of manifold to R/Z. It turns out that there's a natural definition of this from the point of view that the differential forms on a manifold should be thought of (via integration along chains) as exactly its "smooth" cocyles. One motivation for doing this (as mentioned in Peter's talk) is that it allows you to combine the curvature form and Chern character of a principle circle bundle with connection into one datum, but motivations will be left till next week. The source for this talk will be Simons-Sullivan, including the statement of 1.1 and explanation of the main diagram, and hopefully an explanation of at least the map that's used to prove uniqueness and where the technical lemmas come in to show the map is well-defined.

Prerequisites: None worth mentioning. A good warm-up would be to contemplate the one-dimensional case discussed in Peter's talk.

Sept. 23 Nate Watson, Differential Characters, Part 2

Sept. 30 William Slofstra

The goal of my talk will be to explain the early motivation for differential

characters.  I will focus on explaining the connection with Chern-Simons forms

and characteristic classes. The talk will be based on the first five sections

of the paper "Differential Characters and Geometric Invariants" by Cheeger and

Simons. Nate has very kindly scanned this paper, available here: CS.pdf. I also have a photocopied version if you need to make a copy.

It might also be a good idea to take a look at the original paper on

Chern-Simons forms, available from JSTOR

Oct. 7 William Slofstra, Definition of Chern-Simons characters

Oct. 14 Dan Berwick-Evans, Electro-Magnetism and differential characters

Oct. 21 Dan Berwick-Evans and Alan Tarr

Oct. 28 Alan Tarr, Circle-gerbes with connection

Nov. 4 Arturo Prat-Waldron, Deligne Cohomology, Part 1

Nov. 18 Arturo Prat-Waldron, Deligne Cohomology, Part 2

Nov. 25 Dmitri Pavlov, Generalized Smooth Cohomology, Part 1

Dec. 3 Qin Li, Generalized Smooth Cohomology, Part 2