Schedule: MWF 9 - 9:50am, Maloney 560
Office hours: MW 10 - 10:50am
My email: boninger AT bc DOT edu
Textbooks and Papers Referenced (note: some relevant excerpts have been uploaded to links in the next section)
Differential Geometry: Connections, Curvature, and Characteristic Classes, Loring Tu
Differential Geometry: Bundles, Connections, Metrics and Curvature, Clifford Taubes
Characteristic Classes, John Milnor and James Stasheff
"An Introduction to Gauge Theory," John Morgan, from Gauge Theory and the Topology of Four-Manifolds (IAS/Park City Mathematics Series Vol. 4), Edited by Robert Friedman and John Morgan
Instantons and Four-Manifolds, Daniel Freed and Karen Uhlenbeck
Foundations of Differientable Manifolds and Lie Groups, Frank Warner
Morse Theory, John Milnor
Floer Homology Groups for Homology Three-Spheres, Peter Braam
An Instanton-Invariant for 3-Manifolds, Andreas Floer
Tu's book will be our primary text for the first third of the semester. Email me if you're having trouble obtaining a copy of any source material.
The readings and problem sets listed below will be updated on a daily/weekly basis to reflect what we cover in class. Problems are optional, but I recommend trying them out to better understand and absorb the material. Difficulty level will vary.
Readings and Problems
Week 1
Reading
Milnor & Stasheff, Ch. 2, 3
Tu, Sections 7.4, 7.5, 7.7, 10.1--10.3 (see also Prop. 6.3(ii)), 10.6, 10.7, 11.1, 11.2, 29.1 (see also Thm. 13.1 and Sections 14.5--14.7)
Week 2
Reading
Tu, Sections 6.1--6.3, 11.1, 21.1--21.7, 22.2
Week 3
Reading
Tu, 22.1, 23 (all of it), 24 (see also Propositions 11.4 and 11.5), 25, 26
Week 4
Reading
Tu, 27 (and see 15.4--15.7 for a bit of info on Lie groups and Lie algebras), 28.1, 28.2
Week 5
Reading
Tu, 28.3, 31.1, 31.2, 29 (some proofs omitted)
Week 6
Reading
Tu, 30
Week 7
Reading
Tu, 31.3--31.5 (exposition is very different from what we did in class, and some of it appears on the homework. The key points are: curvature is an adP valued form on M, the definition of the covariant derivative on P, and the covariant derivative formulation of the Bianchi identity)
Tu 32
Week 8
Reading
Warner (first few pages)
Week 9
Reading
Freed and Uhlenbeck 2 (up to the Yang-Mills functional)
Week 10
Reading
My notes on curvature and gauge transformations
Week 11
Reading
These notes prove signature is a cobordism invariant; see also the first problem below.
Freed and Uhlenbeck 2 ("Line Bundles" until the end)
Morgan 4 (we just sketched out the broad ideas at the highest level)
Week 12
Reading
Morgan 5 (again, just sketching out ideas at a high level)
Week 13
Reading
Week 14
Reading
Week 15
Reading
Morgan, exercises 4.20--4.24 (the Chern-Simons functional), 5.20, 5.21
Braam
Floer