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18.979: Moduli Spaces

This Fall 2008 graduate geometry seminar is being taught by Prof. Mrowka. This Wiki is maintained by Ben.

References

The main references so far have been

• The Yang-Mills Equations on Riemann Surfaces (Atiyah, Bott),
• Instantons and Four-Manifolds (Freed, Uhlenbeck).

The Plan

Each student registered for a grade is to type up lecture notes for two or so lectures, and also correct mistakes, and ask/answer questions on this Wiki.

Lectures

If anyone is willing to donate their handwritten notes (on loose sheets of paper), I'd be happy to scan them in as a supplement my sloppy notes.

To stake a claim to a lecture, simply write up a draft of that lecture. (All changes will be attributed to your Wiki account.) No permission is required, but the more advanced students should leave the earlier/simpler lectures for the less advanced students.

Wiki	Status	Scans	Description
09-03 Notes	Unclaimed	Ben	Overview of moduli spaces
09-08 Notes	Unclaimed	Ben	Sobolev spaces and gauge theoretic analysis
09-10 Notes	Unclaimed	Ben	$\mathcal{A}/\mathcal{G}$ is a metric space, local structure of a quotient
09-15 Notes	Unclaimed	Ben	Stabilizers and holonomy, outline of slice theorem
09-17 Notes	Unclaimed	Ben	Proof of slice theorem, except injectivity
09-24 Notes	Unclaimed	Ben	Injectivity, and remarks on the Yang-Mills equation
09-29 Notes	Unclaimed	Ben	Classifying space for $\mathcal{G}$
10-01 Notes	Unclaimed	Ben	$\mathcal{G}$-equivariant cohomology of $\mathcal{A}$
10-06 Notes	Unclaimed	Ben	$\mathcal{G}$-equivariant cohomology of $\mathcal{A}$ over $\Sigma$
10-08 Notes	Unclaimed	Ben	Stabilizers and the rank 2 case
10-15 Notes	Draft		Details on $N(2,1)$
10-20 Notes	Unclaimed
10-22 Notes	Unclaimed
10-27 Notes	Unclaimed
10-29 Notes	Unclaimed
11-03 Notes	Unclaimed
11-05 Notes	Unclaimed
11-12 Notes	Unclaimed
11-17 Notes	Unclaimed
11-19 Notes	Unclaimed
11-24 Notes	Unclaimed
11-26 Notes	Unclaimed
12-01 Notes	Unclaimed
12-03 Notes	Unclaimed
12-08 Notes	Unclaimed
12-10 Notes	Unclaimed

Supplementary notes

• Harmonic connections on line bundles

Editing the Wiki

This Wiki is editable by anyone. Just create an account, sign in, and an "edit" tab will appear at the top.

I use a hacked version of the Wikipedia LaTeX rendering engine. You can make commutative diagrams via \xymatrix. Basically, you just enclose all LaTeX code in <math></math>. For an example page illustrating most features, see 10-15 Notes.

Tips:

• Be sure to save your work often! The "Save page" button is your friend. Otherwise, something might navigate you away from the Wiki, causing you to lose your work.
• Edit aggressively. All changes are reversible, so it's impossible to "screw things up."
• Hit the "Show preview" button to see what you're doing.

If you have trouble, feel free to let me know. If you need to use a package, a macro, or some command which isn't supported, I can easily hardcode it for you.