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18.979: Moduli Spaces

The 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 you take decent notes, please let me borrow your loose sheets so I can scan them. Thanks! -Ben

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	Main author	Status	Scans	Description
1	09-03 Notes	Christian	Done	Ben	Overview of moduli spaces,review of G-bundles and connections
2	09-08 Notes	Bhairav	Done	Ben	Sobolev spaces and gauge theoretic analysis
3	09-10 Notes	Tatyana	Draft	Ben	$\mathcal{A}/\mathcal{G}$ is a metric space, local structure of a quotient
4	09-15 Notes	Unclaimed	Unstarted	Ben	Stabilizers and holonomy, outline of slice theorem
5	09-17 Notes	James	Started	Ben	Proof of slice theorem, except injectivity
6	09-24 Notes	Lu	Done	Ben	Injectivity, and remarks on the Yang-Mills equation
7	09-29 Notes	Alex	Done	Ben	Classifying space for $\mathcal{G}$
8	10-01 Notes	Andy	Done	Ben	$\mathcal{G}$-equivariant cohomology of $\mathcal{A}$
9	10-06 Notes	Steven	Done	Ben	$\mathcal{G}$-equivariant cohomology of $\mathcal{A}$ over $\Sigma$
10	10-08 Notes	Nikhil	Done	Ben	Stabilizers and the rank 2 case
11	10-15 Notes	Ben	Draft		Details on $N(2,1)$
12	10-20 Notes	Alex	Done	Ben	Calculation of the Morse index of YMF
13	10-22 Notes	Christian	Started	Ben	Morse Theory
14	10-27 Notes	Bhairav	Done	Ben	Morse-Bott Theory
15	10-29 Notes	David	Draft	Ben	Poincare polynomial of $N(2,1)$
16	11-03 Notes	Nikhil	Unstarted	Ben	$N(2,1)$ as a Symplectic quotient
17	11-05 Notes	Lu	Done	Ben	Uhlenbeck's gauge-fixing lemma
18	11-12 Notes	Unclaimed	Unstarted	Ben	Uhlenbeck's lemma (cont'd)
19	11-17 Notes	Unclaimed	Unstarted	Ben	Uhlenbeck's lemma ($L^n$ estimates)
20	11-19 Notes	Unclaimed	Unstarted	Ben	Uhlenbeck's lemma ($L^n$ estimates cont'd)
21	11-24 Notes	Steven	Draft
22	11-26 Notes	Unclaimed	Unstarted
23	12-01 Notes	Unclaimed	Unstarted
24	12-03 Notes	Unclaimed	Unstarted
25	12-08 Notes	Andy	Started		4-d Yang-Mills
26	12-10 Notes	Tim	Done		Construction of Instantons

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. Most LaTeX commands are supported. You can make commutative diagrams via \xymatrix. This Wiki should recognize most things between \begin{document} and \end{document}. For example pages illustrating most features, see 10-06 Notes or 10-15 Notes.

Tips:

• If you don't already know LaTeX, try LyX. (Version 1.6.0 works well.) It will help you typeset your formulas, help you learn LaTeX, and export your work as a LaTeX document. With some minor modifications, you'll be able to paste that LaTeX into this Wiki.
• 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.