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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
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.
