# Main Page

**Note:** As of 2012, an upgrade to MediaWiki has broken my LaTeX hack. When I get the chance, I will change over everything into a PDF so that it becomes readable. Test edit

## Contents |

## 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.