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'''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
 +
 
== 18.979: Moduli Spaces ==
 
== 18.979: Moduli Spaces ==
  
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The main references so far have been  
 
The main references so far have been  
 
* [http://www.jstor.org/stable/37156 The Yang-Mills Equations on Riemann Surfaces (Atiyah, Bott)],  
 
* [http://www.jstor.org/stable/37156 The Yang-Mills Equations on Riemann Surfaces (Atiyah, Bott)],  
−
* [http://www.amazon.com/Instantons-Four-Manifolds-Mathematical-Institute-Publications/dp/038797377X Instantons and Four-Manifolds (Freed, Uhlenbeck)].
+
* [http://books.google.com/books?id=WxkiAAAACAAJ Instantons and Four-Manifolds (Freed, Uhlenbeck)].
  
 
===The Plan===
 
===The Plan===
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===Lectures===
 
===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.
+
'''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.
 
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.
  
 
{| border="1" cellpadding="5"
 
{| border="1" cellpadding="5"
−
!Wiki  !! Status  !! Scans !! Description
+
! # !!Wiki  !! Main author !! Status  !! Scans !! Description
 
|-
 
|-
−
|[[09-03 Notes]]
+
| 1
−
   || Unclaimed
+
  || [[09-03 Notes]]
 +
   || Christian
 +
  ||  Done
 
   || [../scans/09-03_Ben.pdf Ben]
 
   || [../scans/09-03_Ben.pdf Ben]
−
   || Overview of moduli spaces
+
   || Overview of moduli spaces,review of G-bundles and connections
 
|-
 
|-
−
|[[09-08 Notes]]  
+
| 2
−
   || Unclaimed
+
  || [[09-08 Notes]]  
 +
   || Bhairav
 +
  || Done
 
   || [../scans/09-08_Ben.pdf Ben]
 
   || [../scans/09-08_Ben.pdf Ben]
 
   || Sobolev spaces and gauge theoretic analysis
 
   || Sobolev spaces and gauge theoretic analysis
 
|-
 
|-
−
|[[09-10 Notes]]  
+
| 3
−
   || Unclaimed
+
  || [[09-10 Notes]]  
 +
   || Tatyana
 +
  || Draft
 
   || [../scans/09-10_Ben.pdf Ben]
 
   || [../scans/09-10_Ben.pdf Ben]
−
   || <math>\mathcal{A}/\mathcal{G}</math> is a metric space, local structure of a quotient
+
   || $\mathcal{A}/\mathcal{G}$ is a metric space, local structure of a quotient
 
|-
 
|-
−
|[[09-15 Notes]]  
+
| 4
 +
  || [[09-15 Notes]]  
 
   || Unclaimed
 
   || Unclaimed
 +
  || Unstarted
 
   || [../scans/09-15_Ben.pdf Ben]
 
   || [../scans/09-15_Ben.pdf Ben]
 
   || Stabilizers and holonomy, outline of slice theorem
 
   || Stabilizers and holonomy, outline of slice theorem
 
|-
 
|-
−
|[[09-17 Notes]]  
+
| 5
−
   || Unclaimed
+
  || [[09-17 Notes]]  
 +
   || James
 +
  || Started
 
   || [../scans/09-17_Ben.pdf Ben]
 
   || [../scans/09-17_Ben.pdf Ben]
 
   || Proof of slice theorem, except injectivity
 
   || Proof of slice theorem, except injectivity
 
|-
 
|-
−
|[[09-24 Notes]]  
+
| 6
−
   || Unclaimed
+
  || [[09-24 Notes]]  
 +
   || Lu
 +
  || Done
 
   || [../scans/09-24_Ben.pdf Ben]
 
   || [../scans/09-24_Ben.pdf Ben]
 
   || Injectivity, and remarks on the Yang-Mills equation
 
   || Injectivity, and remarks on the Yang-Mills equation
 
|-
 
|-
−
|[[09-29 Notes]]  
+
| 7
 +
  || [[09-29 Notes]]  
 
   || Alex
 
   || Alex
 +
  || Done
 
   || [../scans/09-29_Ben.pdf Ben]
 
   || [../scans/09-29_Ben.pdf Ben]
−
   || Classifying space for <math>\mathcal{G}</math>
+
   || Classifying space for $\mathcal{G}$
 
|-
 
|-
−
|[[10-01 Notes]]  
+
| 8
−
   || Unclaimed
+
  || [[10-01 Notes]]  
 +
   || Andy
 +
  || Done
 
   || [../scans/10-01_Ben.pdf Ben]
 
   || [../scans/10-01_Ben.pdf Ben]
−
   || <math>\mathcal{G}</math>-equivariant cohomology of <math>\mathcal{A}</math>
+
   || $\mathcal{G}$-equivariant cohomology of $\mathcal{A}$
 
|-
 
|-
−
|[[10-06 Notes]]  
+
| 9
 +
  || [[10-06 Notes]]  
 
   || Steven
 
   || Steven
 +
  || Done
 
   || [../scans/10-06_Ben.pdf Ben]
 
   || [../scans/10-06_Ben.pdf Ben]
−
   || <math>\mathcal{G}</math>-equivariant cohomology of <math>\mathcal{A}</math> over <math>\Sigma</math>
+
   || $\mathcal{G}$-equivariant cohomology of $\mathcal{A}$ over $\Sigma$
 
|-
 
|-
−
|[[10-08 Notes]]  
+
| 10
−
   || Unclaimed
+
  || [[10-08 Notes]]  
 +
   || Nikhil
 +
  || Done
 
   || [../scans/10-08_Ben.pdf Ben]
 
   || [../scans/10-08_Ben.pdf Ben]
 
   || Stabilizers and the rank 2 case
 
   || Stabilizers and the rank 2 case
 
|-
 
|-
−
|[[10-15 Notes]]  
+
| 11
 +
  || [[10-15 Notes]]  
 +
  || Ben
 
   || Draft
 
   || Draft
 
   ||
 
   ||
−
   || Details on <math>N(2,1)</math>
+
   || Details on $N(2,1)$
 
|-
 
|-
−
|[[10-20 Notes]]  
+
| 12
 +
  || [[10-20 Notes]]  
 
   || Alex
 
   || Alex
−
   ||  
+
   || Done
 +
  || [../scans/10-20_Ben.pdf Ben]
 
   || Calculation of the Morse index of YMF
 
   || Calculation of the Morse index of YMF
 
|-
 
|-
−
|[[10-22 Notes]]  
+
| 13
−
   || Unclaimed
+
  || [[10-22 Notes]]  
−
   ||  
+
   || Christian
−
   ||
+
   || Started
 +
  || [../scans/10-22_Ben.pdf Ben]
 +
   || Morse Theory
 
|-
 
|-
−
|[[10-27 Notes]]  
+
| 14
−
   || Unclaimed
+
  || [[10-27 Notes]]  
−
   ||  
+
   || Bhairav
−
   ||
+
   || Done
 +
   || [../scans/10-27_Ben.pdf Ben]
 +
  || Morse-Bott Theory
 
|-
 
|-
−
|[[10-29 Notes]]  
+
| 15
−
   || Unclaimed
+
  || [[10-29 Notes]]  
−
   ||  
+
   || David
−
   ||
+
   || Draft
 +
  || [../scans/10-29_Ben.pdf Ben]
 +
   || Poincare polynomial of $N(2,1)$
 
|-
 
|-
−
|[[11-03 Notes]]  
+
| 16
−
   || Unclaimed
+
  || [[11-03 Notes]]  
−
   ||  
+
   || Nikhil
−
   ||
+
   || Unstarted
 +
  || [../scans/11-03_Ben.pdf Ben]
 +
   || $N(2,1)$ as a Symplectic quotient
 
|-
 
|-
−
|[[11-05 Notes]]  
+
| 17
−
   || Unclaimed
+
  || [[11-05 Notes]]  
−
   ||  
+
   || Lu
−
   ||
+
   || Done
 +
   || [../scans/11-05_Ben.pdf Ben]
 +
  || Uhlenbeck's gauge-fixing lemma
 
|-
 
|-
−
|[[11-12 Notes]]  
+
| 18
 +
  || [[11-12 Notes]]  
 
   || Unclaimed
 
   || Unclaimed
−
   ||  
+
   || Unstarted
−
   ||
+
   || [../scans/11-12_Ben.pdf Ben]
 +
  || Uhlenbeck's lemma (cont'd)
 
|-
 
|-
−
|[[11-17 Notes]]  
+
| 19
 +
  || [[11-17 Notes]]  
 
   || Unclaimed
 
   || Unclaimed
−
   ||  
+
   || Unstarted
−
   ||
+
   || [../scans/11-17_Ben.pdf Ben]
 +
  || Uhlenbeck's lemma ($L^n$ estimates)
 
|-
 
|-
−
|[[11-19 Notes]]  
+
| 20
 +
  || [[11-19 Notes]]  
 
   || Unclaimed
 
   || Unclaimed
−
   ||  
+
   || Unstarted
−
   ||
+
   || [../scans/11-19_Ben.pdf Ben]
 +
  || Uhlenbeck's lemma ($L^n$ estimates cont'd)
 
|-
 
|-
−
|[[11-24 Notes]]  
+
| 21
−
   || Unclaimed
+
  || [[11-24 Notes]]  
 +
   || Steven
 +
  || Draft
 
   ||  
 
   ||  
 
   ||
 
   ||
 
|-
 
|-
−
|[[11-26 Notes]]  
+
| 22
 +
  || [[11-26 Notes]]  
 
   || Unclaimed
 
   || Unclaimed
 +
  || Unstarted
 
   ||  
 
   ||  
 
   ||
 
   ||
 
|-
 
|-
−
|[[12-01 Notes]]  
+
| 23
 +
  || [[12-01 Notes]]  
 
   || Unclaimed
 
   || Unclaimed
 +
  || Unstarted
 
   ||  
 
   ||  
 
   ||
 
   ||
 
|-
 
|-
−
|[[12-03 Notes]]  
+
| 24
 +
  || [[12-03 Notes]]  
 
   || Unclaimed
 
   || Unclaimed
 +
  || Unstarted
 
   ||  
 
   ||  
 
   ||
 
   ||
 
|-
 
|-
−
|[[12-08 Notes]]  
+
| 25
−
   || Unclaimed
+
  || [[12-08 Notes]]  
 +
   || Andy
 +
  || Started
 
   ||  
 
   ||  
−
   ||
+
   || 4-d Yang-Mills
 
|-
 
|-
−
|[[12-10 Notes]]  
+
| 26
−
   || Unclaimed
+
  || [[12-10 Notes]]  
 +
   || Tim
 +
  || Done
 
   ||  
 
   ||  
−
   ||
+
   || Construction of Instantons
 
|}
 
|}
  
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This Wiki is editable by anyone.  Just create an account, sign in, and an "edit" tab will appear at the top.   
 
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 "math mode" commands are supported, meaning anything you would enclose in &#36;'s.  You can make commutative diagrams via <tt>\xymatrix</tt>.  What doesn't work are non-"math mode" commands like <tt>\begin{eqnarray*}</tt>.  For example pages illustrating most features, see [[10-06 Notes]] or [[10-15 Notes]].
+
I use a hacked version of the Wikipedia LaTeX rendering engine.  Most LaTeX commands are supported.  You can make commutative diagrams via <tt>\xymatrix</tt>.  This Wiki should recognize most things
 +
between
 +
<tt>&#92;begin{document}</tt> and <tt>&#92;end{document}</tt>.  For example pages illustrating most features, see [[10-06 Notes]] or [[10-15 Notes]].
  
 
'''Tips:'''
 
'''Tips:'''
 +
* If you don't already know LaTeX, try [http://www.lyx.org/ 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.
 
* 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."
 
* Edit aggressively.  All changes are reversible, so it's impossible to "screw things up."

Latest revision as of 08:38, 22 November 2013

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 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.

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