© | Dror Bar-Natan: Classes: 2013-14: AKT:  < > 

AKT-140228 Video

width: 400 720 download ogg/AKT-140228_400.ogg
Videography by Iva Halacheva troubleshooting

pop-out / sync handout

Notes on AKT-140228:    [edit, refresh]

More on Feynman diagrams, beginning of gauge theory.


[edit navigation bar]

# Week of... Notes and Links
1 Jan 6 About This Class (PDF).
dbnvp Monday: Course introduction, knots and Reidemeister moves, knot colourings.
Tricolourability without Diagrams
dbnvp Wednesday: The Gauss linking number combinatorially and as an integral.
dbnvp Friday: The Schroedinger equation and path integrals.
Friday Introduction (the quantum pendulum)
2 Jan 13 Homework Assignment 1.
dbnvp Monday: The Kauffman bracket and the Jones polynomial.
dbnvp Wednesday: Self-linking using swaddling.
dbnvp Friday: Euler-Lagrange problems, Gaussian integration, volumes of spheres.
3 Jan 20 Homework Assignment 2.
dbnvp Monday: The definition of finite-type and some examples.
dbnvp Wednesday: The self-linking number and framings.
dbnvp Friday: Integrating a polynomial times a Gaussian.
Class Photo.
4 Jan 27 Homework Assignment 3.
dbnvp Monday: Chord diagrams and weight systems.
dbnvp Wednesday: Swaddling maps and framings, general configuration space integrals.
dbnvp Friday: Some analysis of d^{-1}.
5 Feb 3 Homework Assignment 4.
dbnvp Monday: 4T, the Fundamental Theorem and universal finite type invariants.
The Fulton-MacPherson Compactification (PDF).
dbnvp Wednesday: The Fulton-MacPherson Compactification, Part I.
dbnvp Friday: More on pushforwards, d^{-1}, and d^\ast.
6 Feb 10 Homework Assignment 5.
dbnvp Monday: The bracket-rise theorem and the invariance principle.
dbnvp Wednesday: The Fulton-MacPherson Compactification, Part II.
dbnvp Friday: Gauge fixing, the beginning of Feynman diagrams.
R Feb 17 Reading Week.
7 Feb 24 dbnvp Monday: A review of Lie algebras.
dbnvp Wednesday: Graph cohomology and \Omega_{dR}^\ast(\Gamma).
dbnvp Friday: More on Feynman diagrams, beginning of gauge theory.
From Gaussian Integration to Feynman Diagrams (PDF).
8 Mar 3 Homework Assignment 6 (PDF)
dbnvp Monday: Lie algebraic weight systems.
dbnvp Wednesday: Graph cohomology and the construction of Z_0.
Graph Cohomology and Configuration Space Integrals (PDF)
dbnvp Friday: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
9 Mar 10 Homework Assignment 7 (PDF)
dbnvp Monday: The gl(N) weight system.
dbnvp Wednesday: The universal property, hidden faces.
dbnvp Friday: Insolubility of the quintic, naive expectations for CS perturbation theory.
10 Mar 17 Homework Assignment 8 (PDF)
dbnvp Monday: W_{\mathfrak g}\colon{\mathcal A}(\uparrow)\to{\mathcal U}({\mathfrak g}) and PBW.
dbnvp Wednesday: The anomaly.
dbnvp Friday: Faddeev-Popov, part I.
Gaussian Integration, Determinants, Feynman Diagrams (PDF).
11 Mar 24 Homework Assignment 9 (PDF)
dbnvp Monday: {\mathcal A} is a bi-algebra.
dbnvp Wednesday: Understanding and fixing the anomaly.
Friday: class cancelled.
12 Mar 31 Monday, Wednesday: class cancelled.
dbnvp Friday: A Monday class: back to expansions.
E Apr 7 dbnvp Monday: A Friday class on what we mostly didn't have time to do.
AKT-14-ClassPhoto.jpg
Add your name / see who's in!
Dror's Notebook
AKT-14-Splash.png
refresh
panel
Managed by dbnvp: Video frame grabs associated with new comments are queued and depending on load may take a very long time. Hit "refresh panel" until you are bored...

0:00:26 [add] Handout view 2: Wish to understand
0:00:55 [add] Handout view 3: R^n definitions
0:01:47 [edit] 31 seconds, it took.
0:02:36 [add] Handout view 4: Splitting
0:03:42 [add] Handout view 5: Integration
0:04:19 [add] Handout view 7: Perturned Gaussian Integration
0:04:22 [add] Handout view 6: The Fourier Transform
0:07:53 [add] Handout view 7: Perturned Gaussian Integration
0:09:00 [add] Handout view 8: Matching degrees
0:10:32 [add] Handout view 9: A sum over pairings
0:13:16 [add] Handout view 10: Marked Feynman Diagrams
0:15:44 [add] Handout view 11: Unmarked Feynman Diagrams
0:17:37 [add] Handout view 12: Wherefore Aut?
0:27:19 [add] Handout view 13: One more vertex
0:30:32 [add] Handout view 12: Wherefore Aut?
0:30:34 [add] Some diagrams.
0:30:45 [add] Handout view 2: Wish to understand
0:41:45 [edit] 'Showing '(A^g)^h = A^{(gh)}


\begin{align}
A^{(gh)} &= (gh)^{-1}A(gh) + (gh)^{-1}\mathrm{d}(gh) \\
&= (gh)^{-1}A(gh) + (gh)^{-1}\Big((\mathrm{d}g)h +  g(\mathrm{d}h)\Big)\\
&= (gh)^{-1}A(gh) + (gh)^{-1}(\mathrm{d}g)h +  (gh)^{-1}g(\mathrm{d}h)\\
&= h^{-1}(g^{-1}Ag)h + h^{-1}(g^{-1}\mathrm{d}g)h +  h^{-1}\mathrm{d}h\\
&= h^{-1}\Big(g^{-1}Ag + g^{-1}\mathrm{d}g\Big)h +  h^{-1}\mathrm{d}h\\
&= (A^g)^h.

\end{align}

The equality shows that the action is a group action.

0:42:56 [add] Very naive gauge theory.
0:48:41 [add] Very naive gauge theory (2).
0:48:42 [edit] Proof of the claim:

$$d(gs) = (dg)s+gds \implies (dg)s = d(gs)-gds$$

So

\begin{align}
D_{g^{-1} A g+g^{-1} d g} (s)\\
&=ds+(g^{-1} A g+g^{-1} d g)s\\
&= ds + g^{-1} A g s + g^{-1} (d g) s \\
&=ds + g^{-1} A g s + g^{-1} d(gs)-g^{-1}gds\\
&= g^{-1} A g s + g^{-1} d(gs)\\
&= g^{-1} D_A(gs)\\
&= (D_A)^g(s)\\
\end{align}