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AKT-140305 Video

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Graph cohomology and the construction of Z_0.


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# 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.
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Managed by dbnvp: Tip: blackboard shots are taken when the discussion of their content has just ended. To see the beginning of the discussion on a certain blackboard, roll the video to the time of the preceeding blackboard.

0:00:25 [add] Handout view 2: Wishful Thinking
0:02:07 [add] Handout view 3: Theorem
0:03:35 [add] Handout view 4: D
0:04:59 [add] Handout view 5: Details of D
0:05:25 [add] Handout view 6: d
0:05:38 [add] Handout view 7: Details of d
0:05:43 [add] Handout view 5: Details of D
0:06:07 [add] Handout view 4: D
0:06:15 [add] Handout view 5: Details of D
0:07:08 [add] Handout view 6: d
0:08:04 [add] Handout view 7: Details of d
0:12:29 [add] Handout view 8: I
0:13:57 [add] Handout view 3: Theorem
0:15:41 [add] Handout view 7: Details of d
0:15:52 [add] Handout view 9: Signs
0:16:12 [add] Handout view 7: Details of d
0:17:12 [add] Handout view 10: Chords
0:17:20 [add] Handout view 6: d
0:23:36 [add] Handout view 2: Wishful Thinking
0:23:38 [add] Handout view 3: Theorem
0:23:41 [edit] The map I restricted to homological degree m=0.
0:23:51 [add] Handout support.
0:24:57 [edit] H^0(\Gamma) consists of knot invariants.
0:27:29 [edit] Defining Z_0
0:29:03 [edit] The dual of H^0(\mathcal{D}_n).
0:34:01 [edit] A naive way to define d^*.
0:40:54 [edit] The correct definition of d^*.
0:41:17 [add] $Z_0$.
0:41:57 [edit] Showing that (\ker f)^* = V^*/\mathrm{im}f^* for a linear map f : V \rightarrow W, V\; \mathrm{and}\; W are vector spaces. (Leo algknt and Jesse had a discussion.)

Let \iota : \ker f \rightarrow V be the inclusion of \ker f into V, then V^*/\ker \iota^* \cong (\ker f)^*.

We show that \ker \iota^* = \mathrm{im} f^*.


\begin{align} 
\ker \iota^* &= \{\phi \in V^* \;\;|\;\; \iota^*(\phi) = 0\}\\
&= \{\phi \in V^* \;\;|\;\; \phi\circ\iota = 0\}\\
&= \{\phi \in V^* \;\;|\;\; \phi|_{\ker f} = 0, \; \phi = f^*(\alpha), \; \alpha \in W^* \}\\
&= \{\phi \in V^* \;\;|\;\; \phi|_{\ker f} = 0, \; \phi = \alpha\circ f, \; \alpha \in W^* \}\\
&= \{f^*(\alpha) \in V^* \;\;|\;\; (\alpha\circ f)|_{\ker f} = 0, \; \alpha \in W^* \}\\
&= \mathrm{im} f^*.

\end{align}

0:43:06 [add] The adjoint of the graph differential.
0:45:37 [add] The adjoint of the graph differential (2).
0:47:47 [edit] The arising of \mathcal{A}_n.
0:49:28 [add] The adjoint of the graph differential (3).
0:50:03 [add] ${\mathcal A}$ arises!
1:04:09 [add] Chern-Simons and curvatures.