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AKT-140124 Video

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Notes on AKT-140124:    [edit, refresh]

Integrating a polynomial times a Gaussian.


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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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0:05:22 [add] Notices and today's goal.
0:14:26 [add] Gaussian integration of arbitrary quadratics.
0:23:30 [edit] Let \Lambda be a symmetric, positive definite, non-singular square matrix. Then we have the following:

 \langle x - \Lambda^{-1} y, \Lambda(x - \Lambda^{-1}y)\rangle =  \langle x,\Lambda x \rangle -  \langle x, y  \rangle -  \langle \Lambda^{-1}y, \Lambda x  \rangle +  \langle \Lambda^{-1}y,y  \rangle .

We have  \langle \Lambda^{-1}y, \Lambda x \rangle =  \langle x,y\rangle and  \langle \Lambda^{-1}y,y \rangle =  \langle y,\Lambda^{-1}y  \rangle since \Lambda is symmetric.

From the above, we see that -\frac12  \langle x - \Lambda^{-1} y, \Lambda(x - \Lambda^{-1}y) \rangle  + \frac12 \langle y,\Lambda^{-1}y  \rangle = -\frac12 \langle x,\Lambda x \rangle +  \langle x, y  \rangle

0:27:55 [add] Moments of Gaussian integrals.
0:37:36 [add] The second moments.
0:38:21 [edit] It is not clear why Z^{-1} times the integral equals \varphi_1\Lambda^{-1}\varphi_2 and why it is the same as the formula obtained previously
0:47:24 [add] The second moments (2).