|© | Dror Bar-Natan: Classes: 2013-14: AKT:||< >|
|width: 400 720||ogg/AKT-140303_400.ogg||orig/AKT-140303.MOD|
|Videography by Iva Halacheva||troubleshooting|
|#||Week of...||Notes and Links|
|1||Jan 6|| (PDF). |
: Course introduction, knots and Reidemeister moves, knot colourings.
Tricolourability without Diagrams
: The Gauss linking number combinatorially and as an integral.
: The Schroedinger equation and path integrals.
|2||Jan 13||Homework Assignment 1. |
: The Kauffman bracket and the Jones polynomial.
: Self-linking using swaddling.
: Euler-Lagrange problems, Gaussian integration, volumes of spheres.
|3||Jan 20||Homework Assignment 2. |
: The definition of finite-type and some examples.
: The self-linking number and framings.
: Integrating a polynomial times a Gaussian.
|4||Jan 27||Homework Assignment 3. |
: Chord diagrams and weight systems.
: Swaddling maps and framings, general configuration space integrals.
: Some analysis of .
|5||Feb 3||Homework Assignment 4. |
: 4T, the Fundamental Theorem and universal finite type invariants.
: The Fulton-MacPherson Compactification, Part I.
: More on pushforwards, , and .
|6||Feb 10||Homework Assignment 5. |
: The bracket-rise theorem and the invariance principle.
: The Fulton-MacPherson Compactification, Part II.
: Gauge fixing, the beginning of Feynman diagrams.
|R||Feb 17||Reading Week.|
|7||Feb 24||: A review of Lie algebras. |
: Graph cohomology and .
: More on Feynman diagrams, beginning of gauge theory.
|8||Mar 3|| (PDF) |
: Lie algebraic weight systems.
: Graph cohomology and the construction of .
: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
|9||Mar 10|| (PDF) |
: The weight system.
: The universal property, hidden faces.
: Insolubility of the quintic, naive expectations for CS perturbation theory.
|10||Mar 17|| (PDF) |
: and PBW.
: The anomaly.
: Faddeev-Popov, part I.
|11||Mar 24|| (PDF) |
: is a bi-algebra.
: Understanding and fixing the anomaly.
Friday: class cancelled.
|12||Mar 31||Monday, Wednesday: class cancelled. |
: A Monday class: back to expansions.
|E||Apr 7||: A Friday class on what we mostly didn't have time to do.|
Add your name / see who's in!
For the second antisymmetry relation (i.e., the one with last two indices exchanged), we use the fact the metric is invariant. A metric for any finite-dimensional Lie algebra is invariant
Then, with the fact that the metric is symmetric, we write this relation in terms of structural coeffiecnt .
Thus, this shows the antisymmetry relations of . Then, . Similarly, it shows . Thus, the cyclic relation follows.