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Week of...
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Notes and Links
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| 1
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Jan 6
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About This Class (PDF).
 Monday: Course introduction, knots and Reidemeister moves, knot colourings.
Tricolourability without Diagrams
Wednesday: The Gauss linking number combinatorially and as an integral.
Friday: The Schroedinger equation and path integrals.
Friday Introduction (the quantum pendulum)
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| 2
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Jan 13
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Homework Assignment 1.
Monday: The Kauffman bracket and the Jones polynomial.
Wednesday: Self-linking using swaddling.
Friday: Euler-Lagrange problems, Gaussian integration, volumes of spheres.
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| 3
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Jan 20
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Homework Assignment 2.
Monday: The definition of finite-type and some examples.
Wednesday: The self-linking number and framings.
Friday: Integrating a polynomial times a Gaussian.
Class Photo.
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| 4
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Jan 27
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Homework Assignment 3.
Monday: Chord diagrams and weight systems.
Wednesday: Swaddling maps and framings, general configuration space integrals.
Friday: Some analysis of [math]\displaystyle{ d^{-1} }[/math].
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| 5
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Feb 3
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Homework Assignment 4.
Monday: 4T, the Fundamental Theorem and universal finite type invariants.
The Fulton-MacPherson Compactification (PDF).
Wednesday: The Fulton-MacPherson Compactification, Part I.
Friday: More on pushforwards, [math]\displaystyle{ d^{-1} }[/math], and [math]\displaystyle{ d^\ast }[/math].
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| 6
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Feb 10
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Homework Assignment 5.
Monday: The bracket-rise theorem and the invariance principle.
Wednesday: The Fulton-MacPherson Compactification, Part II.
Friday: Gauge fixing, the beginning of Feynman diagrams.
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| R
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Feb 17
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Reading Week.
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| 7
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Feb 24
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Monday: A review of Lie algebras.
Wednesday: Graph cohomology and [math]\displaystyle{ \Omega_{dR}^\ast(\Gamma) }[/math].
Friday: More on Feynman diagrams, beginning of gauge theory.
From Gaussian Integration to Feynman Diagrams (PDF).
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| 8
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Mar 3
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Homework Assignment 6 (PDF)
Monday: Lie algebraic weight systems.
Wednesday: Graph cohomology and the construction of [math]\displaystyle{ Z_0 }[/math].
Graph Cohomology and Configuration Space Integrals (PDF)
Friday: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
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| 9
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Mar 10
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Homework Assignment 7 (PDF)
Monday: The [math]\displaystyle{ gl(N) }[/math] weight system.
Wednesday: The universal property, hidden faces.
Friday: Insolubility of the quintic, naive expectations for CS perturbation theory.
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| 10
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Mar 17
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Homework Assignment 8 (PDF)
Monday: [math]\displaystyle{ W_{\mathfrak g}\colon{\mathcal A}(\uparrow)\to{\mathcal U}({\mathfrak g}) }[/math] and PBW.
Wednesday: The anomaly.
Friday: Faddeev-Popov, part I.
Gaussian Integration, Determinants, Feynman Diagrams (PDF).
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| 11
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Mar 24
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Homework Assignment 9 (PDF)
Monday: [math]\displaystyle{ {\mathcal A} }[/math] is a bi-algebra.
Wednesday: Understanding and fixing the anomaly.
Friday: class cancelled.
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| 12
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Mar 31
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Monday, Wednesday: class cancelled.
Friday: A Monday class: back to expansions.
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| E
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Apr 7
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Monday: A Friday class on what we mostly didn't have time to do.
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Add your name / see who's in!
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| Dror's Notebook
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Here are some thoughts on how to define tricolourability without choosing a diagram.
Another place in which arcs of a diagram come up naturally is in the Wirtinger presentation for the fundamental group of the knot complement. Here is how the presentation is defined: each arc of the knot diagram corresponds with a generator, and each crossing corresponds with a relation between the generators of the incident arcs of the form [math]\displaystyle{ xy=yz }[/math] where [math]\displaystyle{ y }[/math] is the generator corresponding with the overcrossing.
Now if [math]\displaystyle{ \langle S\mid \text{relations}\rangle }[/math] is a Wirtinger presentation for a knot diagram, it's natural to think of a tricolouring as a map [math]\displaystyle{ \phi: S \rightarrow \{R,G,B\} }[/math]. We'd like to try to extend this to a group homomorphism [math]\displaystyle{ \phi:\langle S \mid \text{relations}\rangle \rightarrow \langle R,G,B \mid \text{relations} \rangle }[/math]. This works if target group has the relation [math]\displaystyle{ RG=GB }[/math] along with all other relations obtained by permuting [math]\displaystyle{ R,G,B }[/math]. These relations fix the target group as [math]\displaystyle{ D_{2\cdot 3} }[/math].
Thus, we've associated with each tricolouring a homomorphism from the fundamental group of the knot complement to [math]\displaystyle{ D_{2\cdot 3} }[/math]. Not every such homomorphism gives a tricolouring; for example, take the trivial homomorphism. I believe that the following is a sufficient condition for a homomorphism [math]\displaystyle{ \phi }[/math] to give a tricolouring: for every element [math]\displaystyle{ x }[/math] in [math]\displaystyle{ \pi_1(\R^3\setminus K) }[/math] whose representative as a loop in [math]\displaystyle{ \mathbb{R}^3\setminus K }[/math] has odd linking number with [math]\displaystyle{ K }[/math], [math]\displaystyle{ \phi(x) }[/math] is an order 2 element in [math]\displaystyle{ D_{2\cdot 3} }[/math]. Hence, we can define tricolourings as certain kinds of homomorphisms from [math]\displaystyle{ \pi_1(\R^3\setminus K) }[/math] to [math]\displaystyle{ D_{2\cdot 3} }[/math] without having to choose a diagram.
