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# AKT-140115 Video

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# Week of... Notes and Links
: 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.
Friday Introduction (the quantum pendulum)
2 Jan 13 Homework Assignment 1.
: The Kauffman bracket and the Jones polynomial.
: 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.
Class Photo.
4 Jan 27 Homework Assignment 3.
: Chord diagrams and weight systems.
: Swaddling maps and framings, general configuration space integrals.
: Some analysis of $d^{-1}$.
5 Feb 3 Homework Assignment 4.
: 4T, the Fundamental Theorem and universal finite type invariants.
The Fulton-MacPherson Compactification (PDF).
: The Fulton-MacPherson Compactification, Part I.
: More on pushforwards, $d^{-1}$, and $d^\ast$.
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.
7 Feb 24 : A review of Lie algebras.
: Graph cohomology and $\Omega_{dR}^\ast(\Gamma)$.
: More on Feynman diagrams, beginning of gauge theory.
From Gaussian Integration to Feynman Diagrams (PDF).
8 Mar 3 Homework Assignment 6 (PDF)
: Lie algebraic weight systems.
: Graph cohomology and the construction of $Z_0$.
Graph Cohomology and Configuration Space Integrals (PDF)
: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
9 Mar 10 Homework Assignment 7 (PDF)
: The $gl(N)$ weight system.
: The universal property, hidden faces.
: Insolubility of the quintic, naive expectations for CS perturbation theory.
10 Mar 17 Homework Assignment 8 (PDF)
: $W_{\mathfrak g}\colon{\mathcal A}(\uparrow)\to{\mathcal U}({\mathfrak g})$ and PBW.
: The anomaly.
Gaussian Integration, Determinants, Feynman Diagrams (PDF).
11 Mar 24 Homework Assignment 9 (PDF)
: ${\mathcal A}$ 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.

Dror's Notebook
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0:06:43  I forgot to say that the two manifolds must be "closed" meaning compact and having no boundary.
0:24:32  I am wandering what diagonal means here
0:30:33  Configuration space Given a topological space $X$, the $n$th ordered configuration space of $X$ denoted by $\mathrm{Conf}_n(X)$ is the set of $n$-tuples of pairwise distinct points in $X$, that is $\mathrm{Conf}_n(X):= \prod^n X \setminus \{(x_1, \ldots, x_n) : x_i = x_j \;\mathrm{for} \;i\ne j\}$.

In physics, parameters are used to define the configuration of a system and the vector space defined by these parameters is the configuration space of the system. It is used to describe the state of a whole system as a single point in a higher-dimensional space.

Examples of Configuration space

1. The configuration space of a particle in $\mathbb{R}^3$ is $\mathbb{R}^3$. For $n$ particles in $\mathbb{R}^3$, it is $\mathbb{R}^{3n}$

2. For a rigid body in $\mathbb{R}^3$, the configuration space is $\mathbb{R}^3 \times SO(3)$. Generally, it is $\mathbb{R}^n \times SO(n)$, where $SO(n)$ is the special orthogonal group.

3. The torus with its diagonal removed, $S^1 \times \mathbb{R}$, is the configuration space of two points on $S^1$. This is $C_2(S^1)$

Reference: [1]

0:36:29  The Gauss map.
0:40:30  The sphere $S^2$.
0:52:35 [add] Properties of $sl_1$.