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 ==Algebraic Knot Theory==   ==Algebraic Knot Theory== 
 ===Department of Mathematics, University of Toronto, Spring 2014===   ===Department of Mathematics, University of Toronto, Spring 2014=== 
   
−  '''BA6180 at MWF11'''.  +  '''Agenda.''' Three courses on just one theorem: With <math>\mathcal K</math> the set of knots and <math>\mathcal A</math> something naturally associated to knots and quite related to Lie algebras, ''there exists an expansion <math>Z\colon\mathcal K\to\mathcal A</math>.'' 
   
−  I will be giving an ''Algebraic Knot Theory'' class at the University of Toronto in the spring semester of 2014, and this will be its home page.
 +  '''Instructor:''' {{Home LinkDror BarNatan}}, drorbn@math.toronto.edu, Bahen 6178, 4169465438. Office hours: {{Office Hours}}. 
   
−  '''Agenda.''' Understand "(u, v, and w knots) x (topology, combinatorics, low algebra, and high algebra)". Understand the promise and the difficulty of the notyetexistent "Algebraic Knot Theory".  +  '''Classes.''' Mondays, Wednesdays, and Fridays at 10:1011:00; Mondays and Fridays at Bahen 6180 but Wednesdays at Huron 1018. There will also be a "HW meeting", covering no new material, on Fridays at 6:10PM at Bahen 6180. 
   
−  '''Description.''' There are many types of "knots", and many things to do with them. Specifically, we will talk about three mastertypes of knots  "u" knots which are just the usual knots we are used to seeing in 3space, "w" knots which are 2dimensional knots in 4space, and "v" knots which are "virtual" knots, algebraic creatures that are not specifically embedded anywhere. Each of these mastertypes comes in several shades  there are plain uvwknots, and uvwtangles, and uvwbraids, and uvwknottedgraphs, and there is a rich world of operations that can be applied to these, and a rich world of properties and questions to ask. And then there is the projectivization machine, which converts all these to combinatorial objects, and then algebraic (of two classes, low and high). The "Fundamental Problem" is always to construct a good map (technically, a "homomorphic expansion") from topology to combinatorics/algebra, and in the cases we understand, the solution of the Fundamental Problem involves matters such as the Knizhnik–Zamolodchikov connection, topological quantum field theory and Feynman diagrams and configuration space integrals, Drinfel'd associators and the pentagons and hexagons of category theory, and deep Lie theory. Everything we will mention, but not everything we will cover as deeply as I'd like to.
 +  {{Pensieve linkClasses/141350AKT/About.pdfAbout This Class}} (PDF). 
   
−  '''References.'''
 
−  * ''Introduction to Vassiliev Knot Invariants,'' by S. Chmutov, S. Duzhin, and J. Mostovoy, Cambridge University Press, Cambridge UK, 2012.
 
−  * My own papers and talks and classes as can be found at http://www.math.toronto.edu/~drorbn/.
 
   
−  '''Prerequisites.''' A high level of comfort with vector spaces and algebras and things you can do with them  quotients, tensor products, duals, etc. Basic topology  fundamental groups and van Kampen and rarely a bit of homology. Basic differential geometry  especially differential forms and Stokes' formula. A basic knowledge of Lie groups and algebras and their representations and an appreciation of their value. You will manage with some of that missing, but the more you will be missing the more lost you will be at times.
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Latest revision as of 21:15, 13 January 2014
#

Week of...

Notes and Links

1

Jan 6

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)

2

Jan 13

Homework Assignment 1. Monday: The Kauffman bracket and the Jones polynomial. Wednesday: Selflinking using swaddling. Friday: EulerLagrange problems, Gaussian integration, volumes of spheres.

3

Jan 20

Homework Assignment 2. Monday: The definition of finitetype and some examples. Wednesday: The selflinking number and framings. Friday: Integrating a polynomial times a Gaussian. Class Photo.

4

Jan 27

Homework Assignment 3. Monday: Chord diagrams and weight systems. Wednesday: Swaddling maps and framings, general configuration space integrals. Friday: Some analysis of .

5

Feb 3

Homework Assignment 4. Monday: 4T, the Fundamental Theorem and universal finite type invariants. The FultonMacPherson Compactification (PDF). Wednesday: The FultonMacPherson Compactification, Part I. Friday: More on pushforwards, , and .

6

Feb 10

Homework Assignment 5. Monday: The bracketrise theorem and the invariance principle. Wednesday: The FultonMacPherson Compactification, Part II. Friday: Gauge fixing, the beginning of Feynman diagrams.

R

Feb 17

Reading Week.

7

Feb 24

Monday: A review of Lie algebras. Wednesday: Graph cohomology and . Friday: More on Feynman diagrams, beginning of gauge theory. From Gaussian Integration to Feynman Diagrams (PDF).

8

Mar 3

Homework Assignment 6 (PDF) Monday: Lie algebraic weight systems. Wednesday: Graph cohomology and the construction of . Graph Cohomology and Configuration Space Integrals (PDF) Friday: Gauge invariance, ChernSimons, holonomies. Mar 9 is the last day to drop this class.

9

Mar 10

Homework Assignment 7 (PDF) Monday: The weight system. Wednesday: The universal property, hidden faces. Friday: Insolubility of the quintic, naive expectations for CS perturbation theory.

10

Mar 17

Homework Assignment 8 (PDF) Monday: and PBW. Wednesday: The anomaly. Friday: FaddeevPopov, part I. Gaussian Integration, Determinants, Feynman Diagrams (PDF).

11

Mar 24

Homework Assignment 9 (PDF) Monday: is a bialgebra. Wednesday: Understanding and fixing the anomaly. Friday: class cancelled.

12

Mar 31

Monday, Wednesday: class cancelled. Friday: A Monday class: back to expansions.

E

Apr 7

Monday: A Friday class on what we mostly didn't have time to do.

Add your name / see who's in!

Dror's Notebook



Algebraic Knot Theory
Department of Mathematics, University of Toronto, Spring 2014
Agenda. Three courses on just one theorem: With the set of knots and something naturally associated to knots and quite related to Lie algebras, there exists an expansion .
Instructor: Dror BarNatan, drorbn@math.toronto.edu, Bahen 6178, 4169465438. Office hours: by appointment.
Classes. Mondays, Wednesdays, and Fridays at 10:1011:00; Mondays and Fridays at Bahen 6180 but Wednesdays at Huron 1018. There will also be a "HW meeting", covering no new material, on Fridays at 6:10PM at Bahen 6180.
About This Class (PDF).