Masterclass May 2013

(u,v,w knots)x(topology, combinatorics, low and high algebra)' by Dror Bar-Natan (University of Toronto)

2012.09.18 | Christine Dilling

Date Mon 27 May Fri 07 Jun
Time 09:30    16:00
Location QGM, Aarhus University

Online registration - Deadline: 28 April 2013


Dror Bar-Natan, University of Toronto

Short Abstract. 

When appropriately read using the language of "finite type invariants", "Jacobi diagrams", and "expansions", the combinatorics underlying various knot theories turns out to be the same as the combinatorics underlying various classes of Lie algebras, thus establishing a bijection between certain hard problems in knot theory and certain hard problems in Lie theory, enriching both subjects


My subject is a Cartesian product. It runs in three parallel columns - the u column, for usual knots, the v column, for virtual knots, and the w column, for welded, or weakly virtual, or warmup knots. Each class of knots has a topological meaning and a "finite type" theory, which leads to some combinatorics, somewhat different combinatorics in each case. In each column the resulting combinatorics ends up describing tensors within a different "low algebra" universe - the universe of metrized Lie algebras for u, the richer universe of Lie bialgebras for v, and for w, the wider and therefore less refined universe of general Lie algebras. In each column there is a "fundamental theorem" to be proven (or conjectured), and the means, in each column, is a different piece of "high algebra": associators and quasi-Hopf algebras in one, deformation quantization à la Etingof and Kazhdan in the second, and in the third, the Kashiwara-Vergne theory of convolutions on Lie groups. Finally, u maps to v and v maps to w at topology level, and the relationship persists and deepens the further down the columns one goes.

The 12 boxes in this product each deserves its own set of talks, and the few that are not yet fully understood deserve a few further years of research. Thus my lectures will only give the flavour of a few of the boxes that I understand, and only hint at my expectations for the contents the (2,4) box, the one I understand the least and the one I wish to understand the most.

Closer to the Masterclass futher information and material will be availble at:

Poster (PDF)


Optimistic Rough Tentative Plan

Monday 27 May, PRE-CLASS

The Kauffman bracket and the Jones polynomial (with computations!), the Alexander polynomial, Khovanov homology


Overall introduction: (uvw)x(TCLH). Then the u-column to low algebra

Wednesday 29 May

Micro-introduction: Knot theory as an excuse and it's all about Taylor. Then KZ, Kontsevich, parenthesized tangles, associators

Thursday 30 May

Second introduction: algebraic knot theory. Then KTGs to the pentagon and the hexagon

Friday 31 May

Third introduction: Stonehenge. Then perturbative Chern-Simons theory

Monday 3 June

Fourth introduction: Dalvit on 4D knots. Then w-tangles to the Alexander polynomial

Tuesday 4 June

Fifth introduction: Dalvit on braids, then all about uvw-braids

Wednesday 5 June

Sixth introduction: meta and beta. Then the full Vietnam story

Thursday 6 June

Trivalent vertices, Alekseev-Torossian, Kashiwara-Vergne, and Alekseev-Torossian-Enriquez

Friday 7 June

The v-story in as much as I understand it. (A talk by Karene Chu?)


The daily schedule

10:00 - 10:45 Lecture
11:15 - 12:00 Lecture
12:00 - 14:00 Lunch
14:00 - 14:45 Lecture
15:15 - 16:00 Lecture

Social programme

  • Monday 27/5 at 6pm: Networking dinner. Free of charge.
  • Thursday 6/6 at 6pm: Special dinner at a restaurant in town. You must be signed up for this.

Included in the event

  • Morning and afternoon coffee/tea
  • Lunches on weekdays
  • Social dinner 'first night'
  • Social dinner 'last night'

In special cases, QGM is able to offer limited financial support to junior researchers (PhD students and postdocs). Only applications received before 1 April 2013 will be considered for financial support.typo3/alt_doc.php?edit[tt_content][10667]=edit&columnsOnly=bodytext, rte_enabled&noView=0&returnUrl=/cal/special/2013/mc_uvw/

The list will be regularly updated:


Aceto, Paolo (University of Florence)
Agerholm, Signe (AU)
Andersen, Jørgen Ellegaard (AU)
Andersen, Troels Bak (AU)
Arkhipov, Sergey (AU)
Bar-Natan, Dror (Toronto) - SPEAKER
Calderon, Eran (Technion, Israel)
Chu, Karene (University of Toronto/Fields Institute)
Celoria, Daniele (Università di Firenze)
Dalvit, Ester (Trento University)
Egsgaard, Jens Kristian (AU)
Fuglede Jørgensen, Søren (AU)
Gammelgaard, Niels Leth (AU)
Halacheva, Iva (University of Toronto)
Hansen, Torben (AU)
Humbert, Philippe (University of Strasbourg)
Kanstrup. Tina, (AU)
Kratmann Nissen, Jens-Jakob
Lai, Alan (Caltech)
Lundberg, Emilia (Uppsala University)
Mack, Austin (National Research University - Higher School of Economics)
Mclellan, Brendan (AU)
Mashanova-Golikova, Inna (University of Michigan)
Masulli, Paolo (AU)
Mizusawa, Atsuhiko (Waseda University)
Petersen, Mikkel Stouby (AU)
Selmani, Sam (McGill University)
Sørensen, Mie (AU)
Vo, Huan (UBC, UofT)



Help planning the trip, please see:

Visitor information

Video recordings (will be available after the masterclass)

Staff, Public/media, Master Class