(u,v,w knots)x(topology, combinatorics, low and high algebra)' by Dror Bar-Natan (University of Toronto)
2012.09.18 |
| Date | Mon 27 May — Fri 07 Jun |
| Time | 09:30 — 16:00 |
| Location | QGM, Aarhus University |
Online registration - Deadline: 28 April 2013
Speaker
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
Abstract
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:
www.math.toronto.edu/~drorbn/Talks/Aarhus-1305/
Monday 27 May, PRE-CLASS
The Kauffman bracket and the Jones polynomial (with computations!), the Alexander polynomial, Khovanov homology
Tuesday 28 May, MASTERCLASS STARTS
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
Included in the event
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:
| Participants | ||
Aceto, Paolo (University of Florence) |
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Help planning the trip, please see:
Video recordings (will be available after the masterclass)
