© | << < ? > >> | Dror Bar-Natan:
( Classes: 2012-13 / Talks ):
QGM Master Class, Centre for Quantum Geometry of Moduli Spaces,
Aarhus, Denmark, May 27 - June 7, 2013

(u, v, and w knots) x (topology, combinatorics, low algebra, and high algebra)

Jumps.     plan     reality     pensieve

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 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.

Optimistic Rough Tentative Plan.

Mon May 27, pre-class. The Kauffman bracket and the Jones polynomial (with computations!), the Alexander polynomial, Khovanov homology.
Evening. "Social Networking Dinner".
Tue May 28, day 1. Overall introduction: (uvw)x(TCLH). Then the u-column to low algebra. Wed May 29, day 2. Micro-introduction: Knot theory as an excuse and it's all about Taylor. Then KZ, Kontsevich, parenthesized tangles, associators. Thu May 30, day 3. Second introduction: algebraic knot theory. Then KTGs to the pentagon and the hexagon. Fri May 31, day 4. Third introduction: Stonehenge. Then perturbative Chern-Simons theory.
Mon June 3, day 5. Fourth introduction: Dalvit on 4D knots. Then w-tangles to the Alexander polynomial. Tue June 4, day 6. Fifth introduction: Dalvit on braids, then all about uvw-braids. Perhaps something about surface braids? Wed June 5, day 7. Sixth introduction: meta and beta. Then the full Vietnam story. Thu June 6, day 8. Trivalent vertices, Alekseev-Torossian, Kashiwara-Vergne, a talk about flat knots by Karene Chu.
Evening. "Masterclass Dinner".
Fri June 7, day 9. Alekseev-Torossian-Enriquez, the v-story in as much as I understand it.

The Reality.

Mon May 27, pre-class. The Kauffman bracket and the Jones polynomial (KauffmanBracket-0.pdf, KauffmanBracket.pdf). Penultimate Alexander (pA.html, pAHandout.pdf, Sandbjerg-0810/). Khovanov homology (Hamburg-1208).
Videos: 1, 2, 3.
Tue May 28, day 1. Overall introduction: (uvw)x(TCLH). Then the u-column to low algebra.
Videos: 1, 2, 3, 4.
Wed May 29, day 2. From the Kontsevich integral to associators, but only a tiny bit about the latter (Day2Handout.pdf, Day2.html, KZ.html).
Videos: 1, 2, 3, 4.
Thu May 30, day 3. Second introduction: algebraic knot theory (Day3Handout.pdf, Day3.html). Then all about the theory of associators, but briefly.
Videos: 1, 2, 3, 4.
Fri May 31, day 4. Third introduction: Stonehenge and perturbative Chern-Simons theory, then KTGs to the pentagon and the hexagon (Day4Handout.pdf, Day4.html).
Videos: 1, 2, 3, 4.
Mon June 3, day 5. Dalvit on visualizing 2D in 4D (more), then most of the easier parts of the w-story.
Videos: 1, 2, 3, 4.
Tue June 4, day 6. First the meta/beta story (beta.pdf, beta.html, beta.zip), then towards its relationship with w (twbh.pdf, twbh.html).
Videos: 1, 2, 3, 4.
Wed June 5, day 7. A sketch of the completion of the KBH/meta/beta story (Aarhus_Day_7_Handout.pdf, Day7.html), then braids and a talk by Humbert on surface braids/knots/tangles.
Videos: 1, 2, 3, 4.
Late afternoon: Dalvit movies.
Thu June 6, day 8. Trivalent vertices, Alekseev-Torossian, Kashiwara-Vergne, a talk about flat knots by Karene Chu. (Day8Handout.pdf, Day8.html)
Videos: 1, 2, 3, 4.
Evening: Masterclass Dinner.
Fri June 7, day 9. v-Day: Topology, Combinatorics (18C.pdf, 18C.html), Low Algebra, and High Algebra (Day9Handout.pdf, Day9.html)
Videos: 1, 2, 3.