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.
Appetizer Handout. TheGrandScheme.html, TheGrandScheme.png.
Main Handout. 3x4.html, 3x4.pdf, 3x4.png (source files: 3x4.zip).
Day 3 Handout. Day3.png.
Day 4 Handout. Day4.html, Day4.png.
My scratch work. Pensieve: PSU Talk and Pensieve: 1st Kansas Talk.
Abstract. I'll start my talk by describing the Kashiwara-Vergne Conjecture (1978, Proven Alekseev-Meinrenken, 2006), which states that for any finite dimensional Lie group, certain convolutions on the group are equal to certain convolutions on its Lie algebra, and I'll end my talk in knot theory. Going backward from the hard to the easy, here's the whole thing in one paragraph:
Convolutions are integrals so we need to prove an equality of integrals. This we do (roughly) by finding a Measure Preserving Transformation (MPT) which carries one integrand to the other. This has to work for any Lie group, so the MPT better be given as a universal formula. At this generality all we have to work with is the Lie bracket, and bracket-made formulas can be pictured as certain trivalent diagrams (which in themselves are subject to the Jacobi, or "IHX" relation). Thus we are really seeking a certain big sum F of trivalent diagrams, which, in order to describe our desired MPT, must satisfy certain equations mod IHX. Our space Aw of trivalent diagrams turns out to be the "associated graded" space of the space Kw of ribbon 2-knots in 4-space, and the equations F needs to solve are the equations one needs to solve to get a well-behaved "expansion" Z:Kw-->Aw.
Thus the Kashiwara-Vergne Conjecture, itself a witness to the famed "orbit method", is more or less the same as a natural problem in knot theory, and since knot theory is related to associators and to quantum groups, so is the Kashiwara-Vergne Conjecture. Cool, eh?
Handout. KV.html, KV.pdf, KV.png (source files: KV.zip).
A paper in progress. Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne.
My scratch work. Pensieve/2009-04 (mostly under "Glasgow") and Pensieve/2009-05.