May 10 and 11, 2007. Overview of Khovanov Homology; handouts: KhovanovOverview-1.pdf, KhovanovOverview-2.pdf.
Title. The Virtues of Being an Isomorphism.
Abstract. I'm over forty, I'm a full professor, and it's time that I come out of the closet. I don't understand quantum groups and I never did. I wish I could tell you in my talk about one of the major stumbling blocks I have encountered - I don't understand the amazing Etingof-Kazhdan work on quantization of Lie bialgebras. But hey, I can't tell you about what I don't understand! So instead, I will tell you about how I hope to understand the Etingof-Kazhdan work, one day, as an isomorphism between a topologically defined space and a combinatorially defined one. The former would be the unipotent completion of a certain algebra of virtually-knotted (trivalent?) graphs. The latter would be the associated graded space of the former.
I'll start and spend a good chunk of my time with an old but not well known analogy, telling you why a Drinfel'd associator, the embodiment of the spirits of all quasi-Hopf algebras, is best viewed as an isomorphism between the unipotent completion of the algebra of honestly-knotted trivalent graphs and its associated graded space, a certain combinatorially-defined algebra of chord diagrams. A few words will follow, about the relationship between diagrammatic Lie bialgebras and finite type invariants of virtual knots.
References. Drinfel'd's "Quasi-Hopf algebras", Etingof-Kazhdan's arXiv:q-alg/9506005 (and the rest of the series), Murakami-Ohtsuki's "Topological Quantum Field Theory for the Universal Quantum Invariant", Polyak's "On the Algebra of Arrow Diagrams, Goussarov-Polyak-Viro's arXiv:math.GT/9810073, Haviv's arXiv:math.QA/0211031, DBN's On Associators and the Grothendieck-Teichmuller Group I.
"God created the knots, all else in topology is the work of mortals."
Leopold Kronecker (modified)
See also. Gallery: Places: Kyoto, May 2007.