May 10 and 11, 2007. Overview of Khovanov Homology; handouts: KhovanovOverview1.pdf, KhovanovOverview2.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 EtingofKazhdan 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 EtingofKazhdan 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 virtuallyknotted (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 quasiHopf algebras, is best viewed as an isomorphism between the unipotent completion of the algebra of honestlyknotted trivalent graphs and its associated graded space, a certain combinatoriallydefined algebra of chord diagrams. A few words will follow, about the relationship between diagrammatic Lie bialgebras and finite type invariants of virtual knots.
Handout. DreamMap.pdf.
References. Drinfel'd's "QuasiHopf algebras", EtingofKazhdan's arXiv:qalg/9506005 (and the rest of the series), MurakamiOhtsuki's "Topological Quantum Field Theory for the Universal Quantum Invariant", Polyak's "On the Algebra of Arrow Diagrams, GoussarovPolyakViro's arXiv:math.GT/9810073, Haviv's arXiv:math.QA/0211031, DBN's On Associators and the GrothendieckTeichmuller 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.