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Talks in Kyoto, May 2007

Link Homology and Categorification
Department of Mathematics and RIMS, Kyoto University

Talks at the the Department of Mathematics

May 10 and 11, 2007. Overview of Khovanov Homology; handouts: KhovanovOverview-1.pdf, KhovanovOverview-2.pdf.

Talk at RIMS (May 18, 2007)

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.

Handout. DreamMap.pdf.

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.