##### Department of Mathematics,

University of California San Diego

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### Math 292 - Topology Seminar

## Justin Roberts

#### UCSD

## Introductory Meeting

##### Abstract:

\footnotesize This term's seminar will be on ``Khovanov homology and categorification''.\\ If one wants to show that some quantity takes only non-negative integral values, one of the best ways to do so is to show that it is ``secretly'' the dimension of some vector space. ``Categorification'' is the philosophy that one should look for interesting examples of this kind of thing throughout mathematics, hoping to find that for example: \\ \begin{enumerate} \item Non-negative integers are secretly dimensions of vector spaces \item Integers are secretly virtual dimensions of formal differences of vector spaces (or superdimensions of supervector spaces) \item Integer Laurent polynomials are secretly graded dimensions of Z-graded (super)vector spaces; \item Abelian groups are secretly Grothendieck groups of additive categories \end{enumerate} The Euler characteristic, for example, is an integer-valued invariant with wonderful properties and applications. We can ``categorify'' it by viewing it as the dimension (in the second sense above) of a more powerful vector-space valued invariant, homology. Why is homology more powerful? Because it is \textit{functorial}, capturing information about maps between spaces which the Euler characteristic can't. It's this appearance of functoriality that gives rise to the name ``categorification''.\\ In 1999 Mikhail Khovanov showed that the Jones polynomial for knots in 3-space can be categorified (in the third sense above). He showed how to associate to any knot a bunch of homology groups which turn out to be strictly stronger, as topological invariants, than the Jones polynomial; moreover, they are functorial with respect to surface cobordisms in 4-space between knots! The invention of Khovanov homology has not only had beautiful applications in topology (Rasmussen's proof of Milnor's conjectures about the unknotting numbers of torus knots) but also inspired a lot of work by algebraists which might ultimately explain what quantum groups ``really are''. \\ Our seminar will work through the most important papers about Khovanov homology and knot theory, beginning with those of Dror Bar-Natan, and if there's enough time we'll look at some of the more algebraic work too. \\ The seminar meets Tuesdays in 7218 from 10.30-12. \\ I will give the first talk next Tuesday, and after that we'll try to arrange a schedule of speakers for the rest of the term. Everyone is welcome to attend and/or speak, though

### October 6, 2009

### 10:30 AM

### AP&M 7218

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