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Dror Bar-Natan:
Talks:
I will be giving two talks at UNC:

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On Khovanov's Categorification of the Jones Polynomial

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Colloquium, Phillips 332, 4PM Thursday February 20, 2003

After quickly recalling what is the Euler characteristic and what is
homology and after briefly describing the Jones polynomial, I will move on
to describe in simple terms Khovanov's novel "Categorification of the Jones
Polynomial", which is to the Jones polynomial as homology is to the Euler
characteristic. Namely, it is much more interesting.

With some added and some removed, my talk will be based on my paper
On Khovanov's
Categorification of the Jones Polynomial
and will mostly follow this handout. The only
new material will be on the back side of the handout, which will be
printed with a list of reasons for why is
categorification interestings and with the sketch of the proof that
Khovanov's homology lifts to an invariant of knot cobordisms.

#
Khovanov's Homology for Tangles and Cobordisms

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Seminar, Phillips 330, 3PM Friday February 21, 2003

The minor advantage of homology theory over the Euler characteristic
is that it is a finer invariant. The major advantage is that it is a
functor: Given a map between spaces there is a map between their
homologies. Think of almost any major theorem in algebraic topology and
you'll find that the functoriality of homology is deeply involved. In
my talk, I will explain in elementary terms what seems to be the
corresponding property of Khovanov's homology: that it is a functor
from the category of links and cobordisms to the category of vector
spaces (see Jacobsson's arXiv:math.GT/0206303
and Khovanov's arXiv:math.QA/0207264).
My proof of this property is in the spirit of Khovanov's, but it is
both simpler and more general. It involves the extension of the
theory to the canopoly of tangle cobordisms, with values in several related
canopolies.

What's a canopoly? No, that would go in the talk; not here. It's an
object with a rather messy formal definition but a very simple visual
image.