justin@math.ucsd.edu
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:
1. Non-negative integers are secretly dimensions of vector spaces
2. Integers are secretly virtual dimensions of formal differences of
vector spaces (or superdimensions of supervector spaces)
3. Integer Laurent polynomials are secretly graded dimensions of
Z-graded
(super)vector spaces;
4. Abelian groups are secretly Grothendieck groups of additive
categories
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 _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.
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 there's no obligation to do the
latter.
Oct 6:
Justin Roberts Definition of Khovanov homology
Oct 13:
Lyla Fadali Bar-Natan's reformulation of Khovanov
homology
Oct 20: Justin Roberts Frobenius algebras and Lee homology
Khovanov on Frobenius algebras
Scott Morrison and Dror Bar-Natan on Lee homology
Oct 27:
Ben Wilson Rasmussen's proof of
Milnor's conjecture
Nov 10:
TBA Kuperberg spiders and
canonical bases?