"Recent Developments in Chern-Simons Theory"

Oporto July 2004

In an odd way, this is excellent news to knot theory, to quantum field theory and to quantum algebra: For a long time we thought that practically all of combinatorial knot theory comes in one way or another from Chern-Simons theory and/or from some associated "quantum algebraic" theories. Khovanov's new homology theory has the same flavour as the old combinatorial invariants to a point, but then it diverges. It's presence strongly suggests that more is hidden in Chern-Simons theory, in quantum field theory and in the theory of quantum groups; it leaves us with the challenge of figuring out what this "more" is.

If Khovanov Homology is new, it is best appreciated against some older and better understood background. So we'll only come to our main topic in the third lecture, after spending the first two enjoying the more familiar (though still scenic and beautiful) grounds of Chern-Simons theory and combinatorial knot theory.

In the first lecture we will talk about astrology, cosmic alignments, knots, configuration space integrals and the perturbative expansion of Chern-Simons theory. (Handout: Lecture1.pdf; newer interface at Stonehenge.html).

In the second lecture we will clean up some of the mess left by the first lecture and discuss an assortment of little points related to it - the framing anomaly, finite type invariants, Lie algebras, R-matrix invariants and the Jones polynomial. (Handout: Lecture2.pdf).

In the third lecture we will finally come to Khovanov Homology, and
if time will allow, we will also mention it's newer (and even less
understood) "*sl(n)*" generalization due to Khovanov and
Rozansky. (Handouts: QRG.pdf, NewHandout-1.pdf).

A full and fair treatment of the above topics will take longer than three lectures. So I'll only scratch the surface, telling you about some of the main ideas while pretending that some of the thornier points aren't really there.

**Reality. ** On the evening before, Justin Roberts convinced
me to talk only about Khovanov homology.
The first lecture followed my paper On Khovanov's Categorification of
the Jones Polynomial (handout: QRG.pdf). The second
lecture followed my paper Khovanov's
Homology for Tangles and Cobordisms (handouts: NewHandout-1.pdf,
handout2.pdf, GWU.pdf, transparencies: A.png, B.png, Reid2Proof.pdf, R3Full.pdf). The third
lecture was about the recent Khovanov-Rozansky
paper (handouts: idea.jpg, KRHDefinitions.nb, KRHDefinitions.png).

Knots in Lisboa