Time: First lecture Friday 10am, 6438. But how about Monday,
Wednesday 4-5.30?
Lecturer: Justin Roberts
"It was twenty years ago today, Vaughan Jones taught us how to play!"
This will be a topics course explaining the theory of "Quantum
invariants of three-manifolds" and its relationship with other parts
of low-dimensional topology.
The subject began in 1984 with Jones' discovery of his famous
polynomial invariant of knots in the three-sphere. This discovery
proved to be the tip of the iceberg: several other knot related knot
polynomials (HOMFLY, Kauffman, ...) were quickly discovered, and in
1989 Witten explained how this family of invariants should extend
naturally to give lots of invariants for three-manifolds. The guiding
principle is Topological Quantum Field Theory: an axiomatic
characterisation of the class of invariants resembling the
Eilenberg-Steenrod homology axioms but of an essentially
multiplicative rather than
additive character.
Witten's explanation is via physics, including the notorious
Feynman path integral technique of quantum field theory. It is
still impossible to treat his approach as rigorous, but fortunately
the invariants can be studied (in particular, shown to exist!) by many
other methods. These span quite a large area of mathematics: chiefly
algebra (quantum groups, tensor categories) and differential topology
(the rigorous approach to perturbation theory), but there are also
elementary combinatorial approaches and a whole lot of interesting
diagrammatic methods available.
The course
A provisional outline:
1. Knot polynomials: Jones, HOMFLY, Kauffman, Alexander, and some
applications
2. Tensor categories: braids, R-matrices, quantum groups
3. The Chern-Simons path integral and three-manifold invariants
4. Perturbation theory: the Kontsevich integral and configuration
space integrals
5. Future directions: Rozansky-Witten invariants; hyperbolic volume
conjecture; Gopakumar-Vafa invariants; Khovanov homology
Pre-requisites
It's hard to be very specific. You probably ought to have done the
Topology graduate course and know what homology is. Parts of the
course (especially at the start, talking about knot invariants)
require very little background. Other parts require a bit more
abstract algebra, in particular Lie algebras and category theory. I
will make sure I explain all the algebra I use though.
Books
A few useful references are as follows:
Quantum Invariants, by Tomotada Ohtsuki (World Scientific)
Knot Theory , by W.B.R. Lickorish (Springer)
Lectures on tensor categories and modular functors, by Bakalov and Kirillov
(AMS)
Problems on invariants
of knots and 3-manifolds edited by Ohtsuki.
Various lecture notes from a summer school in Grenoble by
Vogel, Lescop, etc.
Additional Information
My office hours would be after class on Monday, if the 4pm
lecture time goes through.
justin@math.ucsd.edu
