V5A1 - Advanced Topics in Algebra - Vassiliev invariants and related topics
- Lectures: We, Fr, 10:15-12:00, SR 0.008
- First Lecture: Fr, Apr 13 2018
- Instructor: Tobias Dyckerhoff
- Office: HCM, Villa Maria, Endenicher Allee 62
- Email: dyckerho AT math uni-bonn de
Course description
In 1990, Vassiliev introduced a beautiful class of knot invariants that arise via Alexander duality
from a spectral sequence related to a filtration on the space of singular knots. Remarkably, the
essence of (the degree 0 part of) his construction can be captured in rather elementary terms: the
idea is to extend a given knot invariant to singular knots by taking iterated differences of the
various resolutions of a singular knot. Thinking of these iterated differences as discrete versions
of derivatives, Vassiliev proposes to study those knot invariants that are "polynomial" in the sense
that, for some natural number n, their nth derivatives vanish. The resulting theory of what are now
called Vassiliev invariants has striking relations to various other subjects. In this course, we
will focus on the following topics:
- review of basic knot theory
- algebra of chord diagrams
- weight systems
- Kontsevich's integral
- Vassiliev's spectral sequence
- topological quantum field theory
- Chern-Simons theory
- Rozansky-Witten theory
The textbook
- S. Chmutov, S. Duzhin, J. Mostovoy, Introduction to Vassiliev Knot Invariants, Cambridge University Press (2012)
and the following original references
- V.A. Vassiliev, Cohomology of knot spaces, Advances in Soviet Mathematics, Volume 1 (1990)
- M. Kontsevich, Vassiliev's knot invariants, Advances in Soviet Mathematics, Volume 16, Part 2 (1993)
- D. Bar-Natan, On the Vassiliev Knot Invariants, Topology (1995)
- E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989),
- V.A. Vassiliev, On combinatorial formulas for cohomology of spaces of knots, Mosc. Math. J., No. 90 (2001)
- M. Polyak, Feynman diagrams for pedestrians and mathematicians, Proc. Sympos. Pure Math. 73 (2005)
make good starting points. More references will be added to this list as the course progresses.
Familiarity with algebraic topology and homological algebra.
Oral exam