Simons semester, modes of participation:
- Master students based in Hamburg should register via Stine for the lecture course and the exercises.
- PhD students based in Hamburg should register via Geventis for the lecture course and the exercises and send me an email.
- If you are based outside of Hamburg and would like to follow the lectures online, please register via the Simons semester website.
Content:
This course gives an extended introduction to knot homology theories and, more broadly, categorification in quantum topology.
Topics of the course include:
- Basic knot theory
- Review of quantum invariants of knots, links and tangles
- The categorification toolkit
- Introduction to Khovanov homology and its generalizations
- Introduction to triply-graded link homology
- Applications in low-dimensional topology
- Towards topological quantum field theories
Prerequisites: Familiarity with at least 2/3 of the following:
- Algebra (incl. homological) : groups, rings, modules, chain complexes, homotopy equivalence, homology, Ext, Tor
- Topology (differential and algebraic): point-set topology, manifolds, orientations, fundamental group, homology, cohomology
- Category theory: limits, colimits, monoidal structures, enriched categories
Coordinates:
Lectures will be in held in person, streamed and recorded:
- Wednesday, 16:15-17:45, Geomatikum H1, 3rd Apr 24 - 10th Jul 24
- Friday, 12:15-13:45, Geomatikum H4, 5th Apr 24 - 12th Jul 24
Exercise classes will be in person for participants in Hamburg:
- Friday, 16:15-17:45, Room 434, Geomatikum, 12th Apr 24 - 12th Jul 24
Resources:
Video recordings of lectures will be made available for registered participants.
Introductory video ressources:
- Morrison, Scott. Khovanov homology (MSRI introductory workshop on link homology 2010) (Part 1, Part 2)
- Rasmussen, Jacob. Introduction to Knot Theory (IAS/PSMI Lecture Series 2019) (Playlist, see esp. lectures 2-4)
Some background references (to be updated):
- Bar-Natan, Dror. On Khovanov's categorification of the Jones polynomial. Algebr. Geom. Topol. 2 (2002), 337--370 (doi:10.2140/agt.2002.2.337)
- Bar-Natan, Dror. Khovanov's homology for tangles and cobordisms. Geom. Topol. 9 (2005), 1443--1499 (doi:10.2140/gt.2005.9.1443)
- Bar-Natan, Dror. Fast Khovanov homology computations. J. Knot Theory Ramifications 16 (2007), no. 3, 243--255 (doi:10.1142/S0218216507005294)
- Khovanov, Mikhail. A categorification of the Jones polynomial. Duke Math. J. 101 (2000), no. 3, 359--426 (doi:10.1215/S0012-7094-00-10131-7)
- Khovanov, Mikhail. Triply-graded link homology and Hochschild homology of Soergel bimodules. Internat. J. Math. 18 (2007), no. 8, 869--885 (doi:10.1142/S0129167X07004400)
- Morrison, Scott; Walker, Kevin; Wedrich, Paul. Invariants of 4-manifolds from Khovanov-Rozansky link homology. Geom. Topol. 26 (2022), no. 8, 3367--3420 (doi:10.2140/gt.2022.26.3367)