Knot homology, summer semester 2024
This is the website for the
lecture course on Knot homology and its associated exercise class. The course is part of the Simons semester: Knots, Homologies and
Physics.
This course is aimed at (advanced) Master students and PhD students
specializing in algebra, category theory, geometry, topology, and adjacent areas
of mathematical physics. Exceptionally motivated Bachelor students are also
welcome. The purpose of the course is to bridge the gap between a first exposure
to knot homology theories (excellent introductory articles and recorded
mini-lecture series exists, see below) and the level of current research. The
topic continues the theme of the ZMP Seminar on
Knot homology from the winter semester 23/24, but having attended this is not a prerequisite.
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:
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)
Exercise sheets:
The preparatory meeting for the exercise classes will take place on 5th Apr 24, 16:15, Room 434.
Exam:
For participants in Hamburg there will be an oral exam. To qualify for the exam, you should solve
at least 50% of the homework problems and present on the board at least twice.
Contact:
Paul Wedrich.
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