Bordisms and Topological Field Theories WS 2023/24
hosted by Claudia Scheimbauer and Anja Svraka

Time and place

Lectures: Mon 14:15-16:00, Wed 13:15-14:00
Exercises: Wed 12:15-13:00
Seminar Room 02.08.011

Topic

Studying manifolds up to diffeomorphism is very difficult. However, if we instead study manifolds up to “cobordism” and consider the disjoint union we obtain very computable groups. In fact, product of manifolds gives a cobordism ring. In the 1980’s, Atiyah and Segal realized that the notion of cobordism naturally appears when decribing topological field theories mathematically. In the course we will encounter these notions. You can find a syllabus here.

Lecture notes

Handwritten lecture notes will be available at this link and will be regularly updated throughout the semester. A preliminary version of lecture notes can be found here.

Reading assignment December 4

Please read through Lecture 11 in the lecture notes. The main result is also Lecture 16 in Freed, Bordisms Old an New. Find the reading assignment in the lecture notes or here. We will discuss the material on Wednesday, December 6.

Reading assignment December 18

Discuss these questions to see whether you understoood the Classification of 2dTFTs. Reading relevant parts of Kock's book (see references) might help.

Hybrid lecture on December 6

In case you still have trouble getting to TUM, we will have a hybrid class. COnnect via: https://tum-conf.zoom-x.de/j/61613802735?pwd=YTFpdlROSUE0YXd1VDU0djd6SEJXUT09 Passcode: category We will do our best to still make it a lively discussion. For this, if you join online, it is helpful if you join with video (but muted unless you'd like to ask a question). This makes the experience better for all of us.

Knots

See Fabian Roll's beautiful animations: Ingo Runkel's slides contain most of what we did.

References

    Main references

  1. Michael Atiyah, Topological quantum field theories. IHES Publ. Math., (68):175–186 (1989), 1988.
  2. Joachim Kock, Frobenius algebras and 2D topological quantum field theories
  3. Christian Kassel, Marc Rosso, and Vladimir Turaev, Quantum groups and knot invariants
  4. Daniel S. Freed, Bordism: Old and new
  5. Adams, The knot book.
  6. Carqueville, Runkel, Lecture notes on field theory
  7. Schweigert, Lecture notes on Hopf algebras, quantum groups, and topological field theory
  8. John Baez, Some Definitions Everyone Should Know

    9. References for manifolds

  9. Hirsch, Differential Topology
  10. Kosinski, Differential Manifolds (both available via TUM library)

    11. Exercise classes

    All the exercise sheets will be available on this page.