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.

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. References for manifolds

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

  12. Baez, Some Definitions Everyone Should Know (Definition of symmetric monoidal functor) here
  13. Vladimir Hinich, Deformations of homotopy algebras, available here
  14. Vladimir Hinich, DG coalgebras as formal stacks, available here
  15. Maxim Kontsevich, Lecture notes on Topics in algebra. Deformation theory, available here
  16. Maxim Kontsevich and Yan Soibelman, Topics in algebra. Deformation theory, Lecture notes, available here
  17. Marco Manetti, Deformation theory via differential graded Lie algebras, available here
  18. Marco Manetti, Differential graded Lie algebras and formal deformation theory, available here
  19. Marco Manetti, A voyage around coalgebras, available here
  20. Some further topics where these ideas appear

    Koszul duality for algebras
  21. Stewart Priddy, Koszul resolutions, available here
  22. Jean-Louis Loday and Bruno Vallette, Algebraic Operads, available here
  23. Alexander Polishchuk and Leonid Positselski, Quadratic Algebras, available here
  24. Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, available here
  25. Bernhard Keller, Koszul duality and coderived categories (after K. Lefèvre), available here
  26. Gunnar Floystad, Koszul duality and equivalences of categories, available here
  27. Koszul duality for operads
  28. Victor Ginzburg and Mikhail Kapranov, Koszul duality for operads, available here
  29. Jean-Louis Loday and Bruno Vallette, Algebraic Operads, available here
  30. Rational homotopy theory
  31. Daniel Quillen, Rational homotopy theory, available here
  32. Alexander Berglund, Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebras, available here
  33. Alexander Berglund, Lecture notes on Rational homotopy theory, available here
  34. L-infinity algebras and homotopy transfer theorem
  35. Ezra Getzler, Lie theory for nilpotent L-infinity algebras, available here
  36. Marius Crainic, On the perturbation lemma, and deformations, available here
  37. Bruno Vallette, Algebra + Homotopy = Operad, available here
  38. E_n-algebras
  39. V.I. Arnol'd, The cohomology ring of the colored braid group, available here
  40. Frederick Cohen, Thomas Lada, Peter May, The homology of iterated loop spaces, available here
  41. Dev Sinha, The homology of the little disks operad, available here
  42. Alexander Kupers, Talbot pretalk: Kontsevich formality of the little n-disks operad, available here
  43. Factorization version
  44. Jacob Lurie, Lecture 8: Nonabelian Poincare Duality (in topology)., available here
  45. David Ayala and John Francis, Poincaré/Koszul duality, available here
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Exercise classes

All the exercise sheets will be available on this page.

Exam Dates

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