Derived deformation theory and Koszul duality
hosted by Owen Gwilliam and Claudia Scheimbauer

Time and place

Mon 10:30-12:30, Tue 10-12, Seminar Room at MPIM


Many problems and constructions in mathematics come with natural parameters: the coefficients of a polynomial, the metric on a manifold, or the potential in a Hamiltonian. It is fruitful to study how properties vary in these parameters, i.e., over the moduli spaces of such structures. In algebraic geometry, the study of very small variations falls under the heading of deformation theory. In the last thirty years, a rich mix of homological algebra and deformation theory --- sometimes called derived deformation theory -- has influenced a broad range of mathematics and physics. In particular, it was, in part, a motivation for the development of derived geometry. This course aims to explain the approach to derived deformation theory taken in Jacob Lurie's article DAG X: Formal moduli problems, which develops a beautiful framework for studying such formal moduli problems. It provides a unifying perspective on the Koszul duality between Lie algebras and commutative algebras and the Koszul self-duality of associative algebras and E_n-algebras. The 2010 ICM address of Jacob Lurie provides a compelling introduction to and motivation for this approach. We will discuss the relationship with the other work on deformation theory and Koszul duality, which spans areas like pure algebra (e.g., quadratic algebras, cf. Polishchuk-Positselski), algebraic geometry (e.g., deformations of geometric structure, cf. Manetti), and operad theory (cf. Ginzburg-Kapranov, Fresse).

Structure of course

The course will be split into three parts: At the end, we will give an outlook on the simultaneous generalization of Poincaré and Koszul duality in recent work by Ayala-Francis, which provides a striking application of this circle of ideas to topology.

The course will not presume familiarity with higher category theory or higher algebra. We will introduce and explain ∞-category theory and other machinery as needed. Indeed, this subject provides a wonderful place to see how such tools can be used to articulate and prove interesting results in algebraic geometry and topology. A pedagogical goal of the course is thus to practice the yoga of homotopical algebra and derived geometry. Another goal of the course will be to help the students learn to read Lurie's work effectively.

The general format will be the following (although we will deviate, at least in the beginning): In the lectures on Tuesdays we will follow the the main text, giving motivation, the theorems, and proofs. Mondays will be devoted to covering material which could be considered a toolbox useful in higher structures and applications thereof -- this will include model categories, ∞-categories, classical results on Koszul duality, rational homotopy theory, homotopy transfer theorem, and operads. Some of these will be covered by students: possible topics can be found here.



    Primary reference

  1. Jacob Lurie, DAG X: Formal moduli problems, available here
  2. Related references

  3. Jacob Lurie, ICM address 2010, available here
  4. Jacob Lurie, Higher Algebra, available here
  5. Jon Pridham, Unifying derived deformation theories, available here
  6. Mauro Porta, Derived formal moduli problems, Master thesis, available here
  7. Bertrand Toën, Problèmes de modules formels, available here
  8. Some classical references

  9. Vladimir Drinfeld, a letter to V. Schechtman, September 1988, available here
  10. Vladimir Hinich, Deformations of homotopy algebras, available here
  11. Vladimir Hinich, DG coalgebras as formal stacks, available here
  12. Maxim Kontsevich, Lecture notes on Topics in algebra. Deformation theory, available here
  13. Maxim Kontsevich and Yan Soibelman, Topics in algebra. Deformation theory, Lecture notes, available here
  14. Marco Manetti, Deformation theory via differential graded Lie algebras, available here
  15. Marco Manetti, Differential graded Lie algebras and formal deformation theory, available here
  16. Marco Manetti, A voyage around coalgebras, available here
  17. Some further topics where these ideas appear

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