Time and place

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

Topic

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:

• In the main part of the course we will study (formal) moduli problems for commutative algebras and Koszul duality between commutative differential graded algebras and differential graded Lie algebras.
• Generalizing the strategy taken we will study formal moduli problems and Koszul duality of associative algebras.
• Finally we will introduce E_n-algebras and briefly discuss formal E_n moduli problems and E_n Koszul duality.

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.

Calender

• April 11 - Introduction
• April 12 - Classical deformation theory in algebraic geometry
• April 18 - Differential graded Lie algebras
• April 19 - Simplicial sets
• April 25 - Talk on Model categories - Fabbrizzi
• April 26 - Talk on The model category of differential graded Lie algebras - Rigel
• May 2 - Introduction to (∞,1)-categories
• May 3 - Talk on Quasicategories - Stefano
• May 9 - Talk on Rational Homotopy Theory - Albert
• May 10 - Talk on L-infinity Algebras - Nestor
• May 23 - The proof of the main theorem - small dg algebras
• May 24 - The proof of the main theorem - Stating the main theorem and computing homotopy colimits
• May 30 - Talk on Homotopy Transfer and L-infinity Algebras - Aras or here
• May 31 - The proof of the main theorem - the tangent or here
• June 6 - Talk on Koszul duality for associative algebras - Dan
• June 7 - More on ∞-categories -- the adjoint functor theorem part 1, part 2, part 3
• June 13 - Talk on Koszul duality and equivalences of categories - Matthew
• June 14 - CANCELLED
• June 20 - The proof of the main theorem
• June 21 - The proof of the main theorem or here
• June 22, 9-10:30 in seminar room - The proof of the main theorem
• June 27 - Special lecture on Tangent Lie algebra of derived stacks - Benjamin Hennion
• June 28 - Talk on Introduction to Derived Algebraic Geometry - Trang
• July 4 - Talk on Derived algebraic geometry and representation theory I - Galya
• July 5 - Talk on Derived algebraic geometry and representation theory II - Tina
• July 11 - Talk on Koszul duality in representation theory or here - Tim
• July 12 - Formal moduli problems in more general settings or here
• July 18 - Special lecture on Deformations of objects and categories - Anthony Blanc
• July 19 - The final lecture

References

Primary reference

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

Related references

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

Some classical references

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

Some further topics where these ideas appear

Koszul duality for algebras

15. Stewart Priddy, Koszul resolutions, available here
16. Jean-Louis Loday and Bruno Vallette, Algebraic Operads, available here
17. Alexander Polishchuk and Leonid Positselski, Quadratic Algebras, available here
18. Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, available here
19. Bernhard Keller, Koszul duality and coderived categories (after K. Lefèvre), available here
20. Gunnar Floystad, Koszul duality and equivalences of categories, available here

Koszul duality for operads

21. Victor Ginzburg and Mikhail Kapranov, Koszul duality for operads, available here
22. Jean-Louis Loday and Bruno Vallette, Algebraic Operads, available here

Rational homotopy theory

23. Daniel Quillen, Rational homotopy theory, available here
24. Alexander Berglund, Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebras, available here
25. Alexander Berglund, Lecture notes on Rational homotopy theory, available here

L-infinity algebras and homotopy transfer theorem

26. Ezra Getzler, Lie theory for nilpotent L-infinity algebras, available here
27. Marius Crainic, On the perturbation lemma, and deformations, available here
28. Bruno Vallette, Algebra + Homotopy = Operad, available here

E_n-algebras

29. V.I. Arnol'd, The cohomology ring of the colored braid group, available here
30. Frederick Cohen, Thomas Lada, Peter May, The homology of iterated loop spaces, available here
31. Dev Sinha, The homology of the little disks operad, available here
32. Alexander Kupers, Talbot pretalk: Kontsevich formality of the little n-disks operad, available here

Factorization version

33. Jacob Lurie, Lecture 8: Nonabelian Poincare Duality (in topology)., available here
34. David Ayala and John Francis, Poincaré/Koszul duality, available here