Learning seminar on deformation theory

The goal of the seminar is to rigorously understand the statement ''Every deformation problem in characteristic zero is controlled by a differential graded Lie algebra". This statement has long been a philosophy/guiding principle when studying deformations of algebraic or geometric structures. By the end of the seminar we aim to understand the statement and proof of its following modern incarnation:

Theorem (Lurie, Pridham)
There is an equivalence of $\infty$-categories between the $\infty$-category of formal moduli problems and the $\infty$-category of dgLa's over a field of characteristic zero.

Everyone is welcome, whether to give a talk or simply attend. If you think you'd like to give a talk please come to the first meeting or get in touch with one of the organizers.

Contents

In the first part we will see some deformation problems that naturally give rise to dgLa's, and that can also be encoded in deformation functors (also called formal moduli problems). We will see that the two are related by the Maurer-Cartan equation.

In the second part we will study how to construct a deformation functor out of a dgLa using the Maurer-Cartan equation. Conversely, we will build a dgLa out of a deformation functor. For that, we will need to understand some categorical properties of the $\infty$-category of dgLa's - roughly, that we can describe it in terms of generators and relations. This will be done making use of the Chevalley-Eilenberg complex of a dgLa, so that we can work in differential graded local Artinian rings (dgArt) instead.

In the third part we will see that the construction of a dgLa out of a deformation functor is an equivalence of $\infty$-categories between formal moduli problems and dgLa's. Finally, we will see that the inverse of this equivalence is given by the Maurer-Cartan construction.

Talks

1. Motivation and examples
2. Deformation Problems and Moduli Problems  (Notes by Alex [5])
3. Deformation functors - Modern approach and the MC equation (Notes by Joao [6])
4. The Chevalley-Eilenberg complex $C^*$, and how it is related to the Maurer-Cartan equation (Notes by Joost [7])
5. The model category dgLa
6. Koszul duality I - $C^*$ and its adjoint
7. Koszul duality II - $C^*$ is an equivalence (sometimes) (Notes by Sylvain [8])
8. Proof of the equivalence in the Main Theorem, part 1(Notes by Christian [9])
9. Proof of the equivalence in the Main Theorem, part 2 (Notes by David [10])
10. The inverse of the equivalence is given by Maurer-Cartan

References

• Jacob Lurie, DAG X: Formal moduli problems
  http://www.math.harvard.edu/~lurie/papers/DAG-X.pdf [11]
• Bertrand Toën, Problèmes de modules formels
  https://perso.math.univ-toulouse.fr/btoen/files/2012/04/Bourbaki-Toen-2016-final1.pdf [12]

   VIDEO [13]

• Vladimir Hinich, Descent of Deligne groupoids
  https://arxiv.org/abs/alg-geom/9606010 [14]
• Vladimir Hinich, DG coalgebras as formal stacks
  https://arxiv.org/abs/math/9812034 [15]
• Marco Manetti, Deformation theory via differential graded Lie algebras
  https://arxiv.org/abs/math/0507284 [16]
• Damien Calaque, Julien Grivaux, Formal moduli problems and formal derived stacks
  https://arxiv.org/abs/1802.09556 [17]
• Michael Schlessinger, Functors of Artin rings
  https://www.ams.org/journals/tran/1968-130-02/S0002-9947-1968-0217093-3/S0002-9947-1968-0217093-3.pdf [18]
• Daniel Quillen, Rational homotopy theory
  https://www.jstor.org/stable/1970725 [19]
• W. G. Dwyer,  J. Spalinski, Homotopy theories and model categories
  hopf.math.purdue.edu/Dwyer-Spalinski/theories.pdf [20]

On classical/motivating examples:

• Murray Gerstenhaber - On the Deformation of Rings and Algebras
  https://www.jstor.org/stable/1970484?seq=1#page_scan_tab_contents [21]
• Marco Manetti - Lectures on deformations of complex manifolds
  https://arxiv.org/abs/math/0507286 [22]
• Albert Nijenhuis, R. W. Richardson - Cohomology and deformations in graded Lie algebras
  https://projecteuclid.org/euclid.bams/1183527432 [23]

Historical:

• Deligne's letter to Millson (For the philosophy "Every deformation problem in characteristic zero is controlled by a dgla") https://publications.ias.edu/sites/default/files/millson.pdf [24]