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

- Motivation and examples

(Deformations of complex structures, deformations of associative algebras, and the Maurer-Cartan equation)

Speaker: Sylvain Lavau - Deformation Functors

(Classical approaches to deformation functors, and motivation for why do they encode deformations. The talk includes a short reminder on simplicial sets) - Deformation functors - Modern approach and the MC equation

(Deformation functors as functors from local differential graded Artinian rings to simplicial sets. How the Maurer-Cartan equation provides a deformation functor out of a dgLa. The talk includes some discussion on the necessary tools on simplicial sets) - The Chevalley-Eilenberg complex $C^*$, and how it is related to the Maurer-Cartan equation.
- The model category dgLa.
- Koszul duality I - $C^*$ and its adjoint
- Koszul duality II - $C^*$ is an equivalence (sometimes)
- Proof of the equivalence in the Main Theorem, part 1
- Proof of the equivalence in the Main Theorem, part 2
- 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 - Bertrand Toën, Problèmes de modules formels

https://perso.math.univ-toulouse.fr/btoen/files/2012/ 04/Bourbaki-Toen-2016-final1. pdf - Vladimir Hinich, DG coalgebras as formal stacks

https://arxiv.org/abs/math/9812034 - Marco Manetti, Deformation theory via differential graded Lie algebras

https://arxiv.org/abs/math/0507284 - Damien Calaque, Julien Grivaux, Formal moduli problems and formal derived stacks

https://arxiv.org/abs/1802.09556 - 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

**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 - Marco Manetti - Lectures on deformations of complex manifolds

https://arxiv.org/abs/math/0507286 - Albert Nijenhuis, R. W. Richardson - Cohomology and deformations in graded Lie algebras

https://projecteuclid.org/euclid.bams/1183527432

**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

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