Alternatively have a look at the program.

## Deformations of complex structures and associative algebras

## Deformation problems

In this talk we will continue the introduction to deformation problems by taking a different point of view on them:

We will discuss local artinian rings, and how some natural deformation problems can be posed as functors from local artinian rings to sets, or better, to simplicial sets. We will also see what kind of properties such functors satisfy. The talk will include a reminder on the needed definitions and not assume previous knowledge of simplicial sets.

## Deformation functors - a modern approach

We will continue the study of deformation problems by getting to a modern defintion of a deformation functor. These are functors from local Artinian k-algebras to simplicial sets. We will see how the Maurer-Cartan equation provides a such a deformation functor out of a dgLa.

The talk includes an introduction to the necessary tools on simplicial sets.

## The Chevalley-Eilenberg complex of a dgLa

We describe how to associate to any differential graded Lie algebra **g** a differential graded coalgebra C(**g**), whose underlying complex is the Chevalley-Eilenberg complex. We then use this construction to give a new description of the Maurer-Cartan elements of **g****.**

## The model category dgLa

In this talk we will see what is a model category structure, describe a model structure on chain complexes, and how to transfer it to a model category structure on dgLa's via a free-forgetful adjunction.

## Relating dgLa and cdga: $C^*$ and its adjoint

In this talk we will see that the functor $C^*$ from dgLa to cdga admits a right adjoint up to homotopy $D$. In order to understand $D$, we will use the free-forgetful adjunction between dgLa and chain complexes: We will see that up to homotopy, $C^*$ composed with the free functor $F$ has a particularly simple form, which in the following week will let us write a simple expression for $D$ (up to homotopy, when computed on certain nice enough algebras).

## Relating dgLa and cdga: $C^*$ and its adjoint. Part 2

In the previous talk we studied the existence of a right adjoint $D$ to the cohomological Chevalley-Eilenberg functor $C^*$ between the $\infty$-categories $dgLa[W^{-1}]$ and $(cdga^{aug}[W^{-1}])^{op}$. We also discussed that $C^*$ applied to free dgLa's is (homotopically) very simple: It is equivalent to applying the functor that sends a cochain complex $V$ to the augmented cdga $k\oplus V^*[-1]$, equipped with the product determined by $(1,a)(1,b)=(1,a+b)$.

## The main theorem

## Proof of the main theorem

## Deformations as Maurer-Cartan elements

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