Alternatively have a look at the program.

## Homotopy theory of complete Lie algebras

Having as motivation the Deligne's principle by which every deformation functor is governed by a differential graded Lie algebra, we build a homotopy theory for these algebras which extend the classical Quillen approach to any topological space and is based in a new model category structure for (complete) differential graded Lie algebras. The core of this structure lies in the construction of the "Eckmann-Hilton dual" of the classical differential forms on the standard simplices.

## The homotopy type of associative and commutative algebras

Given a (dg) commutative algebra, one can ask how much of its homotopy type is preserved by its associative part. More precisely one can ask if $C$ and $C'$ are commutative algebras connected by a zig-zag of quasi-isomorphisms of **associative** algebras $C\stackrel{\sim}{\longleftarrow} A \stackrel{\sim}{\longrightarrow} C'$, must $C$ and $C'$ be quasi-isomorphic as **commutative **algebras? Despite its elementary formulation, this question turns out to be surprisingly subtle.

## Operads, graph complexes and the rational homotopy of embedding spaces

I will report on joint works with Victor Turchin and Thomas Willwacher on the applications of operads to the study of the rational homotopy type of embedding spaces.

In a first part, I will explain a graph complex description of the rational homotopy of mapping

spaces of $E_n$-operads. Results on the Goodwillie-Weiss calculus implies that this computation gives a description of a delooping of embedding spaces of Euclidean spaces.

## Curved Koszul duality for algebras over unital operads

Koszul duality is a powerful tool that can be used to produce resolutions of algebras in many contexts. In particular, Koszul duality of operads is the tool of choice to define the notion of "homotopy algebras".

In this talk, I will present a framework to study curved Koszul duality for algebras over certain kinds of unital operads (i.e. satisfying $P(0) = \Bbbk$). I will explain how to use it in order to compute the factorization homology of a closed manifold with values in the algebra of polynomial functions on a standard shifted symplectic space.

## An application of A-infinity obstruction theory to representation theory

The existence and uniqueness of enhancements for triangulated categories is an old problem in algebra

and topology, e.g. the stable homotopy category has a unique enhancement up to Quillen equivalence

(Schwede), the derived category of a Grothendieck category too (Canonaco-Stellari), etc. These examples

have a common feature: they are large categories. Triangulated categories of finite type over a perfect

field arise commonly in representation theory. We will show how to use the homotopy theory of operads

## Cosimplicial models for Goodwillie-Weiss manifold calculus

Goodwillie-Weiss manifold calculus is a tool which gives excellent approximations to some topological constructions on manifolds. It has been used for example to obtain many informations on the homotopy type of the space of smooth embeddings of a manifold M into a manifold W, Emb(M,W).

## Galois actions, purity and formality with torsion coefficients

I will explain how to use the theory of weights on étale cohomology to study formality with torsion coefficients, for certain schemes defined over a finite field. As an application, I will give some partial results of formality with torsion coefficients for configuration spaces on the complex space and for the operad of little disks. This is joint work with Geoffroy Horel.

## Homotopy type of the moduli space of stable rational curves

I shall show that the rational cohomology of the moduli space of stable rational curves is a Koszul algebra (answering a question of Yu. I. Manin, D. Petersen and V. Reiner), and explain how this allows one to compute the rational homotopy invariants of this space in a very explicit way. Time permitting, I shall talk about a few classes of spaces for which similar results are available, and a few other conjectural classes of spaces like that.

## Dg Lie algebra models for automorphisms of fiber bundles

I will construct differential graded Lie algebra models for classifying spaces of automorphisms of

bundles and I will discuss how to find Chevalley-Eilenberg cocycles that represent generalized

Miller-Morita-Mumford classes in these models. This leads to a new approach to tautological rings

for simply connected manifolds.

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