# Abstracts for Conference on "Higher Homotopy Algebras in Topology", May 9 - 10, 2019

Alternatively have a look at the program.

## Homotopy theory of complete Lie algebras

Posted in
Speaker:
Aniceto Murillo
Affiliation:
Date:
Thu, 2019-05-09 09:30 - 10:30
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Ricardo Campos
Affiliation:
University of Montpellier
Date:
Thu, 2019-05-09 11:00 - 12:00
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Benoit Fresse
Affiliation:
Université de Lille
Date:
Thu, 2019-05-09 13:45 - 14:45
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Najib Idrissi
Affiliation:
Université Paris Diderot
Date:
Thu, 2019-05-09 15:00 - 16:00
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Fernando Muro
Affiliation:
Date:
Thu, 2019-05-09 16:30 - 17:30
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Pascal Lambrechts
Affiliation:
Université catholique de Louvain-la-Neuve
Date:
Fri, 2019-05-10 09:00 - 10:00
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Joana Cirici
Affiliation:
Universitat de Barcelona
Date:
Fri, 2019-05-10 10:30 - 11:30
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Affiliation:
Trinity College Dublin
Date:
Fri, 2019-05-10 13:00 - 14:00
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Alexander Berglund
Affiliation:
Stockholm University
Date:
Fri, 2019-05-10 14:00 - 15:00
Location:
MPIM Lecture Hall

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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