- This meeting will take place at the University of Barcelona from November 22nd (3pm) to November 24th (12pm)

- Ricardo Campos (Université de Toulouse)
- Joana Cirici (Universitat de Barcelona)
- Geoffroy Horel (Université Sorbonne Paris Nord)

- Alexis Aumonier (University of Cambridge)
- Federico Cantero (Universidad Autónoma de Madrid)
- Pedro Magalhães (Universitat de Barcelona)
- Fernando Muro (Universidad de Sevilla)
- Joost Nuiten (Université de Toulouse)
- Francesca Pratali (Université Sorbonne Paris Nord)
- Victor Roca i Lucio (Ecole Polytechnique Fédérale de Lausanne)
- Bashar Saleh (Stockholm University)
- Anna Sopena (Universitat de Barcelona)

- 22 november, 113 EH (Philology cloister, ground floor)
- 23 november morning, 111 EH (Philology cloister, ground floor)
- 23 november afternoon, S3 (Math cloister, basement)
- 24 november, 111 EH (Philology cloister, ground floor)

22/11 |
113 EH | 15.30-16.30 | Fernando Muro : Hypercommutative algebras and differential forms |

16.30-17 | Coffee break | ||

17-18 | Pedro Magalhães : Formality of Kähler manifolds revisited | ||

23/11 |
111 EH | 9.30-10.30 | Bashar Saleh : Algebraic groups of homotopy classes of automorphisms and operadic Koszul duality |

10.30-11 | Coffee break | ||

11-12 | Anna Sopena : Pluripotential Operadic Calculus | ||

12.15-13.15 | Victor Roca i Lucio : A new approach to formal moduli problems | ||

S3 | 15.30-16.30 | Francesca Pratali : Linear dendroidal oo-operads | |

16.30-17 | Coffee break | ||

17-18 | Joost Nuiten : PD operads and partition Lie algebras | ||

24/11 |
111 EH | 9.30-10.30 | Alexis Aumonier : An h-principle for algebraic bundles and applications |

10.30-11 | Coffee break | ||

11-12 | Federico Cantero : On the cohomology of spaces of non-singular almost-complex hypersurfaces |

A homotopy principle (h-principle for short) often takes the form of a theorem approximating the topology of a space of solutions of a differential equation via homotopical methods. In this talk, I will describe a general h-principle for sections of algebraic vector bundles on projective complex varieties subject to differential relations. I will then emphasise on its applications, mostly through the eyes of rational homotopy theory and homological stability. I will show how it can be used to compute parts of the cohomology of various spaces from classical algebraic geometry. Examples include: the locus of smooth hypersurfaces in a very ample linear system, spaces of smooth complete intersections, and spaces of holomorphic maps to projective spaces.

Twenty years ago, Peters and Steenbrink found that the cohomology of the space of non-singular projective hypersurfaces of degree at least 3 contains a copy of the cohomology of a general linear group. In this talk we will extend their result to the space of almost-complex hypersurfaces, while recovering theirs with different methods coming from rational homotopy theory. This is a joint work with Ángel Alonso (U. Graz).

The interaction of Hodge structures with rational homotopy theory is a powerful tool to provide restrictions on the homotopy types of Kähler manifolds and of complex algebraic varieties. An example is the well-known result of Deligne, Griffiths, Morgan and Sullivan, stating that compact Kähler manifolds are formal. In the simply connected case, it implies, for instance, that the rational homotopy groups of such manifolds are a formal consequence of the cohomology. Despite this fact, the mixed Hodge structure on their rational homotopy groups is not, in general, a formal consequence of the Hodge structures on cohomology. To understand this phenomenon, we will introduce a stronger notion of formality which arises from studying homotopy theory in a category encoding the Hodge structures. We will also introduce obstructions to this strong formality, generalizing the classical ones, and study when are Kähler manifolds formal in this stronger sense.

The de Rham complex of a smooth manifold is a DG commutative algebra with the exterior differential and the exterior product, and the de Rham cohomology is hence a graded commutative algebra. Throughout the last 40 years, the de Rham complex and the cohomology of Poisson manifolds have been endowed with extra algebraic structure, using the Poisson bivector and related differential operators, like the interior product and the Lie derivative. Koszul endowed the de Rham complex with a Lie bracket which turns out to be trivial in cohomology. The Koszul Lie bracket comes from a Batalin-Vilkovisky operator, which is also trivial in cohomology. More recently, the de Rham cohomology of Poisson manifolds has been endowed with a hypercommutative algebra structure by several authors (Barannikov-Kontsevich, Manin, Losev-Shadrin, Park). This hypercommutative algebra structure can be defined at the level of differential forms (Khoroshkin-Markarian-Shadrin), therefore it gives rise to a homotopy hypercommutative algebra structure on de Rham cohomology (Dotsenko-Shadrin-Vallette). We will show that all these (homotopy) hypercommutative algebra structures are trivial, just like the previous ones, i.e. they reduce to the underlying (homotopy) commutative algebra structure existing for all manifolds.

Partition Lie algebras have recently been introduced by Brantner and Mathew as certain homotopy theoretic refinements of dg-Lie algebras that control deformation problems in positive characteristic. In this talk, I will try to explain how partition Lie algebras (and other types of Lie algebras) can be understood as certain types of algebras over a PD operad, a refinement of the usual notion of an operad. A version of Koszul duality for such PD operads gives rise to Lie-algebraic descriptions of various types of formal problems appearing in derived algebraic geometry. Based on joint work with Lukas Brantner and Ricardo Campos

In the last two decades, several equivalent models for the homotopy theory of oo-operads enriched in simplicial sets/topological spaces have been developed. In particular, in Weiss and Moerdijk’s approach, the simplicial model for oo-categories is generalized to the operadic context by replacing the simplex category with a certain category of finite rooted trees. The main purpose of this talk will be to present a similar tree-like approach to the homotopy theory of oo-operads enriched in chain complexes, which we call ‘linear’. We will discuss the combinatorics of the tree category Omega and a Segal-like condition which allows to define linear oo-operads as certain coalgebras over a comonad. Then, by considering a category of ‘trees with partitions’, we will realize linear oo-operads as a full subcategory of a functor category, and discuss what can be said at the oo-categorical level.

The celebrated Lurie—Pridham theorem states that infinitesimal deformations are encoded by dg Lie algebras, over a characteristic zero field. “Infitesimal deformations” are here made precise via the notion of a formal moduli problem. Since then, this theorem has been generalized in many directions. Nevertheless, these generalizations rely on variations of the initial arguments proposed by Lurie. The goal of this talk is to explain a new framework for formal moduli problems, using methods coming from operadic calculus. This allows us to fully characterize when formal moduli problems of some type of algebras are equivalent to their Koszul dual algebras, over a field of any characteristic. And it will give us a new proof of the celebrated Lurie—Pridham theorem, as well as of many of its generalizations. This is joint work with Brice Le Grignou.

Given a simply connected space X, there are several, a priori different, algebraic groups whose groups of Q-points are isomorphic to the group of homotopy classes of homotopy automorphisms of the rationalization of X. We will discuss how the theory of operadic Koszul duality can be used to prove that two of these different algebraic groups are isomorphic.

For complex manifolds, there exists a refined notion of weak equivalence related to both Dolbeault and anti-Dolbeault cohomology. This new class of weak equivalences gives rise to pluripotential formality, a stronger formality concept compared to the classical one. Several questions concerning this formality remain unresolved, such as whether Kähler manifolds are pluripotentially formal. The goal of this talk is to introduce a novel operadic framework designed to address these issues. In particular, I will present the concept of pluripotential A-infinity-algebras and a homotopy transfer theorem based on this strong notion of weak equivalence.