This project is a program of fundamental research in Mathematics, more precisely in Algebra, Geometry, and Topology.

Over the past 20 years,
the new theory of higher structures (operads, homotopy algebras,
infinity-categories) has given rise to powerful tools which made
possible the resolution of open problems and prompted revolutions in
algebraic topology (faithful algebraic invariants of the homotopy type
of spaces), algebraic geometry (derived algebraic geometry), and
deformation theory (formal moduli problems) for instance. The purpose
of the present project is to *apply these higher algebraic methods
in order to bring the same kind of groundbreaking developments*
in algebraic Lie theory, deformation theory, homotopy theory, and
discrete/algebraic geometry. The proposed operadic methods are *effective*
and *algorithmic*, and will thus produce explicit formulas
applicable at a wide scale by other mathematicians.

The project is structured over the following four interconnected topics.

__Higher Lie theory__: integration of curved homotopy Lie algebras with applications to classical problems in Lie theory.__Derived deformation theory__: deformation theory in positive characteristic and universal operadic deformation group.__Operadic homotopy theories__: rational homotopy theory (classical and operadic) and Koszul duality in homotopy theory.__Higher structures on geometrical objects__: diagonal of polytopes, wonderful models, toric varietes, moduli spaces, and higher structures in complex geometry.

The consortium unites experts in each of those fields including permanent faculty members, postdocs and doctoral students.

The project consists of three partners and brings together mathematicians from six universities in France and one university in Spain.

- Administrative center: Laboratoire Analyse, Géométrie et Applications, Université Sorbonne Paris Nord.
- Principal and Local coordinator: Bruno Vallette

- Administrative center: Institut de Recherche Mathématique Avancée, Université de Strasbourg.
- Local coordinator: Vladimir Dotsenko

- Administrative center: Institut de Mathématiques de Toulouse, Université Paul Sabatier (Toulouse).
- Local Coordinator: Ricardo Campos

Two one-year research associate positions are funded by this ANR grant, for early career researchers working in areas of algebra, geometry, and topology closely related to this project. One position will be located in Paris (2022-2023) and the other one in Strasbourg (2023-2024).

Directeur de la publication : Bruno Vallette

Directeurs de la rédaction : Ricardo Campos et Vladimir Dotsenko

Projet ANR-20-CE40-0016*HighAGT*

Directeurs de la rédaction : Ricardo Campos et Vladimir Dotsenko

Projet ANR-20-CE40-0016

Last modified: October 17, 2022.