Higher Algebra, Geometry, and Topology

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.

  1. Higher Lie theory: integration of curved homotopy Lie algebras with applications to classical problems in Lie theory.
  2. Derived deformation theory: deformation theory in positive characteristic and universal operadic deformation group.
  3. Operadic homotopy theories: rational homotopy theory (classical and operadic) and Koszul duality in homotopy theory.
  4. 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 consortium

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

Paris Partner

East Partner

South Partner

Postdoctoral Positions

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 was 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

Last modified: January 15, 2024.