Higher Algebra, Geometry, and Topology – HighAGT

The present proposal is a program of fundamental research in Mathematics, more precisely in Algebra, Geometry, and Topology. Created over the past 50 years, the theory of higher structures (operads, homotopy algebras, infinity-categories) has given rise recently to powerful tools which lead to resolutions of open problems and prompted deep developments, for instance in algebraic topology (faithful algebraic invariants of the homotopy type of spaces), algebraic geometry (derived algebraic geometry), and deformation theory (formal moduli problems). The purpose of the present project is to apply these higher algebraic methods in order to bring the same kind of groundbreaking developments in Lie theory (higher Lie theory), deformation theory (deformation theory in prime characteristic, universal deformation groups, Grothendieck--Teichmüller groups), homotopy theories (rational and tame homotopy theories, Koszul duality), and geometry (algebraic, complex and discrete). Moreover, the proposed operadic methods are effective and algorithmic, and will thus produce explicit formulas applicable at a wide scale.