Preamble. Mathematics becomes suddenly more intricate and exciting when one does not compare anymore objects up to equality or up to isomorphism but under a more relaxed equivalence relation. The most illuminating example comes from topology where one studies topological spaces not up to equality or homeomorphism but up to homotopy equivalence, that is up to continuous deformations. This difficult but rich problem of classifying topological spaces up to homotopy prompted the creation of new domains in mathematics over the last century like algebraic topology, category theory, and homotopical algebra. In the last decades, such ideas have developed vastly past the original topological motivations in various directions.

Scientific context. The aim of the present conference is to gather mathematicians working in Algebra, Geometry, Topology, and Mathematical Physics. Created over the past 50 years, the theory of higher structures (operads, homotopy algebras, infinity-categories) has given rise recently to powerful tools which led 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). More recent, fundamental achievements have been obtained by applying these effective and algorithmic higher algebraic methods in Lie theory (higher Lie theory), derived deformation theory (deformation theory in positive characteristic, universal deformation groups, Grothendieck–Teichmüller groups), homotopy theories (rational homotopy theories), and geometry (algebraic, complex and discrete).

Present conference. We live nowadays a very exciting period of rapid mathematical development of higher structures. The aim of this conference is to cover the new foundational works on higher algebra together with its applications. Beyond the 20 one-hour talks, enough time will be scheduled to allow the various participants to discuss and to interact in order to broadcast their ideas and to initiate further joint collaborations.

This conference will be the closing event the project ANR « Higher Algebra, Geometry, and Topology », which structures the French community working on these topics:

anr-highagt.pages.math.cnrs.fr/index.html