21 jan 2019 - 25 jan 2019
CIRM Luminy Marseille
Algebra. The theory of categories has proven to be a valuable algebraic tool to state and to organise the results in many areas in mathematics. In the same way, the notion of an operad, which emerged from the study of iterated loop spaces in the early 1970’s, now plays a structural role by organising the many operations with several inputs acting on the various objects in algebra, geometry, topology and mathematical physics.
Homotopical algebra. At the end of the last century, it appeared clearly that the introduction of higher algebraic notions was necessary in order to encode higher structures naturally appearing in mathematical problems. Homotopy theory provides us with natural phenomena whose description requires higher maps, that is some notion of higher category. In the same way, the algebraic notion of an operad, while already used to describe homotopy algebras, was proved to be a too strict notion, e.g. for the purposes of deformation theory.
Higher structures. After a long period of research in algebra and topology, the theory of higher structures gave rise recently to radically new and manageable notions, like the homotopy algebras, infinity-categories, and infinity-operads. They allowed renown mathematicians to prove long standing conjectures like the deformation quantisation of Poisson manifolds (Kontsevich) and the fundamental principle of deformation theory (Lurie– Pridham). They also made it possible to develop new domains like derived algebraic geometry (Toen–Vezzosi, Lurie), factorisation algebras (Costello–Gwilliam) and factorisation homology (Ayala–Francis–Tanaka).
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 studies of higher algebra together with its applications to prove some puzzling conjectures and to develop some long-term programs. This conference will be the closing event the project ANR "Higher structures in Algebra and Topology": https://www.math.univ-paris13.fr/∼vallette/ANRSATen.html