Abstract
``Algebra and homotopy theory do not mix well a
priori'' used to say Saunders MacLane because algebraic structures are,
in general, not stable under homotopy operations. On the other hand,
Mathematics is full of algebraic structures on chain complexes for
instance. In algebraic topology, the singular cochain complex of a
topological space is endowed with a associative product. In
differential geometry, the de Rham complex of a manifold carries a
commutative product. And the deformation theory is all based on the
notion of a differential graded Lie algebra.
One should consider MacLane's statement in a positive way: there is
here a fruitful source of beautiful mathematical developments which
would allow us to describe the homotopical properties of types of
algebras, with a view toward applications in Algebra, Geometry,
Topology, etc. The purpose of this course will be to get accustomed to
some of the modern tools of homotopical algebra. We will begin
with a few but crucial examples of differential graded algebras and
their properties, like the homotopy transfer theorem. We will then
study how to encode conceptually the various types of algebraic
structures with the notion of an operad. Operad theory, including the
Koszul duality theory, will provide us with fundamental tools to
describe the homotopical properties of differential graded algebras. We
will conclude this course with recent and elegant applications to
deformation theory (pre-Lie algebras and infiny-groupoid).
This course will directly follow the one of Muriel Livernet on
Homological algebra and algebraic topology (September-October 2017); it will open the doors to the one of Gregory Ginot and Marco Robalo on an
Introduction of homotopy theory (second semester, 2018).
Layout
- Differential graded algebras
- Operads
- Koszul duality
- Algebras up to homotopy
- Deformation theory
References
- Algebraic Operads, Jean-Louis Loday and Bruno Vallette, Grundlehren der mathematischen
Wissenschaften, Volume 346, Springer-Verlag (2012).
- Algebra+Homotopy=Operads, Bruno Vallette, in "Symplectic, Poisson and Noncommutative Geometry", MSRI Publications 62 (2014), 101-162.
Organisation
Les cours auront lieu du 8 novembre au 15 décembre 2017 à
raison de deux séances de 2 heures de cours et 1 heure de travaux
dirigés (exercices, exposés des étudiant-e-s) par semaine.
Prerequisistes
Basic
homological algebra (chain complexes, homotopy, tensor product) and
first notions of category theory (category, functor, monoidal category,
monad).
Professor
Bruno Vallette (Cours/TDs)