An introduction to operads in algebraic topology (SPOC)

This is the SPOC version of the online course "An introduction to operads in algebraic topology". The first (test) session will be open from November 8 until December 17, 2021. Three sessions per year are planned (in Fall, Winter, and Spring). Enrollment is mandatory to attend the SPOC version.

There is a "guest" version of the course, accessible without enrollment for free auditing (with no schedule nor restriction access to the sections), available at: https://moodle.univ-lille.fr/course/view.php?id=30681

Subject and objectives of the course

This course provides an introduction to the theory of operads. We also explain applications of operads in algebraic topology, to serve both for the students intending to pursue in this fields, and to give an illustrative example of application of the ideas of the theory for the students coming from other domains of mathematics.

The notion of an operad is used to govern collections of operations, such as the operations considered in mathematics for the definition of the classical structures of algebra. The theory of operads is used in many branches of mathematics as an effective device to handle multiple algebraic structures in a variety of contexts.

The course comprises three parts. The first part provides a general introduction to the fundamental definitions and constructions of the theory of operads. In the second part, we explain applications of operads to the modelling of operations associated to the cochain complexes that underlie the classical singular cohomology theory of algebraic topology. In the third part, we provide an introduction to Quillen's homotopical algebra and we explain applications of operads to the definition of algebraic models of the homotopy theory of spaces.

Readership

Graduate students and researchers interested in applications of operads in algebra or topology.

Prerequisites
The essential prerequisite, necessary from the very beginning of the course, is a good basis in general algebra (linear algebra over modules, tensor products) and an appetence for the methods of general algebra. The student will also be assumed to be familiar with the language of category theory which will be used all along the course. Nonetheless, notes are provided in the course in order to give an introduction, from scratch, to this subject for the students who do not have this background.
For the second and third parts of the course, a first introduction to the definition of a homology theory (either in algebra or in algebraic topology) will be preferable to follow the motivations of the lectures. Notes are, again, provided in the course on some background of algebraic topology and homotopy theory, but some basic knowledge of the language and fundamental concepts of homological algebra (chain complexes, chain homotopies, ...) will be a necessary prerequisite for the second and third parts of the course.