The course was held on Wednesday afternoons, from January until April, 2012.
The schedule was 14H30-17H45.
**The course begun on January 18, 2012, at 14H30.**

The course was completed by an informal seminar. This seminar was held on Wednesdays at 11H30-12H30, and started on the week following the first course (on January 25).

- January 18:
**Symmetric and braided structures**-- video recording of the lecture - The definition of symmetric groups by generators and relations. Braid groups.
- Symmetric monoidal categories. Braided monoidal categories.
- The mathematical objectives of the course.
- January 25:
**Introduction to operads**-- video recording of the lecture - Introduction to operads. Definitions and fundamental examples (associative, commutative).
- February 1:
**The operad of trees**-- video recording of the lecture - The definition of a tree structure
- The operadic composition of trees
- The magma operads
- February 8:
**Little discs operads, the Boardman-Vogt construction, and the modeling of homotopy structures**-- video recording of the lecture - Homotopy groups and loop spaces.
- Little discs operads. Definition.
- The recognition of iterated loop spaces (statement of the result).
- The Boardman-Vogt construction for the permutation operad.
- February 15:
**Fundamental groupoids of configuration spaces**-- video recording of the lecture - The action of the Boardman-Vogt construction on loop spaces.
- The interpretation of braid groups as fundamental groups of configuration spaces.
- Fundamental groupoids. Operads in groupoids.
- Fundamental groupoids of configuration spaces.
- February 22:
**Fundamental groupoids and the colored braid operad**-- video recording of the lecture - The fundamental groupoids of the little 2-discs operad. The operad of colored braids.
- The colored braid operad governs strict braided monoidal categories.
- February 29, March 7:
**Winter holidays**. - March 14:
**Hopf algebras**-- video recording of the lecture - Fundamental definitions (algebras, coalgebras, and Hopf algebras).
- Symmetric and tensor algebras.
- March 21:
**The structure of Hopf algebras and completions**-- video recording of the lecture - Lie algebras and enveloping algebras.
- Structure theorems (Poincaré-Birkhoff-Witt and Milnor-Moore).
- The completion of Hopf algebras
- March 28:
**Complete Hopf algebras and groups**-- video recording of the lecture - The structure of complete algebras.
- Group like elements.
- The completion of group algebras.
- April 4:
**The Malcev completion of operads in groupoids**-- video recording of the lecture - Summary of the Malcev completion process for groups.
- Groupe like elements as exponentials.
- Extension of the Malcev completion to groupoids.
- Applications to operads in groupoids.
- Short definition of the Grothendieck-Teichmüller group GT(Q).
- April 11:
**The Grothendieck-Teichmüller group GT(Q)**-- video recording of the lecture - The parenthesized braid operad.
- The definition of GT(Q) as the group of automorphisms of the operad of parenthesized braids.
- Drinfeld's explicit definition of GT(Q).
- April 18:
**The Grothendieck-Teichmüller group is the group of homotopy automorphisms of the little 2-disc operad over Q**-- video recording of the lecture - Recollections and complements on the explicit definition of the Grothendieck-Teichmüller group GT(Q).
- Classifying spaces of categories and of operads in groupoids
- The definition of homotopy automorphisms on the little 2-discs operad from the Grothendieck-Teichmüller group. Theorem:
*The Grothendieck-Teichmüller group is the group of homotopy automorphisms of the little 2-disc operad over Q*. Interpretation.

21 septembre 2011 / |