"Operads and Grothendieck-Teichmüller Groups" by B. Fresse (Lille 1)
Planning and synopsis of the lectures
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.