"Operads and Grothendieck-Teichmüller Groups" by B. Fresse (Lille 1)
Planning and synopsis of the lectures
The course will be held on Wednesday afternoons, from January until April, 2012.
The planned schedule is 14H30-17H45.
The course will begin on January 18, 2012, at 14H30.
The course will be completed by an informal seminar. This seminar will be held on Wednesdays at 11H30-12H30, and will start 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.