"Operads and Grothendieck-Teichmüller Groups" by B. Fresse (Lille 1)

Fourth Semester Course, Master Degree Program in Mathematics of Université Lille 1, 2011-12

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.