Operads 2012

Overall introduction:

Operads are algebraic structures formed by collections of operations p = p(x1,...,xn), where the number of variables n runs over N. Operads have been introduced in topology, in the sixties, in order to understand operations controlling associativity and commutativity defects in loop compositions. Since then, operads have been used in other domains fruitfully, and it has become clear that the notion of an operad provides a good conceptual and effective device to handle multiple structures in various contexts.

We offered two courses on operads in 2012: a weekly course, at the Master degree level, from January until April 2012 in Lille, and a series of four lectures, at the Doctoral level, on May 15-22-29 and June 5, 2012, in Paris. The purpose of these courses is to explain new applications of operads. The courses are complementary, and as such, they can be followed together or independently. Both courses may also serve, at different levels, as an introduction to the theory of operads. Indeed, in each course, we review the theory from a new angle, motivated by the applications we aim to explain.

Financial supports:

Limited financial supports are available to cover accommodation and transport expenses of external students who wish to follow these courses. To request such a support, please get in touch with and:

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

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


The idea of the Grothendieck-Teichmüller group appears in Grothendieck's famous program as a geometric picture of the absolute Galois group. The version considered in this course GT(Q) has been introduced by Drinfeld in the domain of quantum algebra. The course focused on applications of the Grothendieck-Teichmüller group in the domains of algebra & topology. Our goal was to give an interpretation of GT(Q) in terms of rational automorphisms of E2 operads, one of the structures introduced at the origin of operad theory in order to understand the internal structures of loop spaces.

Research monograph in preparation "Homotopy of operads and Grothendieck-Teichmüller groups" by B. Fresse:

Part I. From operads to Grothendieck-Teichmüller groups, preprint 2012, 328+xxxviii.

Part II. Homotopy of operads and deformation complexes, to appear.

The ultimate objective of this work is to prove that the Grothendieck-Teichmüller group is the group of homotopy automorphisms of a rational completion of the little 2-discs operad. The full monograph will include two volumes.

The first volume includes: a comprehensive introduction to the fundamental concepts of operad theory; a survey chapter on little discs and En-operads; a detailed study of the connections between little 2-discs and braids; an introduction to the theory of Hopf algebras and the Malcev completion of groups; and a report on the definition of the Grothendieck-Teichmüller group from the viewpoint of algebraic operad theory. Most concepts are carefully reviewed in order to make this account accessible to a broad readership.

The second volume will include: an introduction to the methods of homotopical algebra and model categories; a study of (a cosimplicial version of) the Sullivan model for the rational homotopy of spaces; the definition of a model for the rational homotopy of operads in simplicial sets and topological spaces; the definition of deformation complexes for the study of (Hopf) cooperads; and an account on the application of these deformation complexes for the computation of homotopy automorphism spaces associated to operads, until the proof of the main result of this work asserting that the pro-unipotent Grothendieck-Teichmüller group is isomorphic to the group of homotopy automorphism classes of the rationalization of the little 2-disc operad.

Dates and schedules of the course:

The course was held on Wednesdays, at 14H30-17H45, from 18 January until 18 April 2012, building M5, room A1 (see below for practical information).
The course was completed by an informal seminar. This seminar was held on Wednesdays, at 11H30-12H30. First seminar on January 25, building M1, room Weierstrass.

Lecture planning:

(The detailed synopsis of the lectures is available on the web page http://math.univ-lille1.fr/~operads/LilleCourse2012-Synopsis.html.)


Fundamental notions of algebraic topology (fundamental groups, covering spaces).


The preprint

provides a comprehensive bibliography of the subject, and served as an overall reference for the course. This preprint was made from excerpts from a research monograph in preparation. The first volume of this monograph, which covers the content of this course, and includes several additions and corrections, is now available (see above). The above preprint itself will not be updated. The second volume of the monograph should be made available on this web-page in Spring 2013.

The multimedia service of Université Lille 1 has recorded this course, and made videos are available on lille1.tv (see above for the link to the video recording for each lecture).

Place and transportation:

The course was held at the UFR de Mathématique, Université Lille 1, campus "Cité Scientifique", Lille - Villeneuve d'Ascq (France), building M5, room A1 (see InfosPratiques.html for directions).
The campus "Cité Scientifique" is connected to the main train station of Lille and Lille downtown by an efficient, fully authomatized, subway line (allow 20mn to get the UFR from the train stations). Lille train stations are 1H far from Paris Nord by TGV, 35mn from Bruxelles by TGV, 1H20mn from London, 35mn from Lens or Valenciennes by TER, 40mn from Dunkerque by TER-GV, ... For more comprehensive information (directions, maps, ...), see InfosPratiques.html

"Operads and Homotopy Commutative Structures" by C. Berger (Nice), J. Francis (NWU), B. Fresse (Lille 1), and M. Livernet (Paris 13)

Doctoral Program of the Institut Henri Poincaré


The preceding Master course focuses on internal structures of E2 operads. The overall purpose of the following Doctoral course is to explain applications of En operads in the domain of algebraic topology. The En operads, where n = 1,2,...,∞, are the structures introduced at the origin of operad theory in order to study loop spaces. In the course, we explained that En operads provide a good device to model a scale of homotopy commutative structures, starting with fully homotopy associative, but non-commutative (n=1), and ending with fully homotopy associative and commutative (n=∞). Then we explained how such structures occur in algebra and in algebraic topology.

Dates and schedules:

The course was held on Tuesdays May 15, May 22, May 29, and June 5, 2012, at 13H-17H30 at the Institut Henri Poincaré, room 314 (lectures on May 15-22-29), and room 201 (lecture on June 5). See below for practical information concerning the Institut Henri Poincaré.

Planning and synopsis:


Fundamental notions of topology (manifolds, covering spaces), algebraic topology (fundamental groups, singular homology theory) and homological algebra (homology of chain complexes).


Place and transportation:

The course was be held at the Institut Henri Poincaré, 11 rue Pierre et Marie Curie, in Paris 5ième (France).
The Institut Henri Poincaré is located nearby the subway station "Luxembourg" of RER B, providing a fast connection with the main train stations and Paris airports. For more comprehensive information about transportations and directions, see the web-page http://www.ihp.fr/fr/ihp on the institute web-site.

Financial supports:

See above. Please make your request to before March 31 2012 if you need a support for this course.

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21 septembre 2011 / 21 September 2011