Operads are algebraic structures formed by collections
of operations *p = p(x _{1},...,x_{n})*,
where the number of variables

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.

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:

- give brief curriculum indications,
- specify which lectures you aim to follow, briefly explain your motivations,
- provide information about your needs and other possible funding sources you may request.

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 E_{2} operads,
one of the structures introduced at the origin of operad theory in order to understand the internal structures of loop spaces.

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 E_{n}-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.

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.**

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

- January 18: Symmetric and braided structures -- video recording of the lecture
- January 25: Introduction to operads -- video recording of the lecture
- February 1: The operad of trees -- video recording of the lecture
- February 8: Little discs operads, the Boardman-Vogt construction, and the modeling of homotopy structures -- video recording of the lecture
- February 15: Fundamental groupoids of configuration spaces -- video recording of the lecture
- February 22: Fundamental groupoids and the colored braid operad -- video recording of the lecture
- February 29, March 7: Winter holidays
- March 14: Hopf algebras -- video recording of the lecture
- March 21: The structure of Hopf algebras and completions -- video recording of the lecture
- March 28: Complete Hopf algebras and groups -- video recording of the lecture
- April 4: The Malcev completion of operads in groupoids -- video recording of the lecture
- April 11: The Grothendieck-Teichmüller group GT(Q) -- video recording of the lecture
- 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

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

The preprint

- B. Fresse:
*Operads and Grothendieck-Teichmüller groups*. Preprint hal-00656333, 2012, 200 pages.

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).

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

The preceding Master course focuses on internal structures of E_{2} operads.
The overall purpose of the following Doctoral course is to explain applications of E_{n} operads in the domain of algebraic topology.
The E_{n} 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 E_{n}
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.

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.

**May 15: Introduction to operads and E**_{n}operads, by B. Fresse and M. Livernet.*The purpose of this first lecture is to give a short introduction (or recollections) on the basic definitions of operad theory, and ideas of homotopical algebra.*- 13H-15H: Fundamental ideas of homotopical algebra. Hochschild homology theory as a Quillen derived functor.
- 15H30-17H30: Operads. Definitions and basic examples. The E
_{n}operads of little n-discs.

**May 22: Factorization homology, by J. Francis.***The purpose of this talk is to explain the definition of factorization homology, a homology theory for manifolds generalizing singular homology, mapping space homology and Hochschild homology together.*- 13H-15H: The Eilenberg-Steenrod axioms for factorization homology of topological manifolds.
- 15H30-17H30: Relationship with Hochschild homology and the cohomology of mapping spaces.
- Nonabelian Poincare duality.

**May 29: Homotopical algebra and E**_{n}homology, by C. Berger and M. Livernet.*In this lecture, we will explain some algebraic counterpart of the factorization homology theory, introduced in the previous lecture. Thus, these second and third lectures will be somehow related. Nevertheless, we will make each lecture self-contained, so that these second and third lectures can be followed independently.*- 13H-15H: E
_{n}operads. Models. Cell structures and recognition theorems for E_{n}operads. - 15H30-17H30: The definition of E
_{n}homology as a derived functor. Determination by iterated bar complexes.

- 13H-15H: E
**June 5: Miscellaneous applications and outlook, by C. Berger and B. Fresse.***In this concluding lecture, we provide an introduction to far reaching applications of E*_{n}operads. The program will include:- 13H-15H: The Deligne conjecture and its generalizations (asserting that the derived center of an algebra over an E
_{n}operad is an algebra over an E_{n+1}operad). - 15H30-17H30: Formality of E
_{2}operads and deformation-quantifization, based on the Deligne conjecture. Connection with Grothendieck-Teichmüller theory and graph complexes.

- 13H-15H: The Deligne conjecture and its generalizations (asserting that the derived center of an algebra over an E

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

*Good general references for the background material include:*- G.E. Bredon, Topology and Geometry, Graduate Texts in Mathematics 139, Springer-Verlag, 1993.
- J.-L. Loday, Cyclic Homology, Grundlehren der mathematischen Wissenschaften 301, Springer-Verlag, 1998.
- C. Weibel, An introduction to homological algebra, Cambridge Univ. Press, 1994.

*For further studies, the May 15 lecture covered material excerpted from:*- J. Boardman, R. Vogt, Homotopy invariant algebraic structures on topological spaces, Lectures Notes in Mathematics, Vol. 347. Springer-Verlag, 1973.
- J. Stasheff, Homotopy associativity of $H$-spaces. I, II. Trans. Amer. Math. Soc. 108 (1963), 275-292.
- M. Markl, S. Shnider, J. Stasheff, Operads in algebra, topology and physics. Mathematical Surveys and Monographs 96, American Mathematical Society, 2002.
- J.P. May, The geometry of iterated loop spaces. Lectures Notes in Mathematics, Vol. 271. Springer-Verlag, 1972.

*The May 22 lecture covered material from:*- J. Francis: The tangent complex and Hochschild cohomology of E_n-rings, preprint (2011). Section 3, "Factorization Homology and E_n-Hochschild Theories"
- J. Francis: Factorization homology of topological manifolds, preprint (to appear).
- J. Lurie: Higher Algebra, preprint (2012). Chapter 5, "Little Cubes and Factorizable Sheaves"
- J. Lurie: On the Classification of Topological Field Theories, preprint (2009). Section 4.1, "Topological Chiral Homology"

*The May 29 lecture covered material excerpted from:*- M. Batanin, C. Berger, The lattice path operad and Hochschild cochains, Contemp. Math. 504, Amer. Math. Soc. (2009), 23-59.
- C. Berger, Opérades cellulaires et espaces de lacets itérés, Ann. Inst. Fourier 46 (1996), 1125-1157.
- B. Fresse, Koszul duality of E
_{n}-operads, Selecta Math. (N.S.) 17 (2011), 363–434, and Iterated bar complexes of E-infinity algebras and homology theories. Algebr. Geom. Topol. 11 (2011), 747–838. - M. Livernet, B. Richter, An interpretation of E
_{n}-homology as functor homology. Math. Z. 269 (2011), 193–219.

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.

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

21 septembre 2011 / |