Benoit Fresse
Advanced course "Operads, graph complexes and applications" (scheduled on JanuaryMarch 2025)
Master de mathématiques 20242025  Université de Lille
Introduction
The purpose of this course is to introduce students to current research questions in the domains of algebra and topology.
The first concept that will be considered in the course is the notion of an operad. To explain the idea of this notion, an operad is an object that models the structure formed by composites of operations that govern an algebra structure. The usual categories of algebras, e.g. the category of associative algebras, the category of associative and commutative algebras, the category of Lie algebras, … can be associated to operads. There is a notion of presentation by generators and relations for operads, which reflects the classical definition of the structure of an associative algebra, of a commutative algebra, and of a Lie algebra … in terms of a generating operation (a product, a Lie bracket, …) that satisfies a set of relations. In this context, one of the main devices of the theory of operads is the theory of the Koszul duality, which is used to compute the syzygies (the secondary relations) associated to such presentations.
The definition of the classical categories of algebras and of the associated operads will be studied in the first part of the course. In a second step, we will focus on the study of E_{n}operads: fundamental examples of operads, which are used to model commutativity levels that govern certain algebra structures.
Then we will explain a construction of graph complexes with the aim of computing groups of automorphisms associated to E_{n}operads. To conclude the course, we will outline applications of graph complexes for the operadic interpretation of the GrothendieckTeichmüller group (a group, defined by using ideas of the Grothendieck program in Galois theory, and which models universal symmetries of quantum groups), or the applications of graph complexes for the computation of the homotopy of knot spaces/of embedding spaces of manifolds over the rationals.
Prerequisites: fundamental notions of algebra; fundamental notions of algebraic topology (homology and homotopy)
Synopsis
 The classical categories of algebras and the associated operads
 The Koszul duality of algebras. The Koszul duality of operads
 Models of E_{n}operads in topology, algebra, and category theory
 Graph complexes and homotopy automorphisms of E_{n}operads
 Some applications: the operadic interpretation of the GrothendieckTeichmüller group; the homotopy of knot spaces/of embedding spaces of manifolds
References
 B. Fresse, Homotopy of operads and GrothendieckTeichmüller groups. Mathematical surveys 272, American Mathematical Society, 2017.
 J.L. Loday, B. Vallette, Algebraic operads. Grundlehren der mathematischen Wissenschaften 346, SpringerVerlag, 2012.
 T. Willwacher, M. Kontsevich’s graph complex and the GrothendieckTeichmüller Lie algebra. Invent. Math. 200, no. 3, pp. 671–760 (2015).
Organization
The lessons will happen on Mondays and Wednesdays, on the slot 10h3012h30 (Central European Time), from January until the end of March 2025, with a first lecture scheduled on Wednesday, January 15, 2025, and a oneweek break from February 23 until March 2.
The lectures will be delivered in a traditional classroom format (at the Mathematics Department of the University of Lille, Cité Scientifique campus in Villeneuve d’Ascq) and, simultaneously, as online videoconferences for remote participants (for exterior students). The lessons will be recorded and the recordings will be made publicly available shortly after the lectures.
The lesson sessions (16x2h) will be completed by tutorial/seminar sessions (4x2h), which will take place at regular intervals on the Wednesday slot, and which will be devoted to recollections of notions used in the course, to exercises, to student talks, … (On the first week of the course, an exceptional tutorial session will be organized on Wednesday, January 15, after the first lecture in order to review some prerequisites of the course.) The activities proposed for the tutorials will be made available online, on the moodle site of the course, for the remote participants.
The participation to the course is free, but registration is mandatory for the access to the online activities (videoconferences of the lessons and tutorials). The enrollments will open in September 2024 (check the updates of this site at this moment).
Evaluation: For the students who intend to validate the course for a master degree, the evaluation will be based on a short talk given by the student at a seminar session, and on a final exam, which will take place after the end of the course (participation in person only). The grade obtained in the course will be given by the mean of the grades obtained in both evaluations.
Contact: benoit.fresse.fr / univlillehttps://pro.univlille.fr/benoitfresse/
Provisional schedule (to be confirmed)
Mondays 10:3012:30  Wednesdays 10:3012:30  

 January 15  Lecture 1 + Tutorials (in the afternoon) 
January 20  Lecture 2  January 22  Lecture 3 
January 27  Lecture 4  January 29  Tutorials/Seminar 
February 3  Lecture 5  February 5  Lecture 6 
February 10  Lecture 7  February 12  Lecture 8 
February 17  Lecture 9  February 19  Tutorials/Seminar or Break 
February 24  Break  February 26  Break 
March 3  Lecture 10  March 5  Lecture 11 
March 10  Lecture 12  March 12  Lecture 13 
March 17  Lecture 14  March 19  Lecture 15 
March 24  Lecture 16  March 26  Tutorials/Seminar or Break 
March 31  Exam or Tutorials/Seminar  April 2  End of the course or Exam 