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 Grothendieck-Teichmüller 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

1. The classical categories of algebras and the associated operads
2. The Koszul duality of algebras. The Koszul duality of operads
3. Models of E_n-operads in topology, algebra, and category theory
4. Graph complexes and homotopy automorphisms of E_n-operads
5. Some applications: the operadic interpretation of the Grothendieck-Teichmüller group; the homotopy of knot spaces/of embedding spaces of manifolds

References

1. B. Fresse, Homotopy of operads and Grothendieck-Teichmüller groups. Mathematical surveys 272, American Mathematical Society, 2017.
2. J.-L. Loday, B. Vallette, Algebraic operads. Grundlehren der mathematischen Wissenschaften 346, Springer-Verlag, 2012.
3. T. Willwacher, M. Kontsevich’s graph complex and the Grothendieck-Teichmüller Lie algebra. Invent. Math. 200, no. 3, pp. 671–760 (2015).

Organization

The lessons will happen on Mondays and Wednesdays, on the slot 10h30-12h30 (Central European Time), from January until the end of March 2025, with a first lecture scheduled on Wednesday, January 15, 2025, and a one-week 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 video-conferences 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 (video-conferences 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[chez]univ-lille[point].fr / https://pro.univ-lille.fr/benoit-fresse/

Provisional schedule (to be confirmed)