Isaac Newton Institute for Mathematical Sciences

Grothendieck-Teichmüller Groups, Deformation and Operads

3 January - 26 April 2013

Organisers: Herbert Gangl (Durham), John Jones (Warwick), Pierre Lochak (Paris VI), Bruno Vallette (Nice) and Nick Woodhouse (Oxford)

Scientific Advisors: Giovanni Felder (ETH Zürich), and Alexander Goncharov (Yale)

Lecture Series

The organisers of the GDO programme are planning a series of short courses for students in January, February and March 2013:

There is very limited accommodation available at the Institute - please refer to the list of B&Bs on the Smooth Hound Hotel Guide.

Operads (Bruno Vallette)

Abstract: An operad is an algebraic device which encodes a type of algebras. Instead of studying the properties of a particular algebra, we focus on the universal operations that can be performed on the elements of any algebra of a given type. The information contained in an operad consists in these operations and all the ways of composing them.

The notion of an operad is a universal tool in mathematics and operadic theorems have been applied to prove results in many different fields.

The aim of this course is, first, to provide an introduction to algebraic operads, second, to give a conceptual treatment of Koszul duality, and, third, to give applications to homotopical algebra.

References: Algebraic Operads, Jean-Louis Loday and Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012) [Available for free at]

Algebra+Homotopy=Operad, Bruno Vallette, elementary introduction to operads and their applications in homotopical algebra [Available for free at]

Exercise Sheets:
Operads Exercise Sheet 1
Operads Exercise Sheet 2
Operads Exercise Sheet 3

Wednesday 16 January
10:30-12:00 Lecture 1: Operads: definition, examples and first properties
14:00-15:30 Lecture 2: Koszul duality theory for algebras
16:00-17:30 Exercise session
Wednesday 23 January
10:30-12:00 Lecture 1: Koszul duality theory for operads
14:00-15:30 Lecture 2: Methods for Koszul duality
16:00-17:30 Exercise session
Wednesday 30 January
10:30-12:00 Lecture 1: Homotopy algebras
14:00-15:30 Lecture 2: (Co)homology theories and deformation theory
16:00-17:30 Exercise session

Deformation theory (John Jones, Damien Calaque)

Abstract: The purpose of this course is to give an introduction to deformation theory. The central point is the deformation-quantization of Poisson manifolds (Kontsevich-Tamarkin) in the sense that the course will cover background material and ideas leading to the proof and also subsequent developments following on from the proof. In particular, it will cover the quantization of Lie bialgebras (Etingof-Kazhdan) and also Kontsevich's conjecture about the action of the Grothendieck-Teichmuller group on the space of deformation-quantizations of Poisson manifolds.

Tuesday 5 February
10:30-12:00 Lecture 1: The Hochschild cochain complex, operads, and the Deligne conjecture I
14:00-15:30 Lecture 2: The Hochschild cochain complex, operads, and the Deligne conjecture II
16:00-17:30 Exercise session
Tuesday 12 February
10:30-12:00 Lecture 1: Formality, obstructions to formality, and a key argument in the proof of the formality conjecture I
14:00-15:30 Lecture 2: Formality, obstructions to formality, and a key argument in the proof of the formality conjecture II
16:00-17:30 Exercise session
Tuesday 19 February
10:30-12:00 Lecture 1: Etingof - Kazhdan quantization of Lie bialgebras I
14:00-15:30 Lecture 2: Etingof - Kazhdan quantization of Lie bialgebras II
16:00-17:30 Exercise session
Tuesday 26 February
10:30-12:00 Lecture 1: From associators to the Kashiwara-Vergne conjecture I (after Alekseev-Torossian)
14:00-15:30 Lecture 2: From associators to the Kashiwara-Vergne conjecture II (after Alekseev-Enriquez-Torossian)
16:00-17:30 Exercise session

Grothendieck-Teichmüller groups (Pierre Lochak, Leila Schneps)

Abstract: This course will be organized roughly after the main ingredients and various versions of this theory, still very much in the making.

We will first review the ‘classical’ ([SGA1]) arithmetic (outer) action of the Galois group on the étale (resp. pro-l etc.) fundamental groups of schemes (and DM-stacks).

This will lead naturally to an introduction of the ideas contained in Grothendieck’s ``Esquisse d’un programme'', a very inspiring and still quite relevant piece of mathematical daydreaming.

We will then discuss the original ideas of Y. Ihara, focussing on the thrice punctured projective line and underlining its ubiquity and universality. This will also provide an opportunity, via P. Deligne’s paper devoted to that object, to sketch the connection with mixed Tate motives, MZV's etc.

Then we will move to Drinfeld's famous 1989 paper which in particular first defined the (genus 0) Grothendieck-Teichmüller group as a mathematical object, from the point of view of ‘universal’ deformation theory.

We will complete this tour with some indications on what is known in the case of higher genus.

References: Rather than listing references here, we suggest that the reader take a look at the reference list of the paper entitled Open problems in Grothendieck-Teichmüller theory (by P. Lochak and L. Schneps.; Proc. Symposia in Pure Math. 74, B. Farb editor, AMS Publ., 2006; available on both homepages of the authors). She/he will surely find there a way into the theory tailored to her/his needs and tastes.

Let us stress that all ‘pioneering’ papers (especially [G1], [Dr], [I1] and [De] there, in rough chronological order) which are now about a quarter of a century old, are still very much worth reading. Indeed in some sense they still delineate the contours of the theory in its various versions.

Wednesday 6 February
10:30-12:00 Lecture 1: A brief introduction to the etale fundamental group and to the Galois action
14:00-15:30 Lecture 2: Profinite braid groups; first appearance of GT
16:00-17:30 Special Topic: Belyi Theorem and dessins d'enfants
Wednesday 13 February
10:30-12:00 Lecture 1: Deforming braided categories and a useful dictionary
14:00-15:30 Lecture 2: Infinitesimal braid and the Lie algebra of (the prounipotent version of) GT
16:00-17:30 Special Topic: Associator relations satisfy double shuffle (H. Furusho)
Wednesday 20 February
10:30-12:00 Lecture 1: Moduli spaces of curves, curve complexes
14:00-15:30 Lecture 2: The Teichmueller lego
16:00-17:30 Special Topic: Origamis
Wednesday 27 February
10:30-12:00 Lecture 1: Lie algebras I
14:00-15:30 Lecture 2: Lie algebras II
16:00-17:30 Special Topic: The Grothendieck-Teichmuller Lie algebra and other animals

Multiple Zeta Values (Francis Brown, Herbert Gangl)

Abstract: This series of lectures will be an introduction to the theory of multiple zeta values, the fundamental group of the projective line minus 3 points, and the theory of mixed Tate motives. Multiple zeta values were first introduced by Euler in the 18th century and have recently undergone a huge revival of interest due to their appearance in various different branches of mathematics and physics. As a result, there are two complementary aspects: on the one hand a combinatorial approach studying the structure and relations between multiple zeta values, and on the other hand the motivic approach which connects with Grothendieck-Teichmüller theory.

The first lectures (F. Brown) will cover the theory of iterated integrals, and the Betti and de Rham fundamental group of the projective line minus 3 points. We will derive the action of the motivic Galois group on multiple zeta values and discuss the conjectural relationship with the Grothendieck-Teichmuller group.

The remaining two lectures (H. Gangl) attempt to highlight the connection with double shuffle relations: on the one hand, we want to discuss Furusho's theorem relating generalised double shuffle relations among MZV's to the ones arising from the Drinfel'd associator, on the other hand, we intend to outline how double shuffle relations give rise--already in depth 2--to a surprising connection with periods of modular forms.

Wednesday 6 March
10:30-12:00 Lecture 1: Iterated integrals and de Rham fundamental group of P^1 minus 3 points
14:00-15:30 Lecture 2: Malcev completion and Betti fundamental group of P^1 minus 3 points
16:00-17:30 Discussion session
Wednesday 13 March
10:30-12:00 Lecture 1: The Ihara action on the motivic fundamental group of P^1 minus 3 points
14:00-15:30 Lecture 2: Motivic multiple zeta values and Galois coaction
16:00-17:30 Discussion session
Wednesday 20 March
10:30-12:00 Lecture 1: Relations Among Multiple Zeta Values
14:00-15:30 Lecture 2: Double Zeta Values and Modular Forms
16:00-17:30 Discussion session