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Grothendieck-Teichmüller Groups, Deformation and Operads

3rd January 2013 to 26th 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)

Programme Theme

image of Borromean ring

The main goal of this programme is to bring together the two research communities working on Grothendieck-Teichmüller theory on the one hand, and on operadic deformation theory on the other hand. These themes lie at the forefront of current research in algebra, algebraic and differential geometry, number theory, topology and mathematical physics.

Grothendieck-Teichmüller theory goes back to A. Grothendieck's celebrated Esquisse d'un programme. In 1991, V. Drinfel'd formally introduced two Grothendieck-Teichmüller groups, the former one related to the absolute Galois group and the latter one related to the deformation theory of a certain algebraic structure (braided quasi-Hopf algebra). Introduced in algebraic topology 40 years ago, the notion of operad has enjoyed a renaissance in the 90's under the work of M. Kontsevich in deformation theory. Two proofs of the deformation quantization of Poisson manifolds, one by himself as well as one by D. Tamarkin, led M. Kontsevich to conjecture an action of a Grothendieck-Teichmüller group on such deformation quantizations, thereby drawing a precise relationship between the two themes.

Research in mathematics often alternates breakthroughs and periods of sedimentation of ideas. More than 10 years after those major results, the underlying theories have now been well understood and we can expect new important discoveries. Recently, some results have been proved at the intersection of Grothendieck-Teichmüller groups, deformation theory, operads and multiple zeta values. These fields of research clearly overlap and such a programme should facilitate the cross-fertilization between them.

The main topics of the programme are:

  • Operads (Koszul duality theory, homotopy algebras, moduli space of curves, Deligne conjecture). A graduate course on operads is being given by Bruno Vallette at the University of Copenhagen.
  • Deformation quantization (Lie bialgebras, Hopf algebras, Poisson manifolds, Kashiwara-Vergne conjecture, Duflo isomorphism)
  • Grothendieck-Teichmüller theory (absolute Galois group, GT groups and GT Lie algebras, Drinfel'd associators)
  • Multiple zeta values (mixed Tate motives, polylogarithms, shuffle algebra, Deligne-Ihara algebra)

The programme will include 4 weekly courses, over the course of a month, on the aforementioned topics. The courses will be mainly addressed to students and to non-specialists and will be organised in order to make them accessible to the UK community. Two workshops are also scheduled to cover more advanced results.

The programme is supported by the Clay Mathematics Institute.

Final Scientific Report: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons