Reading Kontsevich's "Operads and Motives in Deformation Quantization", I was wondering about the state of the many conjectures concerning the Grothendieck-Teichmüller group in chapter 4. (Also, where one could read the proof of theorem 6 by M. Nori?) For example I'm interested in the action of GT on deformation quantization mentioned in ch. 4.6., and an update on the issues of chapter 5. Where one could read more about all that?
Edit: The conj.s I'd like to know more about are: That (the algebra generated by) periods of mixed Tate motives come from Drinfeld associators (p.30); That GT comes from the motivic Galois group's action on the Periods (p.30); That GT is $Aut(Chains(C_2))$ (p.31); What is the "universal map $P_{\mathbb{Z},Tate}$$\longrightarrow$$P_{\mathbb{Z},Tate}$" on p. 32?; Where can one read more about the action of GT on deformation quantization (p.32,33)?
Edit: "Theorem 6 (M. Nori)" asked about above is a sheaf version of a theorem by Beilinson. Nori's article about it with the proof (the statement in question is Basic lemma (first form)) and the reference to Beilinson's paper. An article by Morava contains many new ideas, some remarks on a motivic version of the little disk operad, and a possible algebraic topology frame for Kontsevich's ideas on motives and deformation quantization, and how that would fit to Connes' & Kreimer's Galois theory of renormalization.
Edit: Conc. "GT is $Aut(Chains(C_2))$" here and here new articles (communicated by B.V., thanks!).
Edit: Mathilde Marcolli mentions in this article, on her and Connes' work relating renormalization and a "cosmic Galois group", that Kontsevich developed a renormalization theory continuing his article above and relating "in a natural setting" to Connes'/Marcolli's motivic galois action. Her article ends with some remarks on how Beilinson's conjectures may be viewed "extremly suggestive" as something like renormalization, hinting at geometric interpretations of L-values at non-integer points. It would be great if someone knew where to read more about both issues.
Edit: Spencer Bloch's thoughts (video lecture) on motives and renormalization relate to the theme too.
Best Answer
The action of GT on deformation quantization has been developed in http://arxiv.org/abs/1009.1654 (Willwacher) and before in http://arxiv.org/abs/math/0202039 (Tamarkin).
The fact that GT is Aut(Chain(C2)) is true or not ,depending if you work in the unstable homotopy category (Fresse: http://math.univ-lille1.fr/~fresse/E2RationalAutomorphisms.html) or in the stable one (Willwacher: http://arxiv.org/abs/1009.1654).
For new developments concerning the relation between GT and the Galois group of periods of mixed Tate motives over Z, you can take a look at the recent work of Francis Brown.
Damien