ETHZ-UZh-Sbg Arithmetic and Geometry Research Group

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Workshop "Multiple Zeta-Values"

This workshop is organized by J. Ayoub, S.O. Gorchinskiy, G. Wüstholz (Chair) within the ProDoc module. It takes place from September 2-7, 2012 at the Hotel Böglerhof in Alpbach/Tyrol, Austria.

The goal of this workshop is to study the motivic approach to multiple zeta values (MZV's) including the recent advances due to Francis Brown. Basically, we will follow the presentation of Deligne [1]. We also refer to the foundational paper [2] for many preliminary facts concerning the motivic fundamental group of the projective line without three points and the appearence of MZV's in this context.
The program consists of 13 talks by 90 minutes. Each talk is provided with a detailed plan and description in the program below. All statements are expected to be proved or at least explained if the converse is not mentioned. For most of them, hints and ideas can be found in the program as well as in the cited papers. The speakers of the workshop are urged to start preparing their talks as soon as possible. The preparation assumes a creative work with making sort of exercises and compiling various sources and melting them under common notation and concepts of the workshop. The division of talks is tentative, and the speakers of adjacent talks can exchange some of the material to be covered. The difficulty of most talks is rather conecptual than technical. In case help is needed, everbody is welcome to contact: Joseph Ayoub or Sergey Gorchinskiy.
We assume familiarity with the following notions (some if this will be recalled in a pre-workshop, which is scheduled for Tuesday, 28.8., 10am-1pm and Wednesday, 29.8., 3-7pm in HG G19.2; the talks will be given by Brent Doran and Sergey Gorschinskiy): fundamental groups, path spaces, filtered vector spaces, Betti cohomology, (algebraic) de Rham cohomology, mixed Hodge structures, periods, linear algebraic groups (mostly unipotent), Lie algebras, Hopf algebras, dg-algebras, total complexes of bicomplexes. Some familiarity with simplicial and cosimplicial objects, sheaves and local systems will be certainly useful.

Joseph Ayoub
Utsav Choudhury
Brent Doran
Bledar Fazlija
Andrea Ferraguti
Clemens Fuchs (Salzburg)
Javier Fresán (Paris)
Martin Gallauer
Sergey Gorchinskiy (Moscow)
Fritz Hörmann (Freiburg)
Mario Huicochea
Rafael von Känel (Paris)
Alban Krauth
Andrew Kresch
Lars Kühne
Stefan Müller-Stach (Mainz)
Roland Paulin
Simon Pepin Lehalleur
Maximilian Preisinger (Mainz)
Sergey Rybakov (Moscow)
Sonia Samol (Mainz)
Jonathan Skowera
Petra Tadic (Zagreb)
Alberto Vezzani
Konrad Völkel (Freiburg)
Thomas Weissschuh (Mainz)
Gisbert Wüstholz
Jun Yu

The program will start on Sunday evening, the first talk is scheduled for 17:15 - 18:45. For the detailed program of the planned talks see here (version from September 16, 2012): Program-2012

Sunday, 02.09.2012: Arrival day - Introduction
Introduction to multiple zeta values (Mario Huicochea and Roland Paulin)
Monday: Chen's theorem
Iterated integrals (Fritz Hörmann), Pro-unipotent completion (Alberto Vezzani), Bar complex (Thomas Weissschuh)
Tuesday: Hodge structure on the fundamental group
Proof of Chen's theorem (Sergey Rybakov), Mixed Hodge structure of the fundamental group (Javier Fresán), The case of P^1 withouth several points (Rafael von Känel)
Informal discussion, excursions, hiking tours
Thursday: Motivic structure on the fundamental group
Tannakian categories (Konrad Völkel), Mixed Tate motives over Z (Martin Gallauer), Motivic fundamental group (Simon Pepin Lehalleur)
Friday, 07.09.2012: Brown's proof
Zagier's theorem (Lars Kühne), The Proof. Part 1 and Part 2 (Sergey Gorchinskiy)

09:00 - 10:30: First talk
10:30 - 11:00: Coffee break
11:00 - 12:30: Second talk
12:45: Lunch
13:45 - 15:15: Third talk
15:30 - 18:00: Time for informal discussion
19:00: Dinner

Lecture Notes:
Notes taken by J. Skowera during the week are available here.

[1] P. Deligne: Multizetas, d'apres Francis Brown, Seminaire Bourbaki, Janvier 2012, 64eme annee, 2011-2012, Exp. 1048; PDF
[2] P. Deligne, A. Goncharov: Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Ecole Norm. Sup. 38 (2005), 1-56; arXiv:math/0302267; PDF
[3] D. Zagier: Evaluation of the multiple zeta value zeta(2,...,2,3,2,...,2), Ann. Math. 175 (2012), 977-1000; PDF
[4] P. Deligne: Categories tannakiennes, Grothendieck Festschrift Vol. II, Progr. Math. 87 (1990), 111-195
[5] A. Lubotsky, A. Magid: Cohomology of unipotent and prounipotent groups, J. of Algebra 74 (1982), 76-95
[6] C. Peters, J. Steenbrink: Mixed Hodge Structures, Springer, 2008
More references can be found in the detailed program (by J. Ayoub and S. Gorchinskiy): PDF

Additional Material:
M. Levine: Six Lectures on Motives; PDF
Impressum    21.03.2016