Pierre Cartier - cours : Motivic Galois groups and
polylogarithms
Abstract : After a quick review of ordinary Galois theory , I shall
introduce in an informal way the ideas of motives and motivic Galois
group. Then shall come the important distinction between pure and mixed
motives , with a special emphasis on mixed Tate motive . The important
problem was to connect the polylogarithms (and especially the
multizetas) with the periods of the mixed Tate motives . After much
preliminary work by <Deligne , Goncharov and Zagier , F. Brown>
was able to disentangle this connection .

In the second lecture , I plan to describe the motivic fundamental
group of a reasonable algebraic variety , through its main realizations
, namely Betti and de Rham , and also the l-adic realization through
Tate modules . The important notion shall be that of the fundamental
GROUPOID of a space . This brings us in close connection with the
Galois theory of differential equations .

In the third lecture , I shall give a more explicit description
of the motivic Galois group and its action on the fundamental
groupoid of the "sphere minus three points" . It uses the notion of
higher Chow groups introduced by S. Bloch and homotopy of schemes
. I shall describe the method of Marc Levine , and report on the recent
results obtained by I. Soudères (Essen Universität) .

Herbert Gangl - cours : Flavours of polylogarithms
Cours I. The classical case

We first focus on mostly elementary properties of classical
polylogarithms (e.g. differential structure, special values, ladder
relations, functional equations), highlighting them in particular in
the case of the dilogarithm. Then we introduce their "symbols", i.e.
certain algebraic avatars, leading to the famous Bloch group and its
generalisations, and indicate their connection to the algebraic
K-theory of number fields, leading to Zagier's and Lichtenbaum's
Conjectures with a number of explicit examples.

Cours II. The infinitesimal and finite cases

Cathelineau introduced the infinitesimal polylogarithms and a tangent
complex to the Bloch-Suslin complex (for the dilogarithm) and to
Goncharov's motivic complexes (in the higher cases). Meanwhile,
Kontsevich considered a finite variant, dealing with the "1
1/2-logarithm" while leaving equations for higher cases as an open
question. Our task for this lecture is to try to outline a surprising
analogy between the two--seemingly unconnected--topics and how, with
Elbaz-Vincent and using work of Besser relating p-adic and finite
polylogarithms, we were able to relate them and to settle the question
in question.

Cours III. The multiple case

When Goncharov proved Zagier's Conjecture for the trilogarithm ("weight
3"), he also envisioned how to proceed for higher weight cases. For
this, he introduced multiple polylogarithms and, more recently, their
"symbols" and anticipated how an improved understanding of the
interplay between these would help in particular in attacking the
weight 4 case. We indicate how to attach those symbols using polygonal
combinatorics and at the same time relate them to interesting algebraic
cycles (joint with Goncharov and Levin). Moreover, in the spirit of
Goncharov, Spradlin et al. we exploit the symbols for getting a good
grip at the special class of harmonic polylogarithms which occur very
often in Feynman integral calculations (joint work with Duhr and
Rhodes), for finding functional equations in weight 4, and to solve an
old question of Goncharov.

Stéphane Baseilhac - Quantum hyperbolic invariants and the volume conjecture
Abstract : I will introduce q-deformations of the dilogarithm
functions derived from the "generic" representation theory of the
quantum group U

_{q}sl

_{2}. A mainstream of conjectures in quantum
topology relates the asymptotic behaviour of the partition functions of
a topological quantum field theory built out from them, and some values
of the sl

_{2}-Chern-Simons invariants.

David Broadhurst - Polylogarithms from Schlaefli to Schwinger
Abstract : Schlaefli showed how to evaluate volumes in hyperbolic
3-space in terms of dilogarithms. Thurston associated such dilogs to
knot and links. Many classes of Feynman diagram evaluate to dilogs or
higher polylogs, with connections to algebraic geometry revealed by
integration over Schwinger parameters. I shall give examples of such
connections, including recent evaluations in terms of the L-functions
of modular forms.

Rob de Jeu - On K_2 of curves
Abstract: Let C be a curve over a field k and F=k(C) its function
field. We discuss some subgroups of K

_{2}(F) that contain the kernel of
the tame symbol. In computer experiments (performed jointly with Bogdan
Banu) some of those appear to admit an explicit description in terms of
generators and relations, which are themselves subject to a 5-term
relation.

Johan Dupont -
An elementary proof of some dilogarithm identities
Abstract: The physicist A. B. Zamolodchikov conjectured in 1991 that a
socalled Y-system associated to a pair of Dynkin diagrams is periodic
and that the sum over a period of the dilogaritm evaluated on the
Y-system is a certain multiple of π

^{2}. This conjecture was recently
proved in general by B. Keller using representations of quivers and
derived categories. An elementary proof of the periodicity for the pair
(A

_{r}, A

_{r'}) was given in 2007 by A. Yu. Volkov, and we show how this
also imply the associated dilogarithm identities in this case.

Hélène Esnault - Fundamental groups in char. 0 and p>0
Abstract : We show analogies between stratifications in char. 0 and p.

Hidekazu
Furusho - p-adic multiple polylogarithms
Abstract: I will introduce p-adic version of multiple polylogarithm and
explain its basic properties, especially on its special values.

Kevin
Hutchinson - The homology of the special linear group and pre-Bloch
groups of fields
Abstract: The pre-Bloch group, P(F), of a field F is a group presented
by generators and relations which derive from the five-term functional
equation of the dilogarithm. The Bloch group is a subgroup of P(F) which, by a result of
Suslin, is naturally a quotient of the indecomposable K

_{3} of F,
and this in turn is a quotient of the group H

_{3}(SL(2,F),Z). Up to some
possible 2-torsion, the kernel of the map H

_{3}(SL(2,F),Z) to K

_{3}^{ind}(F)
coincides with the kernel of the stabilization map from H

_{3}(SL(2,F),Z)
to H

_{3}(SL(3,F),Z). We will describe how, for fields with valuations, lower bounds - and even exact computations - of
this latter kernel can be expressed as direct sums of pre-Bloch groups
of residue fields.

Zdzislaw Wojtkowiak - l-adic polylogarithms, l-adic iterated integrals, distribution relations and p-adic zeta function
Abstract : We introduce analogues of complex polylogarithms and
iterated integrals in Galois framework. We show that they have the same
functional equations as classical complex ones. We express them as
integrals over (Z

_{l})

^{r} with respect to certain measures. And we get as
expected a byproduct p-adic zeta function of Kubota-Leopoldt (here p=l).