Programme du Séminaire MAT 14-16 juin 2012

Salle de conférence du laboratoire J-A. Dieudonné

Résumés des exposés plus bas

Jeudi
Vendredi Samedi
8h45 Accueil
9h15 - 10h15
P. Cartier
Motivic Galois groups and polylogarithms - 1

café
10h35 - 11h35
St. Baseilhac
Quantum hyperbolic invariants and the volume conjecture


11h40 - 12h40
H. Gangl
Flavours of polylogarithms - 1


 
9h15 - 10h15
H. Gangl
Flavours of polylogarithms - 2


café
10h35 - 11h35
D. Broadhurst
Polylogarithms from Schlaefli to Schwinger

11h40 - 12h40
P. Cartier

Motivic Galois groups and polylogarithms - 2
 
9h15 - 10h15
P. Cartier
Motivic Galois groups and polylogarithms - 3

café
10h35 - 11h35
H. Gangl
Flavours of polylogarithms - 3



11h40 - 12h40
H. Furusho
p-adic multiple polylogarithms


13h - 14h  Repas AURAIN 13h - 14h  Repas AURAIN 13h - 15h  Repas Union
14h30 - 15h30
J. Dupont
An elementary proof of some dilogarithm identities

café
16h00 - 17h00
K. Hutchinson
The homology of the special linear group and pre-Bloch groups of fields



17h05 - 18h05
R. de Jeu
On K2 of curves




14h30 - 15h30
H. Esnault
Fundamental groups in char. 0 and p>0
café
16h00 - 17h00
Z. Wojtkowiak
l-adic polylogarithms, l-adic iterated integrals, distribution relations and p-adic zeta function


17h05 - 18h05
P. Elbaz-Vincent
De la géométrie à la géométrie - un panorama des travaux de Jean-Louis Cathelineau



19h30 Dîner de la conférence




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 Uqsl2. 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 sl2-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 K2(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 π2. 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 (Ar, Ar') 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 K3 of F, and this in turn is a quotient of the group H3(SL(2,F),Z). Up to some possible 2-torsion, the kernel of the map H3(SL(2,F),Z) to K3ind(F) coincides with the kernel of the stabilization map from H3(SL(2,F),Z) to H3(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 (Zl)r with respect to certain measures. And we get as expected a byproduct p-adic zeta function of Kubota-Leopoldt (here p=l).