• Spencer Bloch

    University of Chicago, USA
  • Guido Kings

    Universität Regensburg, Germany
  • Jörg Wildeshaus

    Université de Paris XIII, Villetaneuse, France
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The kk-th polylogarithm function is defined on z<1|z|<1 by

Lik(z)=n1znnk.Li_k(z) = \sum_{n\geq 1} \frac{z^n}{n^k}.

In the past 25 years or so, polylogarithms have appeared in many different areas of Mathematics. The following list is taken, for the most part, from \cite{Oe}: volumes of polytopes in spherical and hyperbolic geometry, volumes of hyperbolic manifolds of dimension 3, combinatorial description of characteristic classes, special values of zeta functions, geometry of configurations of points in P1\mathbb P^1, cohomology of GLn(C)GL_n(\mathbb C), calculation of Green's functions associated to perturbation expansions in quantum field theory, Chen iterated integrals, regulators in algebraic KK-theory, differential equations with nilpotent monodromy, Hilbert's problem on cutting and pasting, nilpotent completion of π1(P1{0,1,})\pi_1(\mathbb P^1-\{0,1,\infty\}), Bethe's Ansatz in thermodynamics, and combinatorial problems in quantum field theory.\\ Of course, these problems are not all unrelated. One common thread is that values of polylogarithms appear naturally as periods of certain mixed Hodge structures associated to mixed Tate motives over cyclotomic fields. How these periods are related to special values of LL-functions is a part of the Beilinson con\-jec\-tures, which were discussed in a previous Arbeitsgemeinschaft. Since that time, the general picture has clarified. A number of talks are devoted to aspects of this more general philosophy (talk 2-5, 9, 10 13-17). The pp-adic aspects of the theory \mbox{have} been studied and will be explained in the eleventh talk. In addition, a vast generalization, multiple polylogarithms of the form

Lis1,,sk:=n1>>nk1z1n1zknkn1s1nksk,Li_{s_1,\dotsc,s_k} := \sum_{n_1>\ldots > n_k \geq 1} \frac{z_1^{n_1}\cdots z_k^{n_k}}{n_1^{s_1}\cdots n_k^{s_k}},

have come to play a role as presented in the fourth and twelfth talk. \\ The connection between volumes of hyperbolic manifolds and polylogarithms is described in the sixth talk. The seventh talk introduces higher torsion and explains why polylogarithms occur in that setting.\\ Polylogarithms play a role in physics. The eighth talk explains how zeta values, polylogarithms, and multiple polylogarithms appear in calculations of perturbative expansions in quantum field theory. \\ The final talks 13-17 return to the relationship with special values of LL-functions. The thirteenth and fourteenth talk are devoted to the Zagier conjectures for number fields. The fifteenth talk concerns the elliptic polylogarithm \mbox{sheaves} and Zagier's conjecture for elliptic motives. Finally, talk 16 and 17 describe how polylogarithms relate to Euler systems and to the Bloch-Kato conjectures on special values of LL-functions.\\ The {\it Arbeitsgemeinschaft} was organized by Spencer Bloch (University of Chicago), Guido Kings (Universit\"{a}t Regensburg) and J\"{o}rg Wildeshaus (Universit\'{e} Paris~13). It was held October 3rd -- October 9th, 2004 with 46 participants.

Cite this article

Spencer Bloch, Guido Kings, Jörg Wildeshaus, Polylogarithms. Oberwolfach Rep. 1 (2004), no. 4, pp. 2539–2590

DOI 10.4171/OWR/2004/48