We define complexes analogous to Goncharov’s complexes for the K–theory of discrete valuation rings of characteristic zero. Under suitable assumptions in K–theory, there is a map from the cohomology of those complexes to the K–theory of the ring. In case the ring is the localization of the ring of integers in a number field, there are no assumptions necessary. We compute the composition of our ...

Historically the polylogarithm has attracted specialists and non specialists alike with its lovely evaluations Much the same can be said for Euler sums or multiple harmonic sums which within the past decade have arisen in combinatorics knot theory and high energy physics More recently we have been forced to consider multidimensional extensions encompassing the classical polylogarithm Euler sums...

Two-loop MHV amplitudes in planar ${\cal N} = 4$ supersymmetric Yang Mills theory are known to exhibit many intriguing forms of cluster-algebraic structure. We leverage this structure upgrade the symbols eight- and nine-particle complete analytic functions. This is done by systematically projecting onto components these that take different functional forms, matching each component an ansatz mul...

In these notes, we give an overview of our paper [BKT] which gives an explicit description of the de Rham and p-adic realizations of the elliptic polylogarithm, for a general elliptic curve defined over a subfield of C in the de Rham case and for a CM elliptic curve defined over its field of complex multiplication and with good reduction at the primes above p ≥ 5 in the p-adic case. As explaine...

A bstract We derive the connected tree-level part of 4-point holographic correlators in AdS 3 × S $$ \mathcal{M} M (where is T 4 or K 3) involving two multi-trace and single-trace operators. These are obtained by studying a heavy-heavy-light-light correlation function formal limit where heavy operators become...

We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral representation for the general multiple polylogarithm, and develop a q-analogue of the shuffle product.

In this paper, we review a number of basic results about so-called Hopf algebras. We begin by giving a historical account of the results obtained in the 1930's and 1940's about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to Hopf, Samelson, Leray and Borel. The main part of this paper is a thorough analysis of the relations betwee...

Cohen, Lewin and Zagier found four ladders that entail the polylogarithms Lin(α −k 1 ) := ∑ r>0 α −kr 1 /r n at order n = 16, with indices k ≤ 360, and α1 being the smallest known Salem number, i.e. the larger real root of Lehmer’s celebrated polynomial α + α − α − α − α − α − α + α + 1, with the smallest known non-trivial Mahler measure. By adjoining the index k = 630, we generate a fifth ladd...

In this paper, we introduce a new type of degenerate Genocchi polynomials and numbers, which are called poly-Genocchi by using the polylogarithm function, derive several properties these systematically. Then, also consider unipoly-Genocchi attached to an arithmetic investigate some identities those polynomials. particular, give explicit expressions unipoly related special numbers