Multiple polylogarithms and mixed Tate motives

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Multiple polylogarithms and mixed Tate motives

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Multiple polylogarithms and mixed Tate motives

08 Mar 2001-arXiv: Algebraic Geometry-

TL;DR: The theory of multiple polylogarithm Hopf algebras from an analytic, Hodge and motivic point of view has been studied in this article, where the authors define the category of mixed Tate motives over a ring of integers in a number field.

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Abstract: We develop the theory of multiple polylogarithms from analytic, Hodge and motivic point of view. Define the category of mixed Tate motives over a ring of integers in a number field. Describe explicitly the multiple polylogarithm Hopf algebra.

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Noncommutative Geometry, Quantum Fields and Motives

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Alain Connes¹, Matilde Marcolli²•Institutions (2)

Collège de France¹, Max Planck Society²

01 Jan 2007

TL;DR: In this article, the Riemann zeta function and non-commutative spaces are studied in the context of quantum statistical mechanics and Galois symmetries, including the Weil explicit formula.

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Abstract: Quantum fields, noncommutative spaces, and motives The Riemann zeta function and noncommutative geometry Quantum statistical mechanics and Galois symmetries Endomotives, thermodynamics, and the Weil explicit formula Appendix Bibliography Index.

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573 citations

Journal Article•DOI•

The Multiple Zeta Value data mine

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Johannes Blümlein, David J. Broadhurst¹, Jos Vermaseren•Institutions (1)

Open University¹

01 Mar 2010-Computer Physics Communications

TL;DR: The “pushdown” mechanism, whereby the ornate enumeration of primitive MZVs, by weight and depth, is reconciled with the far simpler enumerations of primitive Euler sums is elucidated.

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376 citations

Cites background from "Multiple polylogarithms and mixed T..."

...We call the number of indices of the Euler sums and MZVs their depthd and w = d ∑ k=1 |ck| (1.3) their weight....
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Journal Article•DOI•

Galois symmetries of fundamental groupoids and noncommutative geometry

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Alexander Goncharov¹•Institutions (1)

Max Planck Society¹

01 Jun 2005-Duke Mathematical Journal

TL;DR: In this article, the authors define a Hopf algebra of motivic iterated integrals on the line and prove an explicit formula for the coproduct Δ in this hopf algebra.

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Abstract: We define a Hopf algebra of motivic iterated integrals on the line and prove an explicit formula for the coproduct Δ in this Hopf algebra. We show that this formula encodes the group law of the automorphism group of a certain noncommutative variety. We relate the coproduct Δ to the coproduct in the Hopf algebra of decorated rooted plane trivalent trees, which is a plane decorated version of the one defined by Connes and Kreimer [CK]. As an application, we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. These formulas play a key role in the mysterious correspondence between the structure of the motivic fundamental group of ℙ 1 - 0 ∞ ∪ μ N , where μ N is the group of all N th roots of unity, and modular varieties for GL m (see [G6], [G7]). In Section 7 we discuss some general principles relating Feynman integrals and mixed motives. They are suggested by Section 4 and the Feynman integral approach for multiple polylogarithms on curves given in [G7]. The appendix contains background material.

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357 citations

Journal Article•DOI•

Jet Substructure at the Large Hadron Collider: A Review of Recent Advances in Theory and Machine Learning

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Andrew J. Larkoski¹, Ian Moult², Ian Moult³, Benjamin Philip Nachman²•Institutions (3)

Reed College¹, Lawrence Berkeley National Laboratory², University of California, Berkeley³

27 Jan 2020-Physics Reports

TL;DR: A comprehensive review of state-of-the-art theoretical and machine learning developments in jet substructure is provided in this article, which is meant both as a pedagogical introduction and as a comprehensive reference for experts.

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340 citations

Journal Article•DOI•

Derivation and double shuffle relations for multiple zeta values

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Kentaro Ihara¹, Masanobu Kaneko², Don Zagier³•Institutions (3)

Nagoya University¹, Max Planck Society², Collège de France³

13 May 2006-Compositio Mathematica

TL;DR: In this paper, extended double shuffle (EDS) relations for multiple zeta values (MZVs) are derived and derived algebraic structures of MZVs, as well as a linearized version of EDS relations are also studied.

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Abstract: Derivation and extended double shuffle (EDS) relations for multiple zeta values (MZVs) are proved. Related algebraic structures of MZVs, as well as a ‘linearized’ version of EDS relations are also studied.

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336 citations

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References

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Journal Article•DOI•

Iterated path integrals

[...]

Kuo-Tsai Chen

01 Sep 1977-Bulletin of the American Mathematical Society

930 citations

"Multiple polylogarithms and mixed T..." refers background in this paper

... This map is an isomorphism after tensoring by C. c) The maps Ps are compatible with the natural projections. Proof. The part a) follows from (131). The part b) is the special case of Chen’s theorem ([Ch]) for XZ. In fact in our case it follows trivially from (131). Indeed, both vector spaces in (132) are of the same dimension. Choose simple loops σi around zi based in zk+1. Then composing a given pat...
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Book Chapter•DOI•

Values of Zeta Functions and Their Applications

[...]

Don Zagier¹•Institutions (1)

Max Planck Society¹

01 Jan 1994

TL;DR: In this article, the authors give a highly idiosyncratic and prejudiced tour of a number of these applications, making no attempt to be systematic, but only to give a feel for some of the ways in which special values of zeta functions interrelate with other interesting mathematical questions.

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Abstract: Zeta functions of various sorts are all-pervasive objects in modern number theory, and an ever-recurring theme is the role played by their special values at integral arguments, which are linked in mysterious ways to the underlying geometry and often seem to dictate the most important properties of the objects to which the zeta functions are associated. It is this latter property to which the word “applications” in the title refers. In this article we will give a highly idiosyncratic and prejudiced tour of a number of these “applications,” making no attempt to be systematic, but only to give a feel for some of the ways in which special values of zeta functions interrelate with other interesting mathematical questions. The prototypical zeta function is “Riemann’s” (math) and the prototypical result on special values is the theorem that ζ(k) = rational number × π k (k > 0 even), (1) which Euler proved in 1735 and of which we will give a short proof in Section 1. (The “applications” in this case are the role which the rational numbers occurring on the right-hand side of this formula play in the theory of cyclotomic fields, in the construction of p-adic zeta functions, and in the investigation of Fermât’s Last Theorem.)

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755 citations

"Multiple polylogarithms and mixed T..." refers background or result in this paper

...ee however the work of M. Hoﬀman [Hof] and references there. In the very end of 80’th they were resurrected as coeﬃcients of Drinfeld’s associator [Dr]; couple of years later rediscovered by D.Zagier [Z] who, in particular, found and studied the double shuﬄe relations for the multiple zeta numbers; appeared in the works of M. Kontsevich [K1] on knot invariants, and the author [G0-1] on mixed Tate mot...
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...k2 + X k1>k2> 1 km 1 k n 2 New relations were found by D.Zagier. Surprisingly the dimension of the space of cusp forms for SL 2(Z) showed up in his study of the depth 2 double shuﬄe relations [Z], [Z1]. 2 Shortly after Kontsevich [K1] showed that the new relations are coming from the product formula for the iterated integrals (4). The main problem is to describe explicitly the structure of th...
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...)∨ • = UF(3,5)∨ • The rule is clear from the pattern (π2) 3f 3(f 7)3(f 5)2 −→pg 3(g 3p2)3(g 3p)2. In particular if dk := dimZk then one should have dk = dk−2 + dk−3. Computer calculations of D.Zagier [Z] conﬁrmed this prediction for k≤12. Later on much more extensive calculations were made by D. Broadhurst [Br]. One may reformulate 1.1 as a statement about the space of irreducible multiple zeta value...
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Journal Article•DOI•

Multiple polylogarithms, cyclotomy and modular complexes

[...]

Alexander Goncharov¹•Institutions (1)

Brown University¹

01 Jan 1998-Mathematical Research Letters

684 citations

"Multiple polylogarithms and mixed T..." refers methods in this paper

...shuﬂe relations in a diﬀerent way and worked out several regularization procedures. In this subsection we present an approach to the regularization of the ﬁrst shuﬄe relations developed in the end of [G3]. Another approach was recently developed in the theses of G. Racinet [R]. In particular Racinet provides an explicit formula relating his regularization with the canonical regularization. I left to t...
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Journal Article•DOI•

Operads and Motives in Deformation Quantization

[...]

Maxim Kontsevich¹•Institutions (1)

Institut des Hautes Études Scientifiques¹

14 Nov 1999-Letters in Mathematical Physics

TL;DR: The operad of chains of the little discs operad is formal, and from this fact and from Deligne's conjecture on Hochschild complexes follows almost immediately my formality result in deformation quantization as discussed by the authors.

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Abstract: The algebraic world of associative algebras has many deep connections with the geometric world of two-dimensional surfaces. Recently, D. Tamarkin discovered that the operad of chains of the little discs operad is formal, i.e. it is homotopy equivalent to its cohomology. From this fact and from Deligne's conjecture on Hochschild complexes follows almost immediately my formality result in deformation quantization. I review the situation as it looks now. Also I conjecture that the motivic Galois group acts on deformation quantizations, and speculate on possible relations of higher-dimensional algebras and of motives to quantum field theories.

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664 citations

"Multiple polylogarithms and mixed T..." refers background in this paper

...xed Tate motives. Later on, in the mid of 90’th D. Broadhurst and D. Kreimer [Kr], [BK] discovered them in quantum ﬁeld theory. As coeﬃcients of Drinfeld’s associator appear in quantization problems ([K2]). In fact in [Dr] and [K1] appeared not power series (3) but the so-called Drinfeld integrals related to the multiple ζ-values by the following formula ﬁrst noticed by Kontsevich ([K1]): ζ(n1,...,nm)...
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Journal Article•DOI•

Higher regulators and values of L-functions

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Alexander Beilinson

01 Jul 1985-Journal of Mathematical Sciences

TL;DR: In this paper, conjectures regarding the value of L-functions of motives are formulated and some computations are presented corroborating them, and some conjectures are formulated regarding the L-values of motives.

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Abstract: In the work conjectures are formulated regarding the value of L-functions of motives and some computations are presented corroborating them.

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540 citations