Iterated Integrals and Algebraic Cycles: Examples and Prospects

@article{Hain2001IteratedIA,
title={Iterated Integrals and Algebraic Cycles: Examples and Prospects},
author={Richard M. Hain},
journal={arXiv: Algebraic Geometry},
year={2001}
}
• R. Hain
• Published 25 September 2001
• Mathematics
• arXiv: Algebraic Geometry
This paper is for the proceedings of the Chen-Chow Conference held in Tianjin, China in October 2000. The goal of the paper is to produce and survey evidence for a connection between Chen's work on iterated integrals on the one hand, and algebraic cycles and motives on the other. The paper is expository, and begins with an introduction to Chen's work. Topics covered include loop space de Rham theorems; iterated integrals of currents; the works of Carlson-Clemmens-Morgan and Bruno Harris on…
41 Citations

Figures from this paper

ITERATED INTEGRALS, DIAGONAL CYCLES AND RATIONAL POINTS ON ELLIPTIC CURVES by

• 2011
— The theme of this article is the connection between the pro-unipotent fundamental group π1(X; o) of a pointed algebraic curve X, algebraic cycles, iterated integrals, and special values of

Periods of Limit Mixed Hodge Structures

This paper is an expanded version of a talk given at the Current Developments in Mathematics Conference last November (2002) on the work of Wilfred Schmid on periods of limits of Hodge structures.

Iterated Shimura integrals

In this paper, I continue the study of iterated integrals of modular forms and noncommutative modular symbols for Γ ⊂ SL(2, Z) started in [Ma3]. The main new results involve a description of the

Iterated Shimura integrals

. In this paper I continue the study of iterated integrals of modular forms and noncommutative modular symbols for Γ ⊂ SL (2 , Z ) started in [Ma3]. Main new results involve a description of the

The Hodge-de~Rham Theory of Modular Groups

This paper is an exposition of the completion of a modular group with respect to its inclusion into SL_2(Q) and the connection with the theory of modular forms and variations of mixed Hodge structure

A twisted tale of cochains and connections

Abstract Early in the history of higher homotopy algebra [Stasheff, Trans. Am. Math. Soc. 108: 293–312, 1963], it was realized that Massey products are homotopy invariants in a special sense, but it

Iterated integrals in holomorphic foliations

Abstract In this article we study the iterated integrals in holomorphic foliations. We define the corresponding Petrov/Brieskorn type modules, give a formula for the Gauss-Manin connection of

Towards algebraic iterated integrals on elliptic curves via the universal vectorial extension

• Mathematics
• 2020
For an elliptic curve $E$ defined over a field $k\subset \mathbb C$, we study iterated path integrals of logarithmic differential forms on $E^\dagger$, the universal vectorial extension of $E$. These

On periods: from global to local

Complex periods are algebraic integrals over complex algebraic domains, also appearing as Feynman integrals and multiple zeta values. The Grothendieck-de Rham period isomorphisms for p-adic algebraic

Diagram complexes, formality, and configuration space integrals for braids

• Mathematics
• 2018
We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a

References

SHOWING 1-10 OF 63 REFERENCES

The Arithmetic and Geometry of Algebraic Cycles

• Mathematics
• 2000
Preface. Conference Programme. Conference Picture. List of participants. Authors' addresses. Cohomological Methods. Lectures on algebro-geometric Chern-Weil and Cheeger-Chern-Simons theory for vector

The currents defined by analytic varieties

This paper is concerned with integration on complex analytic spaces and the (De Rham) currents defined by such integration. I t contains results about continuity of fibering and intersection of such

Exponential Iterated Integrals and Solvable Completions of Fundamental Groups

We develop a class of integrals on a manifold called exponential iterated integrals, an extension of K. T. Chen’s iterated integrals. It is shown that these integrals can be used to compute the

Extensions of mixed Hodge structures

According to Deligne, the cohomology groups of a complex algebraic variety carry a generalized Hodge structure, or, in precise terms, a mixed Hodge structure [2]. The purpose of this paper is to

Knots, groups, and 3-manifolds : papers dedicated to the memory of R.H. Fox

There is a sympathy of ideas among the fields of knot theory, infinite discrete group theory, and the topology of 3-manifolds. This book contains fifteen papers in which new results are proved in all

Addition theorem of Abel type for hyper-logarithms

• K. Aomoto
• Mathematics
Nagoya Mathematical Journal
• 1982
Several kinds of generalizations of classical Abel Theorem in algebraic curves are known, for example see [12] and [13]. It seems to the author these are all regarded as local relations among

Algebraic K-Theory and Algebraic Topology

• Mathematics
• 1993
Preface. Conductors in the Non-separable Residue Field Case R. Boltje, G.-M. Cram, V.P. Snaith. On the Reciprocity Sequence in the Higher Class Field Theory of Function Fields J.-L. Colliot-Thelene.

On the Structure of Hopf Algebras

• Mathematics
• 1965
induced by the product M x M e M. The structure theorem of Hopf concerning such algebras has been generalized by Borel, Leray, and others. This paper gives a comprehensive treatment of Hopf algebras

Values of Zeta Functions and Their Applications

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