Related papers: Iterated Integrals and Algebraic Cycles: Examples …

Higher order automorphic forms have recently been introduced to study important questions in number theory and mathematical physics. We investigate the connection between these functions and Chen's iterated integrals. Then using Chen's…

K.T. Chen showed that iterated integrals give comparison isomorphisms between the cohomologies of bar complexes and fundamental group rings. This led to the development of an algebraic-geometric approach to studying periods given by…

The algebraic structure of iterated integrals has been encoded by Chen. Formally, it identifies with the shuffle and Lie calculus of Lyndon, Ree and Sch\"utzenberger. It is mostly incorporated in the modern theory of free Lie algebras.…

We give a definition of higher dimensional iterated integrals based on integration over membranes. We prove basic properties of this definition and formulate a conjecture which extends Chen's de Rham Theorem for iterated integrals to the…

Chen's iterated integrals may be generalized by interpolation of functions of the positive integer number of times which particular forms are iterated in integrals along specific paths, to certain complex values. These generalized iterated…

These notes are for the author's lectures, "Integral Reduction and Applied Algebraic Geometry Techniques" in the School and Workshop on Amplitudes in Beijing 2016. I introduce the applications of algebraic geometry methods on multi-loop…

Let X be a simply connected Riemannian manifold. Until now, quantitative topology has used Sullivan's rational homotopy theory as the bridge between geometric information on X and torsion-free homotopy theoretic information on X. In this…

We describe several applications of the theory of cycles to questions in Commutative Algebra. The main topic is the use of the theory of local Chern characters to answer some questions on modules of finite homological dimension. This paper…

We dicuss some curious aspects of the Rogers dilogarithm and the functional relations in integrable systems in two dimensions. This is for the proceedings of the XX1 Differential Geometry Methods in Theoretical Physics, Tianjin, China, 5-9…

Chen's iterated integrals are treated within synthetic differential geometry. The main result is that iterated integrals produce a subcomplex of the de Rham complex on the free path space as well as based path spaces.

Gian-Carlo Rota mentioned in one of his last articles the problem of developing a theory around the notion of integration algebras, which should be dual to the one of differential algebras. This idea has been developed historically along…

We extend the notion of regularized integrals introduced by Li-Zhou that aims to assign finite values to divergent integrals on configuration spaces of Riemann surfaces. We then give cohomological formulations for the extended notion using…

We close a gap appearing at the same time in the author's thesis "Iterated rings of bounded elements and generalizations of Schm\"udgen's theorem" [1] and in the author's article "Iterated rings of bounded elements and generalizations of…

This volume of the Electronic Proceedings in Theoretical Computer Science (EPTCS) contains extended abstracts of talks to be presented at the Seventh International Conference on Computability and Complexity in Analysis (CCA 2010) that will…

We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalisation of multiple polylogarithms, we construct our set of integration kernels ensuring…

We use Chen iterated line integrals to construct a topological algebra ${\cal A}_p$ of separating functions on the {\it Group of Loops} ${\bf L}{\cal M}_p$. ${\cal A}_p$ has an Hopf algebra structure which allows the construction of a group…

We deduce from the work of Chen, that the restriction morphism from closed free iterated integrals to closed iterated integrals on loops is onto. We use this to show that the module of higher order invariants of smooth functions is…

We derive the full system of canonical differential equations for all planar two-loop massless six-particle master integrals, and determine analytically the boundary conditions. This fully specifies the solutions, which may be written as…

Gian-Carlo Rota suggested in one of his last articles the problem of developing a theory around the notion of integration algebras, complementary to the already existing theory of differential algebras. This idea was mainly motivated by…

Following the work of Gangl, Goncharov and Levin in [GGL], we will give a combinatorial framework for motivic study of iterated integrals on the affine line. We will show that under a certain genericity condition these combinatorial objects…