Related papers: Iterated Integrals in Quantitative Topology

A simplicial analogy of Chen's iterated integral was introduced in another paper. However, its properties were hardly investigated in the paper. In particular, no mention is made of whether it coincides with Chen's iterated integral as a…

Let $\Gamma$ be a connected bridgeless metric graph, and fix a point $v$ of $\Gamma$. We define combinatorial iterated integrals on $\Gamma$ along closed paths at $v$, a unipotent generalization of the usual cycle pairing and the…

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…

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…

We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are complexes whose differentials are modeled on an arbitrary simplicial set much in the same way the ordinary Hochschild differential is modeled on…

In this paper, we extend the iterated integrals from smooth manifolds to digraphs and develop the associated algebraic and geometric structures. Iterated integrals on a digraph naturally give rise to the iterated path algebra and the…

This paper investigates how the geometry of a cycle in the loop space of a Riemannian manifold controls its topology. For fixed $\beta \in H^n(\Omega X; \mathbb{R})$ one can ask how large $|\langle \beta, Z \rangle|$ can be for cycles $Z$…

Gromov's band-width conjecture gives a precise upper bound for the width of a compact Riemannian band with positive scalar curvature lower bound, assuming that the cross-section of the band admits no positive scalar curvature metrics.…

We use an index-theoretic technique of Hitchin to show that the space of complete Riemannian metrics of nonnegative sectional curvature on certain open spin manifolds has nontrivial homotopy groups in infinitely many degrees. A new…

We construct a parallel transport on higher loop spaces of a manifold in term of a higher dimensional generalization of iterated path integrals. Under mild assumptions, we define a de Rham complex on higher loop spaces and we recover a…

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…

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M…

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.…

Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory,... All these topics share certain relations, called "loop equations" or…

Let E be a circle-equivariant complex-orientable cohomology theory. We show that the fixed-point formula applied to the free loopspace of a manifold X can be understood as a Riemann-Roch formula for the quotient of the formal group of E by…

Let $\{X_i\}$ be a sequence of compact $n$-dimensional Alexandrov spaces (e.g. Riemannian manifolds) with curvature uniformly bounded below which converges in the Gromov-Hausdorff sense to a compact Alexandrov space $X$. In an earlier paper…

We establish enhanced bounds on Cheeger-Gromov rho-invariants for general 3-manifolds and yet stronger bounds for special classes of 3-manifold. As key ingredients, we construct chain null-homotopies whose complexity is linearly bounded by…

For each positive integer Q there exists a path connected metric compactum X such that the Qth-homotopy group of X is compactly generated but not a topological group (with the quotient topology).

We consider the problem of homotopy-type reconstruction of compact subsets $X\subset\R^N$ that have the Alexandrov curvature bounded above ($\leq$ $\kappa$) in the intrinsic length metric. The reconstructed spaces are in the form of…