TITLE: Understanding iterated integrals
QUESTION [18 upvotes]: I have encountered iterated integrals on papers dealing with multizeta values, polylogarithms etc.. Since then I am trying to figure out the motivations and purpose of the theory.
It seems the defintions and methods go back to K.-T. Chen. The integrals seem to converge like an exponential series. He published many papers on this topic. Some of these(as seen in his collected works) seem to relate to path spaces, loops spaces etc., and their homology/cohomology. Many notions in algebraic topology seem to be carried out in this context using the analytic tool of iterated integral. He calls it a "de Rham theoretical approach" to the fundamental group, etc.. Is this a "de Rham homotopy theory"? Are we able to capture topological properties by repeated integration? In particular I have in mind the "iterated path integrals" paper of K.-T. Chen in mind. There are lots of others too, and some of them are in the Annals; so one cannot question the mathematical importance of the topic.
I am sorry for asking a vague question. I am a beginner on a topic struggling to understand the concepts and motivations behind them. I will be grateful for any pointers towards more understanding, so that I can get started.

REPLY [11 votes]: One topological significance of these iterated integrals is that they can be used to model the bar complex on the de Rham cochains of a manifold, and thus in some sense the "de Rham complex" of the manifold's loop space since $H_*(Bar(\Lambda^* X)) \cong H^*(\Omega X)$ [Here I am using $\Lambda$ for the de Rham complex and $\Omega X$ to denote loops].  For a simply-connected space, this bar complex gives a reasonable way to encode the rational homotopy type, though Sullivan models have been more popular.  So in particular, one can understand a complete set of homotopy functionals (that is, the linear dual of homotopy groups) using iterated integrals: Milnor and Moore showed that $\pi_*(X) \otimes {\mathbb Q}$ is isomorphic to the Lie algebra of indecomposibles of $H_*(\Omega X; {\mathbb Q})$; thus the linear dual of homotopy is the "Lie coalgebra of coindecomposibles" of the rational loopspace cohomology of $X$ (again given by Chen integrals if you wish).  Chen showed early on that these integrals give information about $\pi_1$ as well (through its nilpotent completion).  
My coauthor Ben Walter and I have developed a model for rational homotopy types which is close to both the Chen and Quillen models (and compatible with the Sullivan model as well), and clarified the story of functionals on homotopy groups as well.  Because of Chen's work, we know that we will have plenty of information to mine in the non-simply connected setting.