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.



Understanding iterated integrals
The theory of iterated integral gives a mixed Hodge structure on rational homotopy of a variety. In the case of the fundamental group, as far as I know, one can only detect the
nilpotent completion of the fundamental group. (At least, this is the only part of $\pi\_1$ that people work with in a motivic context --- see e.g. Deligne's paper on the thrice punctured sphere, many papers of Dick Hain, and perhaps Sullivan's original paper from 1977 on rational homotopy theory.)


See recent work of Minhyong Kim for application of these ideas to studying rational points
on curves over number fields. 


