# Lecture: Stieltjes integration and iterated integrals

Semester schedule:

Lecture Monday 12:15-13:45 (starting April 17)

Exercise Class Wednesday 12:15-13:45 (starting April 26)

Potsdam-Golm Mathematics Institute (Haus 9), Room 1.22

or Zoom

The exercise class is of course optional for researchers joining us, students wishing to take the exam are expected to regularly present and discuss their exercise solutions in the class on Wednesday. As the first lecture will be devoted to an overview of the contents and to a repetition of integration against a one-dimensional variable, it is unproblematic to start with the second lecture on April 24. No exercise class in the first semester week.

Abstract:

We’ll develop the deterministic theory of Stieltjes integration in the Lebesgue and Riemann framework, the smooth regime as well as the rough Young regime where a uniquely well-defined integration theory is still available. After an introduction into the algebraic preliminaries of the shuffle algebra and the free Lie group, the core of the lecture will focus on the four main theorems of the iterated integral signature, which beautifully connect algebra, analysis in one time variable, path geometry and path topology. We’ll conclude with how to find geometric invariants of paths in the iterated integral signature and with an outlook towards the theory of rough paths, which seamlessly continues the theory to the full rough Hölder continuous regime.

The figure in the poster illustrates the notion of tree-like equivalence.

Table of Contents:

- Lebesgue and Riemann Stieltjes integration
- Sewing lemma
- Young integration theory
- Shuffle algebra, concatenation product and coproducts
- Free Lie group
- Iterated integrals: Chen’s concatenation identity and the halfshuffle integration relation
- Iterated integrals: Characterization of paths up to tree-like equivalence
- Iterated integrals: Chen-Chow theorem on the image of the truncated signature map
- Iterated integrals: Invariants under rotation-reflection
- A brief introduction to Rough Paths

If you want to attend online, please attend the lecture on April 17 or April 24 via Zoom or inform me by email to preiss * at * math.uni-potsdam.de

Exercise Sheets: Sheet 1