# The signature of iterated integrals: algebra, analysis and machine learning

The signature of iterated integrals was introduced by Chen in the early part of the 20th century.
It maps smooth enough curves \(X: [0,T] \rightarrow \mathbb{R}^d\) into an infinite collection of numbers
$$
\int_0^T \int_0^{r_n} \int_0^{r_2} dX^{i_1}_{r_1} .. dX^{i_n}_{r_n}.
$$
We will cover
- How analytic properties of integration translate to algebraic properties of the signature.
This opens the door to studying the free associative algebra,
the free Lie algebra
and related Hopf algebraic concepts.
- How \(X\) is (almost) completely determined by its signature.
- How the signature has found application in stochastic analysis, via the theory of rough paths.
- How the signature is used in machine learning as a means for feature extraction.

The lecture is at 9:00h on Thursdays in A3 02.

Lecture notes (2018-07-10)
(will be updated, as we go along)
#### Lecture 1, 2018-04-12

- introduction of the signature of iterated integrals
- its state space: the free associative algebra (the tensor algebra)

#### No lecture on 2018-04-19

#### Lecture 2, 2018-04-26

- Chen's relation
- the redundancy of the signature (shuffle identity)

#### Lecture 3, 2018-05-02

- Changed date, and time 16:00
- Hopf algebraic setup

#### No lecture on 2018-05-10

#### No lecture on 2018-05-17

#### Lecture 4, 2018-05-24

- Hopf algebraic setup
- grouplike elements
- primitive elements

#### Lecture 5, 2018-05-30

- a basis for Lie polynomials indexed by Lyndon words

#### Lecture 6, 2018-06-07

- at 9:15, Special discussion round for people interested in the signature, tensors, etc

#### Lecture 7, 2018-06-07

- wrapping up Lie polynomials indexed by Lyndon words

#### No lecture on 2018-06-21

#### Lecture 7, 2018-06-28

#### Lecture 8, 2018-07-05

#### Lecture 9, 2018-07-12

- the Lie group structure of the truncated signature