This seminar at TU Chemnitz for the winter semester 2018 is geared towards **Bachelor and Master students**.
Below I sketch some of the possible topics, with samples of the literature.

Interested students please write me a short email, with their status (Bachelor/Master), background and maybe even with a subtopic they are interested in: joscha.diehl@gmail.com

The first meeting has taken place. If you still want to get a topic, please contact me via email.

Iterated integrals of the form \[\
\int_0^T \int_0^{r_n} \int_0^{r_2} dX^{(i_1)}_{r_1} .. dX^{(i_n)}_{r_n},
%=
%\int_0^T \int_0^{r_n} \int_0^{r_2} \dot X^{i_1}_{r_1} .. \dot X^{i_n}_{r_n} dr_1 .. dr_n\] appear in a plethora of mathematical situations. Here, \(X:[0,T]\to\mathbb{R}^d\) is a curve, \(n \ge 1\) and \(i_1, .., i_n \in \{1, .., d\}\).

If \(X\) is differentiable, these integrals are classically well-defined and encode a lot of information about the curve. Indeed, it is (almost) possible to reconstruct the curve \(X\) given the values of all these integrals (\(T\) is fixed!)

*Boedihardjo, Horatio, et al. "The signature of a rough path: uniqueness." Advances in Mathematics 293 (2016): 720-737.*

These integrals play an important role in control theory

*Brockett, R. W., and Liyi Dai. "Non-holonomic kinematics and the role of elliptic functions in constructive controllability." Nonholonomic motion planning. Springer, Boston, MA, 1993. 1-21.*

There are many interesting algebraic aspects; commutative and non-commutative

*various Sections of Reutenauer, Christophe. "Free lie algebras." Handbook of algebra. Vol. 3. North-Holland, 2003. 887-903.*

**What can we do if \(X\) is not differentiable?**

In the case where \(X\) is \(\alpha\)-Hölder continous, with \(\alpha \in (1/2,1]\), the integration theory of Young applies.

*Young, Laurence C. "An inequality of the Hölder type, connected with Stieltjes integration." Acta Mathematica 67.1 (1936): 251.*

*Section 4.3 - Friz, Peter K., and Martin Hairer. A course on rough paths: with an introduction to regularity structures. Springer, 2014.*

In stochastic analysis, the most fundamental stochastic process is the **Brownian motion** (or Wiener process)

It turns out that its sample paths are only Hölder continous for \(\alpha < 1/2\), so that Young’s theory is not applicable. Even more can be said: there is no linear space carrying Brownian motion that allows for a sensible integration theory

*Section 1.5.1/1.5.2 - Lyons, Terry J. "Differential equations driven by rough signals." Revista Matemática Iberoamericana 14.2 (1998): 215-310.*

There is nonetheless a *stochastic* way to define integration against a Brownian motion \(X\): via Itôs stochastic calculus. Following a slightly different approach (Stratonovich integration) a larger class of Gaussian processes can also be integrated

*Chapter 10 - Friz, Peter K., and Martin Hairer. A course on rough paths: with an introduction to regularity structures. Springer, 2014.*

Not only does this procedure define these iterated integrals. We actually get a **rough path** lift of the Gaussian process.
This rough path can now be used to solve differential equations (which in this setting corresponds to **stochastic differential equations**

*various Sections - Friz, Peter K., and Martin Hairer. A course on rough paths: with an introduction to regularity structures. Springer, 2014.*

Recently, iterated integrals have found application in the analysis of **time series** and more generally in **machine learning**.
They are used as a feature extraction mechanism (think: "nonlinear Fourier transform")

*Chevyrev, Ilya, and Andrey Kormilitzin. "A primer on the signature method in machine learning." arXiv preprint arXiv:1603.03788 (2016).*

*Diehl, Joscha, and Jeremy Reizenstein. "Invariants of multidimensional time series based on their iterated-integral signature." arXiv preprint arXiv:1801.06104 (2018).*

*Pfeffer, Max, Anna Seigal, and Bernd Sturmfels. "Learning Paths from Signature Tensors." arXiv preprint arXiv:1809.01588 (2018).*

*Kiraly, Franz J., and Harald Oberhauser. "Kernels for sequentially ordered data." arXiv preprint arXiv:1601.08169 (2016).*