Consider a $\mathbb{R}^d$-valued continuous function on $[0,1]$ that has finite length (i.e. finite variation). One can define integration with respect to such a function via the classical Riemann-Stieltjes integral.

Rough path theory enables us to define integration with respect to functions of infinite length. It turns out that these paths must first be endowed (non-canonically) with more information, which leads to paths not taking values in $\mathbb{R}^d$, but some bigger space (a certain Lie group).

One important area of application is stochastic analysis, where most processes are of infinite variation. Nonetheless this talk will focus on deterministic aspects and aims to be understandable with knowledge of only undergraduate mathematics.