**60: Probability theory,...** There is even more movement in the stochastic PDE community with Hairer,s regularity structures. The groups appearing there are character groups of certain Hopf algebras and there is the massive work by 
Bruned, Hairer and Zambotti to uncover the algebraic framework, leading them to certain Hopf algebras: Algebraic renormalisatiom of regularity structures arxiv:1610.08468.

Also the whole topic of rough paths (Now Msc2020 60Lxx) is quite connected, as rough paths can be viewed as paths again in character groups of certain Hopf algebras. See e.g.

https://mathoverflow.net/questions/300846/what-does-the-group-action-of-a-rough-path-in-a-lie-group-look-like/302173#302173

This can be found In most modern treatments in the guise of dealing with the tensor algebra and shuffle products. Some modern more algebraic developments are also included in the works of Ebrahimi-Fard, Manchon et al.