Commonly, the regularity of a function describes how well it is approximated by polynomials. This understanding breaks down when we consider solutions to PDEs perturbed by a highly irregular stochastic object called white noise: Polynomials are simply too crude to be the right approximating quantity in this case, but what is?
To answer this question, I will introduce the notion of a regularity structure pioneered by Martin Hairer. We will see that it is a far-reaching generalisation of Taylor series that is robust enough to set up a solution theory for afore-mentioned (stochastic) PDEs.