Hopf and pre-Lie algebras in regularity structures

These lecture notes aim to present the algebraic theory of regularity structures as developed in arXiv:1303.5113, arXiv:1610.08468, and arXiv:1711.10239. The main aim of this theory is to build a systematic approach to renormalisation of singular SPDEs; together with complementary analytic results, these works give a general solution theory for a wide class of semilinear parabolic singular SPDEs in the subcritical regime. We demonstrate how "positive" and "negative" renormalisation can be described using interacting Hopf algebras, and how the renormalised non-linearities in SPDEs can be computed. For the latter, of crucial importance is a pre-Lie structure on non-linearities on which the negative renormalisation group acts through pre-Lie morphisms. To show the main aspects of these results without introducing many notations and assumptions, we focus on a special case of the general theory in which there is only one equation and one noise. These lectures notes are an expansion of the material presented at a minicourse with the same title at the Master Class and Workshop "Higher Structures Emerging from Renormalisation" at the ESI, Vienna, in November 2021.