Renormalisation in Regularity Structures of Multi-indices

This talk will be concerned with some aspects of the renormalization of the $\Phi^4$ equation in the singular but subcritical (also called super-renormalizable) regime. My goal will be to explain how what is called a "model", indexed by multi-indices, naturally arises from considering the geometric structure of the solution manifold. The notion of model is central in regularity structures and one of the crucial tasks is to robustly estimate it. Time permitting I would like to give some insights into the proof of the estimates and discuss some further research directions.