The long term goal of my project is to prove the renormalizability of non minimally coupled scalar phi⁴ theories on curved Riemannian Manifolds with boundaries. The method used in establishing perturbative renormalization of such models is based on the renormalization group flow equations. It is mainly inspired by the renormalizability proof of the scalar phi⁴ theory on a Riemannian half space by Kopper and Borji. Their proof requires a detailed knowledge of the propagators, or equivalently the heat kernels of the underlying theory. Hence, a first significant step of my work is to obtain an accurate understanding of the heat kernels in generically curved and compact Riemannian manifolds with boundaries, as well as their properties. In this talk I will give a brief overview on how one can obtain a local expansion of the heat kernel in terms of the curvature, discuss several of their properties and, most importantly, what one would need in order to generalize the proof of Kopper and Borji to the case of curved manifolds with boundary.
In this talk, we give an overview of the proof of perturbative renormalization of a (massive) scalar field theory with a quartic self-interaction on the four dimensional Euclidean half-space. For a particular choice of the renormalization conditions, we explain how the effective action associated to the Robin and Neumann boundary conditions can be expressed as the sum of the effective action corresponding to the translationally invariant theory, plus a remainder which consists of surface counter-terms that renormalize the Robin parameter. In the case of Dirichlet boundary conditions, no surface counter-terms are required to render the theory finite.
We will describe the connection between the Polchinski equation and the Boue-Dupuis formula. We will also explain applications of this to the construction of some Gibbs measures and couplings of these measures to the Free Field.
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.
In this talk I will describe at a high level the approach of Hairer's theory of regularity structures for semilinear subcritical singular SPDEs. As a particular focus, the talk will aim to highlight how the problem of renormalisation arises in that context alongside some recent progress in the construction of appropriately renormalised ‘models’. Finally, time permitting, I will discuss some open problems pertaining to renormalisation in this setting.
