Ajay Chandra (London), Patrícia Gonçalves (Lisboa), Martina Hofmanová (Bielefeld), Roman Kotecký (Warwick), Antti Kupiainen (Helsinki), Jérémie Unterberger (Nancy), Lorenzo Zambotti (Paris).

Ajay Chandra (London): Renormalization in Regularity Structures.

The inception of the theory of regularity structures transformed the study of singular SPDE by generalizing the notion of "Taylor expansion" and classical notions of regularity in a way flexible enough to accommodate renormalization.
In the years since then our understanding of how to implement renormalization in regularity structures has developed rapidly. I will describe some of these developments, namely the use of multiscale analysis for stochastic estimates in the theory and the identification of how to tune renormalization constants to obtain solutions with good invariance properties.

Patrícia Gonçalves (Lisboa): Deriving the stochastic Burgers equation from weakly asymmetric interacting particle systems. pdf

In this talk I will explain how to derive energy solutions of the stochastic Burgers equation from the analysis of the equilibrium fluctuations of the unique conserved quantity of a weakly asymmetric microscopic dynamics. I will present the strategy of the proof and I will discuss recent results for models in contact with reservoirs and for dynamics conserving more than one quantity.

Martina Hofmanová (Bielefeld): A PDE construction of the Euclidean $\Phi^4_3$ quantum field theory. pdf

We present a self-contained construction of the Euclidean $\Phi^4$ quantum field theory on $\mathbb{R}^3$ based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on $\mathbb{R}^3$ defined on a periodic lattice of mesh size $\varepsilon$ and side length $M$. We introduce an energy method and prove tightness of the corresponding Gibbs measures as $\varepsilon \rightarrow 0$, $M \rightarrow \infty$. We show that every limit point satisfies reflection positivity, translation invariance and nontriviality (i.e. non-Gaussianity). Our argument applies to arbitrary positive coupling constant and also to multicomponent models with $O(N)$ symmetry. Joint work with Massimiliano Gubinelli.

Roman Kotecký (Warwick): Vector gradient fields as microscopic models of nonlinear elasticity.

I will discuss how to employ random vector gradient fields with non-convex interaction in the role of microscopic models of displacement of atoms of a crystal structure. Using RG techniques, strict convexity of the free energy, as a function of an affine boundary deformation, is proven for a class of models at low temperatures. Based on a joint paper with S. Adams, S. Buchholz, and S. Muller.

Antti Kupiainen (Helsinki): Renormalization Group, Quantum fields and Stochastic PDEs.

I will discuss the problem of divergencies in quantum field theory and stochastic PDEs and how these can be addressed using the renormalisation group.

Jérémie Unterberger (Nancy): pdf

Lorenzo Zambotti (Paris): Renormalisation of geometric SPDEs. pdf

We consider a natural class of $\R^d$-valued one-dimensional stochastic PDEs driven by space-time white noise that is formally invariant under the action of the diffeomorphism group on $\R^d$. This class contains in particular the KPZ equation, the multiplicative stochastic heat equation, the additive stochastic heat equation, and rough Burgers-type equations. This includes the natural noisy perturbation of the harmonic map flow with values in an arbitrary Riemannian manifold, that we conjecturally identify the process associated to the Markov dynamics of the Dirichlet form corresponding to the $L^2$-gradient flow for the Brownian bridge measure. (joint work with Yvain Bruned, Franck Gabriel and Martin Hairer).