"One of the main challenges of modern mathematical physics is to understand the behaviour of systems at or near criticality. In a number of cases, one can argue heuristically that this behaviour should be described by a nonlinear stochastic partial differential equation. Some examples of systems of interest are models of phase coexistence near the critical temperature, one-dimensional interface growth models, and models of absorption of a diffusing particle by random impurities. Unfortunately, the equations arising in all of these contexts are mathematically ill-posed. This is to the extent that they defeat not only ""standard"" stochastic PDE techniques (as developed by Da Prato / Zabczyk / Röckner / Walsh / Krylov / etc), but also more recent approaches based on Wick renormalisation of nonlinearities (Da Prato / Debussche / etc).
Over the past year or so, I have been developing a theory of regularity structures that allows to give a rigorous mathematical interpretation to such equations, which therefore allows to build the mathematical objects conjectured to describe the abovementioned systems near criticality. The aim of the proposal is to study the convergence of a variety of concrete microscopic models to these limiting objects. The main fundamental mathematical tools to be developed in this endeavour are a discrete analogue to the theory of regularity structures, as well as a number of nonlinear invariance principles.
If successful, the project will yield unique insight in the large-scale behaviour of a number of physically relevant systems in regimes where both nonlinear effects and random fluctuations compete with equal strength."
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