Martin Hairer wins Breakthrough Prize

The theory of regularity structures has its roots in the theory of rough paths for SPDEs and represents its generalization to higher dimensions. It is a novel theory developed by the author, for which he was awarded the Fields Medal in 2014. This paper represents a complete self-contained survey on the theory, on its background, motivations, applications and interconnection with other theories. Written in a lucid style with perfect exposition and well-structured into ten sections, it may serve as a book as well.

The purpose of regularity structures is to build a framework that allows one to solve and study the solutions to semilinear SPDEs with highly irregular random input. It results in an algebraic framework that allows one to describe functions and generalized functions by a kind of Taylor expansion around each point in space-time. The main novel idea is to replace the classical polynomial model by arbitrary models that involve other functions and that are purpose-built for the problem at hand. A regularity structure is thus a triplet $(A,T, G)$ where $A\subset \bf R$ denotes an appropriate index set, $T$ denotes the model space which is a graded vector space of the form $T=\bigoplus_{\alpha\in A}T_\alpha$ whose elements describe the local expansion of a function at any given point, and $G$ denotes a group of linear operators acting on $T$ which help to translate the coefficient of a local expansion around a given point into coefficients of an expansion around a different point.

The next step is to build a calculus incorporating the operations of multiplication, composition with smooth functions and integration against singular kernels. Concerning the problem of multiplication of generalized functions, the author provides deep insights, as well as a connection to alternative theories: Colombeau’s algebras of generalized functions, Hida’s white noise theory, Bony’s paraproduct and Lyons’ rough path theory are reflected within the ideas of regularity structures.

In order to solve SPDEs, one has to find an appropriate regularity structure that is rich enough to allow the formulation of a fixed point problem associated to the SPDE. By solving this fixed point problem one arrives at an abstract solution map to the original equation which can also be interpreted as a limit of classical solutions to regularized problems, usually modified by the addition of diverging counterterms that arise through the action of a renormalization group.

Thus, the theory of regularity structures makes it possible to give a pathwise meaning to ill-posed SPDEs that appear when describing the macroscopic behaviour of physical models near a critical point. One of the reasons why the theory is efficient at providing precise small-scale features of solutions to semilinear SPDEs is that it comes with very sharp Schauder estimates.

Some of the examples that illustrate the theory include the continuous parabolic Anderson model, the KPZ-type equations, the Navier-Stokes equation with singular forcing, stochastic quantization equations, etc. As a novel application the author solves the long-standing problem of building a natural Markov process that is symmetric with respect to the finite volume measure describing the $\Phi_3^4$ Euclidean quantum field theory.

Reviewed by Dora Seleši