Martin Hairer

fl. 1975, Geneva, Switzerland

Probability and statisticsDifferential equations and dynamical systemsFields Medal 2014Breakthrough Prize in Mathematics 2021

Austrian-British mathematician who created the theory of regularity structures, giving rigorous meaning to singular stochastic PDEs such as the KPZ equation; Fields Medal 2014 and Breakthrough Prize 2021.

Martin Hairer was born in Geneva in 1975, the son of the numerical analyst Ernst Hairer, and grew up speaking several languages while his father taught at the University of Geneva. He studied mathematics at the University of Geneva, completing a PhD in physics in 2001 on stochastic dynamics. After research stays he joined the University of Warwick in 2002, became professor there, moved to Imperial College London in 2017, and in 2023 took a chair at EPFL in Lausanne.

Hairer began with ergodic properties of stochastic Navier–Stokes equations, proving with Jonathan Mattingly that two-dimensional turbulence driven by noise has a unique invariant measure — a rigorous foundation for fluid mixing and noise-induced transitions. His decisive breakthrough was the theory of regularity structures (2014): an algebraic-topological machinery extending rough paths that allows singular SPDEs to be solved locally despite ill-defined nonlinearities, with renormalization built in. The framework has grown into a field of its own, spawning paracontrolled calculus and applications to gauge theories and quantum field theory models such as the stochastic quantization of the Φ4 equation in three dimensions.

He received the Fields Medal in 2014 in Seoul, alongside Artur Avila, Manjul Bhargava and Maryam Mirzakhani, and the Breakthrough Prize in Mathematics in 2021. He was knighted in 2022 and continues at EPFL on stochastic PDEs, quantum field theory and stochastic quantization.

Workplaces: École Polytechnique Fédérale de Lausanne (EPFL), Imperial College London, University of Warwick

United Kingdom, Austria

Award-winning work

Fields Medal 2014The medal cites his outstanding contributions to the theory of stochastic partial differential equations, and in particular to the theory of regularity structures. Equations like the Kardar–Parisi–Zhang (KPZ) equation of interface growth or the stochastic quantization equation involve nonlinear operations on distributions, products that classical analysis cannot even define. Hairer's theory of regularity structures, developed in a single landmark 2014 paper, gives a systematic framework that enlarges the solution space so these equations have a rigorous local meaning, including a renormalization procedure that produces finite, physically correct answers. He thereby proved that KPZ is well-posed and that a whole class of singular SPDEs can be solved, and the theory has since been used to connect these equations to the KPZ universality class studied in probability and statistical physics.

Contributions, linked to the library

Library topicStochastic processes and Markov chainsBuilt regularity structures and rough-path tools that turned singular stochastic PDEs into a solvable class.Library topicSystems of differential equationsProved well-posedness for the KPZ equation and related singular SPDEs.Library topicThe Navier–Stokes equationsWith Mattingly, proved ergodicity of the two-dimensional stochastic Navier–Stokes system.