Wendelin Werner

fl. 1968, Cologne, West Germany

Probability and statisticsMathematical physicsFields Medal 2006

German-born French probabilist who, with Lawler and Schramm, rigorously described the conformally invariant scaling limits of planar random curves, receiving the Fields Medal in 2006.

Wendelin Werner was born in Cologne in 1968; his family moved to France when he was an infant and he was raised and educated there. A student of the École Normale Supérieure (Ulm), he received his PhD from Université Pierre-et-Marie-Curie in 1993 under Jean-François Le Gall, with a dissertation on Brownian motion. After postdoctoral work at Cambridge, he became a CNRS researcher and then a professor at Université Paris-Sud in Orsay in 1997. In 2013 he moved to ETH Zürich.

Werner trained as a probabilist specializing in the fine geometry of Brownian paths. His decisive contribution, in a celebrated series of papers with Greg Lawler and Oded Schramm beginning in 2000, was to make Schramm's newly invented SLE curves into a complete calculus for two-dimensional conformally invariant random phenomena. They proved the existence and conformal invariance of scaling limits for loop-erased random walks and uniform spanning trees, computed Brownian intersection exponents, and determined the 4/3 dimension of the Brownian frontier — results that put the physicists' conjectured critical exponents on firm ground.

He received the Fields Medal in 2006 in Madrid. Werner was also a leading actor in popularizing the mathematics of critical phenomena; he has held visiting positions including at the University of Cambridge, was elected to the Royal Society in 2020, and continues at ETH Zürich on random geometry, including the Gaussian free field and its connections to SLE and quantum gravity approaches.

Workplaces: ETH Zürich, Université Paris-Sud, CNRS

France, Germany

Award-winning work

Fields Medal 2006The medal cites his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory. With Greg Lawler and Oded Schramm, Werner proved that the Schramm–Loewner evolution (SLE) describes the scaling limits of discrete random curves, resolving long-standing conjectures: the scaling limit of the loop-erased random walk is SLE of parameter 2, and the outer boundary of planar Brownian motion has Hausdorff dimension 4/3, confirming a prediction by Mandelbrot. His work also computed the intersection exponents of Brownian motion, solving a problem posed by physicists. This built a rigorous bridge between probability and the conformal field theory used by physicists to describe two-dimensional critical phenomena.

Contributions, linked to the library

Great problemCritical exponents of self-avoiding walksShowed, with Lawler and Schramm, that SLE(8/3) is the conjectured scaling limit of self-avoiding walks, making their critical exponents computable.Library topicBrownian motion and stochastic calculusDetermined the fine geometry of planar Brownian motion, including intersection exponents and the 4/3 dimension of its outer boundary.Library topicStochastic processes and Markov chainsDeveloped stochastic Loewner evolution into the standard framework for conformally invariant random processes.