01 · A life in mathematics
The person behind the work
Wendelin Werner was born in Cologne and grew up in France. After studying at ENS and completing his doctorate in Paris, he worked at the CNRS and became a professor at Université Paris-Sud in 1997, also teaching at ENS. His 2006 Fields Medal recognized work on planar random curves and Brownian motion. With Gregory Lawler and Oded Schramm he developed stochastic Loewner evolution into a rigorous tool for statistical physics, computing critical exponents that describe fractal boundaries. His research showed that apparently erratic random paths can have exact geometric laws, connecting probability with conformal maps and scale-invariant phenomena.
The contribution, explained
Planar Brownian motion is an idealized random path with fluctuations at every scale. Its full trace has Hausdorff dimension two, but the frontier seen from outside is thinner. Physicists predicted that this outer boundary has dimension four-thirds. Werner, Gregory Lawler and Oded Schramm used conformal methods and stochastic Loewner evolution to prove the prediction. SLE describes growing random curves by conformal maps driven by a one-dimensional Brownian motion. By linking this tractable description to planar probability, they computed intersection exponents and derived exact geometric information. The achievement showed that random fractals can obey rigorous quantitative laws despite their endlessly complicated appearance.
Why it matters
The work made SLE a central bridge between probability, complex analysis and statistical physics. It proved critical exponents and fractal dimensions, helping explain why different planar models can exhibit closely related large-scale random geometry.