01 · A life in mathematics
The person behind the work
Stanislav Smirnov was born in Leningrad and studied mathematics in Saint Petersburg before receiving his doctorate at Caltech in 1996. He joined the University of Geneva in 2003 and was a professor there when awarded the 2010 Fields Medal. His work established conformal invariance in the scaling limits of important two-dimensional models of statistical physics, including critical percolation on the triangular lattice and the planar Ising model. He found discrete observables whose properties survive the passage to a continuum. These results made rigorous several powerful predictions about random interfaces, linking probability with complex analysis and geometry.
The contribution, explained
Color sites of a triangular lattice randomly with equal probabilities, and ask whether one color crosses a given region. At the critical value, clusters appear at every scale. Physicists predicted that as the lattice spacing shrinks, crossing probabilities and interfaces should respect conformal maps, transformations preserving local angles. Smirnov found special discrete observables that satisfy analogues of complex-analytic relations. Their continuum limits made conformal invariance rigorous for critical triangular-lattice percolation and, through related work, the planar Ising model. This explains how a discrete model with only a few lattice symmetries can develop a much richer symmetry in its large-scale limit.
Why it matters
These proofs established rigorous foundations for key predictions of two-dimensional statistical physics. They connect lattice models to complex analysis and stochastic Loewner evolution, allowing exact formulas and universal-looking behavior to be understood mathematically.