Stanislav Smirnov

Native spelling: Станислав Смирнов

fl. 1970, Leningrad, Soviet Union

Probability and statisticsMathematical physicsClay Research Award 2001Fields Medal 2010

Russian-Swiss mathematician who proved conformal invariance of critical percolation and the planar Ising model, laying rigorous foundations for two-dimensional statistical physics; Fields Medal 2010.

Stanislav Smirnov was born in Leningrad (now Saint Petersburg) in 1970. He studied mathematics at Saint Petersburg State University, then went to the California Institute of Technology, earning his PhD in 1996 under Nikolai Makarov on complex analysis and spectral problems. After a period at Yale University and the Institute for Advanced Study, he became a professor at the Royal Institute of Technology (KTH) in Stockholm and moved in 2003 to the University of Geneva. He later also took a position at the newly founded Skolkovo Institute of Science and Technology (Skoltech) in Moscow.

Smirnov's work opened the era of rigorous conformal invariance for lattice models. His 2001 proof for percolation showed that the probabilities of crossing a domain are described by Cardy's formula and that the discrete exploration path converges to SLE(6). To handle the Ising model he introduced discrete holomorphic observables — functions on lattice edges satisfying a form of the Cauchy–Riemann equations — whose continuum limits are genuine holomorphic functions. Together with Dmitry Chelkak and others this program proved conformal covariance of fermionic observables, energy densities and spin correlations, and supplied many of the missing mathematical underpinnings of two-dimensional conformal field theory.

He received the Fields Medal in 2010 in Hyderabad. Smirnov leads a research group in Geneva and a laboratory at Skoltech, working on universality: extending conformal invariance and SLE descriptions to other lattices and to models such as spin models, dimers and self-avoiding walks.

Workplaces: University of Geneva, Skoltech and Saint Petersburg State University

Russia, Switzerland

Award-winning work

Fields Medal 2010The medal cites his proof of conformal invariance of percolation and the planar Ising model. In 2001 Smirnov proved that critical site percolation on the triangular lattice has a conformally invariant scaling limit, establishing Cardy's formula for crossing probabilities rigorously — the first demonstration that a lattice model of statistical physics converges to the conformal field theory predicted by physicists. He then identified the discrete structures now called Smirnov fermionic observables (s-holomorphic functions) and used them to show that interfaces in the critical Ising model and the Fortuin–Kasteleyn random-cluster model converge to Schramm–Loewner evolution curves of specific parameters. This confirmed the physicists' picture of two-dimensional critical phenomena and made the Ising model's critical exponents, such as the magnetization exponent 1/8, mathematically accessible.

Contributions, linked to the library

Great problemCritical exponents of self-avoiding walksHis discrete observables and SLE framework drove progress on critical exponents of planar lattice models.Library topicStochastic processes and Markov chainsProved that interfaces of critical lattice models converge to Schramm–Loewner evolution curves.