Stanislav Smirnov

From the SPG Mitteilungen:

"In 2001 he actually proved the Cardy formula from first principles, for a particular lattice percolation model. The essentials of his argument are relatively simple: he showed that a certain local observable of percolation is discretely holomorphic: that is, it is approximately given by the real part of an analytic function. Moreover the approximation gets better and better as the lattice spacing approaches zero. Complex analytic functions are well-known to be linked to conformal mappings in two dimensions: as undergraduates we learn how to use them to solve problems in electrostatics. Once one knows the behaviour of such a function on the boundary, one knows it everywhere. This line of argument led Smirnov to his proof of the Cardy formula. Moreover, he showed it to be true in an arbitrary region, and that, it turns out, is the main requirement to reversing Schramm’s argument and showing that the boundaries of percolation clusters are indeed described by SLE.

This was not his only achievement cited for the Fields Medal. In 2006 he announced a similar result for curves in the two-dimensional Ising model, giving at last a firm mathematical basis for Polyakov’s 1970 hypothesis."

Cardy, John. "Conformal Invariance and the Scaling Limits of Lattice Models," (2010)

Fields Medalist, 2010