Phase transitions are natural phenomena when a small change of external parameter, like temperature, leads to a drastic change of the properties of the material: ice melting to water, ferromagnets becoming paramagnets above the Curie temperature, etc. To study these phenomena, one introduces lattice models – particles are placed in a lattice and only adjacent particles interact. In two dimensions, this leads to beautiful conjectures of universality and conformal invariance, with random fractal SLE curves appearing in the limit.

The talk aims to describe some of the classical results and the state of the art in the area now. If time permits, we will also discuss the main ideas in the proofs.