Critical and near-critical percolation
Critical and near-critical percolation is well-understood in dimension 2 and in high dimensions. The behaviour in intermediate dimensions (in particular 3) is still largely not understood, but in recent years there was some progress in this field, with contributions by van den Berg, Cerf, Duminil-Copin, Tassion and others. We will survey this recent progress (and a few older but not sufficiently known results).
Prerequisits: The Fortuin-Kasteleyn-Ginibre (FKG) and van den Berg-Kesten (BK) inequalities. See e.g. Chapter 2 of Duminil-Copin's lecture notes or almost any book on percolation theory.