The 2016 Nobel Prize in Physics was awarded half to Thouless and half to Haldane and Kosterlitz “for theoretical discoveries of topological phase transitions and topological phases of matter”. I shall review a part of this development that may be attractive for a general mathematical audience. In particular
The context in statistical mechanics: the rotator model on $Z^2$.
The role of cohomology of complexes in this model.
How cohomology brings in the electrostatics of charged point particles.
How electrostatics in two dimensions causes the discontinuous change in charge-density correlations called the KT phase transition.
Theorems (old and new) on the existence and nature of the KT transition.