We consider the quantum Heisenberg ferromagnet in three dimensions.
We present the first rigorous proof of validity of the spin-wave approximation
at low temperatures, at the level of the system's free energy. The proof
combines a bosonic formulation of the model induced by the Holstein-Primakoff
representation with probabilistic estimates, localization bounds and functional
inequalities. Joint work with Michele Correggi and Robert Seiringer.