Exact Results in Quantum Theory

In the talk I will present the survey of recent results concerning the justification of statistical mechanics for systems described by finite dimensional Hilbert spaces. Theorems I will discuss take their origin in the early paper of John von Neumann from 1929. Yet, they use tools of modern mathematics: Lie groups, homogeneous spaces and the phenomenon of concentration of measure. A the beginning I shall introduce the necessary mathematics that in the latter part will allow me to state the theorems in the compact and clear fashion.