  Horn's conjecture, which given the spectra of two Hermitian matrices
describes the possible spectra of the sum, was recently settled in the
affirmative. In this survey we discuss one of the many steps in this, which
required us to introduce a combinatorial gadget called a {\em honeycomb}; the
question is then reformulable as about the existence of honeycombs with certain
boundary conditions. Another important tool is the connection to the
representation theory of the group U(n), by ``classical vs. quantum''
analogies.
