\indent the eigenvalues of the laplacian (or laplacian with potential) on a smoothly bounded domain domain are very natural quantities, arising as the fundamental tones of a drum, the rates of decay of diffusions, and the energy levels of quantum systems. i will discuss some of the history relating to inequalities for low eigenvalues, leading up to the proof of a conjecture of yau and van den berg for the `fundamental gap' or excitation energy of a convex domain.