In solid state physics one considers a lattice of ions in ${\bf R}^d$ $(d \leq 3)$ with electrons moving in a field generated by a common potential $q(x)$ (independent electron model). The usual mathematical approach consists of determining the period spectrum of the Schrödinger operator $-\Delta + q(x)$. This then leads to a certain analytic $d$–dimensional subvariety of $({\bf C}^*)^d \times {\bf C}$ fibered over the complex numbers via the second projection. Its fibres for $d = 2$, resp. $d = 3$ are the complex Fermi curves, resp. Fermi surfaces (in solid state physics one considers only the “real” points, e.g. the intersection with $(S^1)^d \times {\bf R}^1$, where $S^1 \subset {\bf C}^*$). To isolate the geometrical aspects of the problem one discretizes the Schrödinger operator, leading to algebraic Fermi curves and surfaces. It turns out that for $d = 2$ generically the potential is completely determined by a physically measurable function of the energy, the so-called density of states function. To prove this not only a very detailed study of various degenerations is necessary, but also a substantial amount of algebraic geometry is needed, such as the Torelli theorem for curves and Deligne's theorem on the fixed part.