Abstract: The discrete periodic Schrodinger operator, a graph Laplacian plus a periodic potential, arises from the tight-binding approximation model for crystals in solid state physics. Through Floquet theory, the spectrum of this operator can be studied as the projection of a real algebraic variety, known as the Bloch variety or dispersion relation. This variety is described by the vanishing set of the dispersion polynomial, a Laurent polynomial whose algebraic properties encode various spectral information. In this talk, we will introduce the relevant background and highlight some recent results, with particular emphasis on generic properties of the Bloch variety.