The upcoming Thematic Einstein Semester entitled "Geometric and Topological Structure of Materials" will focus on recent advancements in computational materials science. In this talk, we will present two interesting areas where algebra can play a role in future research. First, in the design and analysis of novel materials, questions of rigidity of frameworks arise. For generic configurations, matroid-theoretic and combinatorial techniques can be used to study rigidity. But the frameworks arising in applications are rarely generic, therefore deciding local rigidity presents a difficult computational problem which connects to the numerical algebraic geometry of varieties and semi-algebraic sets. Second, both point configurations and also polycrystalline materials give rise to cellular decompositions of 3-dimensional space. We will discuss how a Fourier-type analysis of these cells can be used to understand their approximate symmetry. The representation theory of SO(3) and its finite subgroups can be used to extract Fourier coefficients from the surface normal density of a given cell.