This book brings together two areas which are regarded as separate disciplines: topology and martensitic transformations. Topology is the mathematical study of continuity, particularly through the transformation of surfaces, in this case, between what are called triply-periodic minimal surfaces (TPMS). Martensitic transformations are solid state structural transformations without significant diffusion or change in chemical composition. A particular type of such transformation is the shape-memory effect, where materials recover a pre-defined shape upon heating after being deformed at a lower temperature. Shape-memory materials have seen many technological applications. The main contributions of this thesis are: 1. A direct connection between TPMS and density-functional theory, which is more fundamental and versatile than previous approaches based on the electrostatics of point charges. 2. Showing that the martensitic transformation is a topological transformation between TPMS. Surfaces related by topological transformations retain several characteristics, known as topological invariants. For martensitic transformations, this is the genus, which is the number of holes in the surface. 3. For the shape-memory effect, showing that lattice sites must remain as flat points of the TPMS along the transformation path. This provides a fast-screening method for identifying shape-memory materials which can be used for more detailed studies of candidate materials.
