A flat torus is defined by a lattice in Euclidean space. The "length spectrum" of the torus is the collection of lengths of lattice vectors, counted with multiplicities. In two and three dimensions, the length spectrum determines the torus (i.e., determines the lattice) up to congruence. However, in higher-dimensions, there exist non-congruent flat tori with the same length spectrum. We will first review these results and then raise the following unsolved problems: In addition to the length spectrum, suppose you also know the collection of all parallelograms spanned by pairs of lattice vectors. (I.e., you know the abstract collection of such parallelograms and how many times each one appears.) Is this information enough to determine the lattice up to congruence? One can also throw in higher-dimensional parallelopipeds and ask the analogous question. Or one can ask whether the length spectrum and "area spectrum" (areas of all parallelograms spanned by lattice vectors) is enough, etc.

This lecture will use only linear algebra and basics from group theory.