Brigham Young University

**Title**: Material Lattices

**Abstract**:

The Materials Genome Initiative is the White House’s push to accelerate the development of new materials to enable 21st century technologies. The main thrust of our MURI project is to leverage new mathematics and algorithms to develop materials computationally rather than experimentally. Lattices are an interesting and productive example of how mathematics can play a key role, since many materials can be described as simple lattices, decorated by atoms. For example, if a distribution of atoms is in the form of a lattice, and it’s periodicity is described by a sublattice, note that a lattice is a group, and a sublattice is a subgroup — and the quotient group is necessarily a finite abelian group, which can be described as a direct sum of (at most) three cyclic groups. Thus we are effectively being asked to “color” or “decorate” the quotient group — subject to the symmetries.