Quantum Dots are nano-scale faceted pyramidal structures that self-assemble
from a planar semiconductor template. The discovery of these remarkable objects
has provoked a renaissance in the theoretical study of faceted crystal surfaces.
The associated mathematical models generally involve a partial differential
equation (PDE) governing the crystalline surface, with possible couplings to
adjacent bulk fields, such as a melt or a non-local elastic stress.
We will consider two such classes of PDE, one variational and the other not.
In each case, I will present a theoretical characterization of the associated
surface morphometry and coarsening dynamics. This novel theory incorporates
ideas that span the geometric theory of PDE, matched asymptotic expansions
and the calculus of variations. In addition, it provides a mathematical
framework for the simulation of million-facet surfaces, thereby providing
unique morphometric capabilities in Surface Science, as well as introducing
new paradigms in the theory of non-equilibrium statistical mechanics.