  The remarkable range of biological forms in and around us, such as: the
undulating shape of a leaf or flower in the garden, the coils in our gut, or
the folds in the brain, raise a number of questions at the interface of
biology, physics and mathematics. How might these forms be described, how can
their shapes be predicted, and how can they eventually be controlled for
function? We review our current understanding of the mathematical aspects of
this problem, that brings together analysis, geometry and mechanics in the
description of the morphogenesis of low-dimensional objects.
  Starting from the view that shape is the consequence of metric frustration in
an ambient space, we examine the links between the classical Nash embedding
problem and biological morphogenesis. Then, motivated by a range of
experimental observations and numerical computations, we revisit known rigorous
results on curvature-driven patterning of thin elastic surfaces, especially the
asymptotic behaviors of the solutions as the (scaled) thickness of the surfaces
becomes vanishingly small and the local curvature can become large. We close
with a brief description of some open problems in this area from a mathematical
perspective, and how they link to technological questions driven by the allure
of being able to manipulate and design soft surfaces in science and engineering
on scales from a few Angstroms to meters.
