### Optimal Geometry

My research in optimal geometries involves a combination
of mathematical theory and numerical experiments. Many
real-world problems can be cast in the form of optimizing some
feature of a shape; mathematically, these become variational problems
for geometric energies. A classical example is the soap bubble which
minimizes its area while enclosing a fixed volume; this leads to the
study of the constant mean curvature surfaces found in foams. Cell
membranes are more complicated bilayer surfaces, and seem to minimize an
elastic bending energy known as the Willmore energy; this bending energy
can also be used for mathematical purposes like turning a sphere inside-out.
I have also studied energies for knots (which may lead to algorithms for
simplifying and recognizing knots), and singularities in bubble clusters
and foams.
More details about my research efforts in
optimal geometry and related areas of
fundamental mathematics are online here; this work
has been supported by the National Science Foundation.
I have also been involved with more interdisciplinary research efforts,
studying foams (with support from NASA),
and studying issues in computational geometry for the simulation of
casting and extrusion of metals at
CPSD.
More information about my grant support
can be found here online.