minimize electronic paths (to avoid recombination). This technology is used in remote
locations where cost-effective access to local power grids is not possible.
Quantum Dots: One of the energy goals of nanomaterials is to achieve better
energy conversion efficiency in portable power, solar cells and solid state lighting. The
devices include composite materials and quantum dots, both areas of intense
mathematical interest, from the description and understanding of wetting phenomena to
the rigorous prediction of island shapes (pyramids, domes, barns) which, in turn,
determine the technological properties of the material (see Fonseca, et al 2007 ).
Shape Memory Materials: The typical procedure in the mathematical analysis of
materials is to start with a material, describe its constitutive laws and equilibrium states,
predict the microstructure and macroscopic material behavior, and then compare with
experiment. The inverse procedure leads to the development of new materials. Indeed,
new ferromagnetic shape-memory materials have been created in this way, beginning
with a theoretical concept of an interesting property or effect, formulating the material
response via energy minimization or a dynamical theory, proposing a hypothetical
material and going to the laboratory to actually make the material. This inverse
procedure can lead to entirely new materials that might not have been anticipated by
purely experimental approaches. James and Wuttig followed this approach to produce a
new material that exhibits, under moderate field, about 50 times the field-induced strain
of giant magnetostrictive materials (see Bhattacharya , et al, 2009, James and Wuttig,
1998).
Ultracapacitors: These are electrochemical capacitors that have an unusually
high energy density when compared to common capacitors (e.g. on the order of
thousands of times greater than a high capacity electrolytic capacitor), and have a
variety of commercial applications (e.g. as energy storage devices used in vehicles). The
underlying mathematical modeling uses the Nernst-Planck-Poisson equation, commonly
applied in describing the ion-exchange kinetics in solids, and is still poorly understood.
Uncertainty and Energy Investment Portfolios
As another example containing challenging mathematical problems, we consider
the problem of choosing investments for production of new sources of energy, for
example low-greenhouse gas methods of generating electricity. This is an example of
the very general problem of constructing research and investment portfolios.
Assume all sources under consideration are substitutable (e.g. that they are all
roughly the same in terms of their environmental impact). Assume a given set of
probabilistic forecasts for the cost of each technology as a function of the cumulative
production investment made in each technology, and that there is uncertainty in both the
parameters as well as the future costs if the parameters are known. The goal is to
construct an investment schedule that maximizes the time-discounted utility, where the
time discounted utility function gives greater weight to costs on shorter time horizons and
prefers certainty over uncertainty.
This problem is inherently nonlinear due to the dependence of the cost on the
investment. Unlike standard portfolio theory as normally applied in finance,
diversification is not necessarily favorable. This is because the rate of progress for a
given technology increases as more is invested in that technology, and so if the cost as
a function of investment is known, one should simply invest everything in the best
technology. But with uncertainty one does not want to take the risk of making a bad bet