it can not be managed by human intelligence alone; we have to have good
models to guide decision-making. Improved models can also catch problems
before they occur, guiding the placement of new lines or generating stations or
extra maintenance on key ones that are vulnerable. Today’s “smart grid” allows
us to monitor the health of the power system with greater precision and much
more rapidly than before. However, this calls for new and more powerful
algorithms and new and more powerful statistical tools to rapidly detect
anomalies from the massive amount of data generated about the grid, and to
take corrective actions before dangers cascade throughout the system.
Mathematical scientists have only recently begun to get seriously involved in
modeling the power grid, and an enormous amount of work remains to be done.
Another major challenge is to find clean sources of energy and effectiveways to store that energy. This will require new materials to be created, and math
can dramatically speed up the process of finding materials with the particular
properties we need. For example, a recently created material can turn low-grade
heat (which is typically lost as waste) into usable electricity. The material when
cool is an ordinary, non-magnetic metal that seems like nothing special. But
when it heats up, it undergoes a phase transformation and becomes strongly
magnetic.
As Faraday’s Law describes, this change in the magnetic field creates anelectric current. Many of us as kids created a transformation like this by rubbing a
magnet along a nail, aligning the electrons and turning the nail magnetic. Unlike
such a nail, which stays magnetized after the magnet goes away, this new
material goes back to being almost perfectly non-magnetic once it cools. The
removal of the magnetic field also induces an electric current, and the material is
ready to be used again.
This reversibility of the magnetic field is an extremely rare property, andresearchers would have had great difficulty finding a material that can do this
without guidance from mathematics. By analyzing the macroscopic properties
they were looking for, they were able to deduce the microscopic structure the
material would need to have, and then they could go into the lab and create it.
The mathematical ideas used are based in the “calculus of variations,” which in
principle can be used to reveal almost all the properties of interest about a
material. Realizing this potential fully would lead to a true revolution in materials
science, but it will require significant advances in the theory.