The Institute for Mathematics and Its Applications (IMA) recognized early that materials science, a critical technology area for the nation, also represented an important opportunity for the mathematical sciences. The IMA helped build a mathematical research community in this area more than five times the size of the community 10 years earlier, and research advances at the IMA were significant. Based on work done at the 1995–1996 IMA Year on Mathematics in Materials Science (which in turn built on a 1984–1985 IMA program focusing on continuum physics and partial differential equations), microstructure theories of martensitic materials were developed to the point that the behavior of new materials could be predicted. A particular advance was the new concept for a hypothetical material combining two types of transformations: a ferromagnetic transition and a martensitic transformation. This research led to a successfully implemented strategy for developing this class of materials. As a result, new alloys have been produced that exhibit this magnetostrictive effect to a degree some 50 times greater than what had been observed in the previous record holders, the so-called giant magnetostrictive materials. Today, nearly every area of active materials science research includes research mathematicians, and the mathematical sciences are playing a crucial role in several large federal thrusts in materials research. The IMA had a major influence in coalescing and nurturing this development.
In the mid-1980s a deliberately planned juxtaposition of two research programs at the Mathematical Sciences Research Institute was the catalyst for interaction between Vaughan Jones, whose research concerned operator algebras, and researchers in low-dimensional topology. That interaction led Jones to notice that a sequence of algebraic relations discussed was similar to those that define a mathematical object called the braid group. Jones then found that this correspondence was more than just an analogy, and he obtained a general invariant for characterizing knots and links that was different from and more useful than a previously known classical one. This insight rapidly led to many advances and, ultimately, to a deep pairing of these ideas with another mathematical area, algebraic K-theory. Jones received a Fields Medal (the mathematical equivalent of a Nobel Prize, but which is awarded only once every 4 years) in part for this development.
A new research approach to drug discovery was opened through sequential drug design strategies developed at the National Institute for Statistical Sciences. As a consequence, the high-throughput screening of hundreds of thousands of drug compounds is now feasible, in a unique combination of computational chemistry, computer science, and statistics. |

considerable boon for the mathematical sciences both in the United States and abroad. Frequently, U.S. mathematical scientists form a large cohort in foreign institutes just as foreign mathematical scientists are frequent visitors to U.S. mathematical institutes. It is vital to preserve these opportunities for mixing different mathematical cultures. Fundamental mathematical research in the United States will continue to profit by having a significant proportion of the most active U.S. mathematical researchers exposed to the best experts worldwide.

Increasingly, mathematical institutes are providing a bridge linking academic mathematical scientists and industries keen on finding new mathematical tools for their work. For instance, the industrial postdoctoral research fellow positions developed by the Institute for Mathematics and Its Applications have given considerable encouragement to mathematical areas that are directly applicable to industry and have also heightened recognition in the marketplace of