This book reports the findings of a committee commissioned by the National Academy of Sciences. Their task was to examine the broad field of mathematical sciences and to make recommendations for how the field needs to evolve to best serve the country (i.e., the United States) by 2025. Lest the reader be inclined to stop reading immediately, please hold on and read a little further. This is not another piece of turgid prose from a government bureaucracy. Those who contributed to this work have evidently given the subject a good deal of thought, and they have worthwhile insights and provocative recommendations to convey.

One observation in particular forms a foundation for the committee’s subsequent recommendations: they “found that the discipline is expanding and that the boundaries within the mathematical sciences are beginning to fade as ideas cross over between subfields and the discipline becomes increasingly unified”. The authors note furthermore that the boundaries between the mathematical sciences and other disciplines are also breaking down. At the same time, work in many fields (theoretical physics, computer science, and biology, mathematical biology, bioinformatics) is often essentially indistinguishable from research done in mathematics. This leads the committee to argue that the mathematical sciences should be defined broadly and very inclusively.

What follows from this? One element is purely financial — the funding for the mathematical sciences has not matched their growth or the breadth of their influence, and has depended very heavily on the National Science Foundation. Consequently, the committee finds that funding needs both a substantial adjustment and a broader base across government agencies. Many of the other consequences address recruitment and education to meet expected demands for talented and well trained people.

The committee makes one striking recommendation about the training of mathematical scientists. They suggest there would be considerable value in increasing the number of mathematical scientists who are:

· knowledgeable across a broad range of disciplines beyond their own area of expertise,

· capable of communicating well with researchers in other disciplines,

· cognizant of the role of the mathematical sciences in the wider world of science, engineering, medicine, defense and business, and

· competent with computation.

By no means do they argue that this is desirable for all mathematical scientists. They just want to see more of the kind they describe.

To meet the perceived needs of the next couple of decades, the committee recommends that “the typical educational path in the mathematical sciences needs adjustment.” They go on to say:

Although there is a very long history of discussions about this issue, the need for a serious reexamination is real, driven by changes in how the mathematical sciences are being used. For example, someone who wants to study bioinformatics ought to have a pathway whereby he or she can learn probability and statistics; learn enough calculus to find maxima and minima and understand ordinary differential equations, get a solid dose of discrete mathematics; learn linear algebra; and get an introduction to algorithms.

The committee recognizes that the trends they see and the responses that they recommend may be seen as disruptive (or irrelevant) to those they call “core mathematicians”. To begin to address that, the report includes an extended personal reflection from Mark Green, vice-chairman of the committee and professor of mathematics at UCLA. The committee also adds that “support for basic science is always fragile, and this may be especially true of the core mathematical sciences. In order for the whole mathematical sciences enterprise to flourish long term, the core must flourish.”

There is a good deal more in this report, and I’d recommend that anyone concerned about the future of the profession take a look at it. One part in particular illustrates the current vitality of mathematics with a summary of fourteen areas with recent advances (ranging from the fundamental lemma and the topology of three-dimensional spaces to protein folding and compressed sensing).

Copies of this book have been sent to all mathematical science departments at two- and four-year colleges and universities in the United States. In addition, access to the book online is available here at no cost.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.