From the Director

Opportunity Unlimited

Doug Arnold,

IMA Director

We believe that mathematics is a field of almost unlimited opportunity—provided that it looks outward towards its interfaces with other fields.

This quote, from the influential 1998 Odom report, elegantly encapsulates the primary motivation for the IMA and its mission of increasing the impact of mathematics by fostering interdisciplinary mathematical research. The strengthening of interdisciplinary research is now widely recognized as crucial—crucial to the vitality of mathematics, crucial to the progress of an increasing number of areas of science and engineering, and crucial to the health, prosperity, and security of our society. However, it is important to remember that connecting mathematics with external disciplines and problems has not always been so highly valued by the mainstream of mathematical researchers. In this column, I hope to highlight the major cultural change that has taken place in mathematics research over the last quarter century, and, especially, the role that the IMA played—and continues to play—in bringing this about.

In 1982, when the IMA was founded, the academic mathematics community had largely withdrawn from its historically close relationship with the sciences, a relationship which had served as a primary source of motivation and inspiration for mathematics for centuries. Instead, the community was highly directed towards internally generated problems, making little effort to communicate mathematical results outside the discipline, or even outside a particular subdiscipline within mathematics, and was generally isolated within the scientific landscape. I believe that the mathematical science community owes an immense debt of gratitude to the founders of the IMA for taking the lead in recognizing and helping to reverse this isolation, guiding the mathematics community from detachment and introspection to centrality and connectivity.

It is interesting to compare the 1978 National Science Board memorandum which laid out the motivation for the establishment of a mathematical sciences research institute, with the visionary proposal submitted by former IMA directors Hans Weinberger and Willard Miller in response to it the following year.

The NSB memorandum states the motivation for a national math institute thus:

American research in mathematics is today in a golden age. But there is within the mathematical community a general consensus that, in order to maintain and even further to stimulate the unequaled pace of research of the past dozen years, another mathematical sciences research institute, similar but not identical to the famous Institute for Advanced Study, should be established…By the term "research institute" we mean an organization where mathematical scientists on leave from their institutions of permanent employment, can congregate in appropriate numbers for extended periods for the sole purpose of performing research…The aim of the Institute is, of course, the promotion of research in the mathematical sciences.

Nowhere in the 8,500 word document is there any suggestion that a non-mathematician would ever set foot in the institute. Words like "interdisciplinary" do not appear.

Now read the opening of the original IMA proposal:

The most notable feature [of contemporary mathematics]…is the isolation of the areas of growth not only from each other but also from other scholarly disciplines and areas of concern. This aspect of the problem needs to be addressed if mathematics is to remain both viable and worthy of support by society as a whole. It is in response to such problems that we are proposing the establishment of a National Institute of Mathematics and its Applications.

No talk of a golden age here! Instead the proposal emphasizes the importance of "exposing the participants to a variety of problems which arise in the biological, physical, engineering, and social sciences in order to spawn new and challenging areas of mathematical research." They emphasize that "no serious program for communicating new problems from other areas of science to mathematicians can be successful without the active participation of research leaders in the field of application under study," and propose to bring in non-mathematician scientists for both long and short visits. They repudiate the artificial dichotomy between pure and applied mathematics which had resulted in a very circumscribed applied mathematics subdiscipline, disconnected from the exciting recent advances in mathematics. Instead, they state that "the emphasis is to be on new applications rather than old ones," and speak of "the importance of including a major proportion of 'pure' mathematicians," with "the particular field of expertise not nearly important as the willingness to explore new problems and think in new ways." They argue that "the very best mathematical minds, most of whom have not been trained in what is called 'applied mathematics'" will be needed to successfully attack the variety of challenging problems which arise, and declare "interdisciplinary research," "substantive collaborative efforts," and the involvement of "mathematicians with diverse fields of interest present at the same time" as the foundation of the institute.

The appreciation of the importance of interdisciplinary research is now widespread, an integral part of the plans of many universities, colleges, departments, and funding agencies. This is great news for mathematicians, because mathematics can assume a central role in bringing together ideas from many disciplines in the assault on challenging and important problems. As Rockefeller University biologist Joel E. Cohen wrote recently, speaking of biology, "mathematics can help biologists grasp problems that are otherwise too big (the biosphere) or too small (molecular structure); too slow (macroevolution) or too fast (photosynthesis); too remote in time (early extinctions) or too remote in space (life at extremes on the earth and in space); too complex (the human brain) or too dangerous or unethical (epidemiology of infectious agents)." (PLoS Biol 2(12), 2004). And biology, although a major source of opportunities, is just one of many fields where math can play a key role. There seems to be no end to the new problems waiting to challenge mathematicians willing to enter into interdisciplinary collaborations. Fortunately many math research institutes in this country and abroad have adopted the goal of fostering interdisciplinary mathematical research, and many research mathematicians have embraced such research. Ours is a field of unlimited opportunity—as long as we continue to look outward to other fields.