INTRODUCTION

I. INTRODUCTION

Mathematics in a Time of Rapid Change

For the mathematical sciences, the past fifty years have been a golden era of great discoveries and new developments spanning the entire spectrum from basic theory to real-world applications. Accomplishments in basic theory have been wide ranging, including the development of sympletic geometry, mirror symmetry, and quantum groups; the discovery of solitons; the proof of Fermat’s theorem some 350 years after it was stated; the classification of all simple finite groups (the building blocks for groups generally). Subfields, once viewed as quite disjoint, are now seen as part of a whole. Striking examples of applications also occurred, such as; the designing of the Boeing 777 airliner, which relied on mathematical theory, computation modeling, and powerful simulation techniques to replace physical testing and to speed the design process; the use of wavelets as a fundamental tool for fingerprint analysis; the multipole algorithm in electromagnetic computation; and neural network algorithms in pattern matching applications. U.S. mathematicians have been at the forefront of these and other world developments in mathematics.

The mathematical sciences -- and all other sciences -- are performed in a world that is changing rapidly. The exploding importance of information to all sectors of society and the pervasive role of technology in maintaining the security and prosperity of the nation have placed the mathematical sciences in a position of central importance. Mathematics provides the context for communication and discovery in many other disciplines. Competitive pressures throughout business and government, coupled with the broad expansion of computer analysis and data management, have extended the applications of mathematics to every domain of human activity.

The fundamental changes taking place in many areas of science and technology–especially biology, communications, and computation – are accompanied by important new problems that cannot be solved without new mathematics. The modern desires to improve decision-making (for example, to make real-time stock market or hedging decisions) and to understand very complex problems (for example, to model the impact of human activities on the environment) will require original mathematical techniques. These deep challenges, which are vitally important to the nation, offer novel opportunities for research in mathematics. Without increased openness by mathematicians to problems of other disciplines, mathematics may miss opportunities to contribute to and gain from these developments.

The missed opportunities would extend beyond mathematics. Excellent mathematical ideas developed in biology would probably not be helpful to financial modeling, say, because the jargon used in the two fields is very different. Mathematics can help standardize developments in one field for use by others. Two examples include finite element methods, developed by structural engineers, and sparse matrix methods, developed by power systems engineers and economists. In both cases, standardization and generalization of the results by mathematicians have led to applications in many other fields. The following graphics further illustrate this point.

In an ideal world, mathematics has a clean flow from its core out to applications and from applications back to the core. This flow facilitates the adaptation of mathematical concepts from one field, such as physics, to economics, and vice versa.

When the mathematical sciences retreat from such multidisciplinary involvements, mathematics suffers from lack of enrichment by the ideas and challenges of other disciplines. The other disciplines also suffer, for two reasons:

The disciplines lose the expertise and easy access to the vast knowledge base developed by the mathematical sciences; and,

The disciplines develop overly specialized mathematical languages and tools that inhibit multidisciplinary communication.

In 1993, there were 22,820 doctoral mathematical scientists employed in the United States. It was estimated(see Endnote 1) that of these mathematical scientists, 14,670 (64.3%) were employed at universities and 4-year colleges (6,427 at doctorate granting universities), 5,160 (22.6%) in industry, and 960 (4.2%) by the Federal government. Of those active in research, 35% reported receiving Federal funding(see Endnote 2). It is worthwhile to compare with other sciences?

1993 Fulltime Academic Doctorate Faculty
	Total Number	Active in Research	Receiving Fed Support	% Active with Fed Support
Biological Sciences	68194	51767	36143	69%
Physical Sciences	28644	20029	13463	67%
Mathematical Sciences	15475	9517	3250	35%

The mathematical sciences are divided into two largely independent groups: 1) academic mathematicians, and 2) users of mathematics, both inside and outside the university community. The weak coupling between these two groups is a central problem for mathematics worldwide (as it is for some other areas of science and technology). To enhance the vigor of both groups, it is vital that the creators and users of mathematics be more strongly connected. Excellent mathematics, however abstract, leads to practical applications. In turn, hard problems in nature stimulate the invention of new mathematics.

Traditionally, abstract mathematicians follow natural paths of inquiry toward the development of new concepts and new theories. They are often influenced by problems arising outside of mathematics, but as often, perhaps more often, they are driven by the inherent beauty and inner consistency of the results. It might be years or decades before such concepts find application — if they ever do. The physicist Eugene Wigner marveled as to "the unreasonable effectiveness of [abstract] mathematics in the natural sciences;" nowadays, one would add finance and management. Arthur Jaffe (David I (see Endnote 3), p. 120) explains this as follows: "Mathematical ideas do not spring full grown from the minds of researchers. Mathematics often takes its inspirations from patterns in nature. Lessons distilled from one encounter with nature continue to serve as well when we explore other natural phenomena." Even if one values mathematics only for its role in applications, one must value these basic abstract investigations because they provide the foundation upon which applied and computational mathematics, as well as statistics and computer science, are based.

The United States excels in abstract mathematical research. To make the best use of this strength, there is a need for a faster flow of knowledge between the creators and users of mathematics. The progress of science deteriorates when mathematicians or the users of mathematics must develop new knowledge "on demand." Only when the doors of communication are open wide can the mathematical enterprise function at full potential. Users benefit from quick access to known mathematics, and mathematicians are challenged by new formulations and questions from users.

Strengthening the connections between the creators and the users of mathematics, while maintaining historical proficiency in pure mathematics, is the most important opportunity now open to the National Science Foundation in its support of the field. It is imperative to find new ways to speed the flow of mathematical discoveries between academic mathematical scientists and those who use their results, and between different fields of the mathematical sciences.

The third community of the mathematical enterprise consists of the students. During this time of rapid change, the way students are educated must keep pace with quickly shifting realities of vocation and employment. There is a need to broaden and extend the curriculum for future mathematical scientists, as well as for the users of that science, to make them more flexible and ensure that the two groups are equipped to interact effectively in the interests of their disciplines and of society. It will require ingenuity on the part of the mathematical scientists and of other disciplinarians to do so, without losing depth.

The importance of the mathematical sciences to society dictates that we adapt the way we prepare the next generation of mathematical scientists to face new realities, which include increasingly multidisciplinary work and the extension of the mathematical sciences into other fields. Tomorrow’s mathematical scientists must be educated in new ways if they are to contribute to the mathematical enterprise and to society across the full range of employment opportunities and professional challenges. (see Endnote 4)

The Purpose of This Report

This report is part of the National Science Foundation’s response to comply with the Government Performance and Results Act (GPRA). The act requires an evaluation of how well the Foundation has met its strategic goals, which are:

to enable the United States to uphold a position of world leadership in all aspects of science, mathematics, and engineering;

to promote the discovery, integration, dissemination, and employment of new knowledge in the service of society; and

to achieve excellence in U.S. science, mathematics, engineering, and technology education at all levels.

In March 1997, the Division of Mathematical Sciences (DMS) of the National Science Foundation (NSF) convened a Senior Assessment Panel and charged it to undertake an assessment of the Mathematical Sciences in the United States. The Panel was asked to undertake the following tasks: to assess the health and position of leadership of the United States mathematical sciences; to evaluate the connections of mathematics with the other sciences, technology, education, commerce, and industry; to appraise the performance of mathematics in the education and training of professional mathematical scientists; and to make recommendations for action. This report describes the results of the Panel’s work.

The principal strategy used by NSF to achieve its objectives is to support research and education in universities. The report therefore focuses on NSF's performance in supporting the performance and teaching of academic mathematical sciences and in encouraging interactions between academic mathematicians and the users of mathematics.

The Process Used by the Panel

The Panel consisted of leading mathematicians drawn largely from outside the United States and individuals from important U.S. stakeholder communities that are strongly dependent on mathematics (science, technology, education, government, and finance). None had received recent NSF funding in mathematics. Members who were mathematicians brought to the Panel their expertise in the various subdisciplines, the progress of international research, and the means used by other nations to support mathematical research. Members of the stakeholder communities provided judgments on their mathematical needs, on opportunities for mathematicians in these communities, and on the effectiveness of mathematical knowledge in service to society.

The Panel enjoyed staff support from the Division of Mathematical Sciences, which provided data, analysis of data, and a wealth of reports. Members of the Panel met four times (March 20-22, June 5-7, September 5, and September 22, 1997) at the National Science Foundation Headquarters in Arlington, Virginia to study data and reports, to discuss appropriate criteria for making assessment, and to formulate recommendations of a qualitative nature.

The Panel also discussed, at considerable length, the vital importance of mathematics in K-12 education in the United States. Not without regret, the Panel concluded that, given its composition and expertise, it should refrain from making assessments or recommendations in this area. However, the Panel wishes to emphasize its sense of the essential importance of K-12 mathematics to the well-being of the United States, to underscore the assessments and recommendations made in previous reports, and to affirm that much needs to be done in this area. The education of present, and, more importantly, future, teachers, will be the key to the improvement of K-12 mathematics. Because the education of teachers is the task of current and future university and college mathematicians, the quality of graduate students in mathematics and the education they receive will be crucial to the improvement of K-12 mathematics.

The Structure of the Report

This report includes a benchmarking comparison of U.S. mathematics with mathematics in Western Europe and the Pacific Rim. The report combines detailed benchmarking of the individual fields of the mathematical sciences with a strategic analysis of the role of mathematics in building and maintaining U.S. strength across the range of fields that comprise and use mathematics. This range extends from fundamental discoveries in mathematics to the application of mathematics in other scientific and engineering disciplines and in such "user" areas as government, finance, and manufacturing.

The methodological sections of the report describe the procedure followed by the Panel (Chapter III) and the means and data used for making benchmarking comparisons (Chapter IV and Appendix 2).

The substantive results of the Panel’s work are presented as follows:

The mathematical sciences, their contributions to the nation, and their relationship to other disciplines and professions that use mathematics (Chapter II);

The findings of the panel (Chapter V);

Objectives for NSF, recommendations for achieving those objectives, and suggested milestones against which to measure progress (Chapter VII).

The report also includes (in Appendices) data which support the findings and materials that address many elements of the report in greater detail.