## Mathematical Sciences Investment

The National Science Foundation has proposed a major increase in funding for research in the mathematical sciences. An interagency team has been formed to lead a national effort to address the needs of broad, nationally important applications and to exploit emerging opportunities. The National Science Foundation will take the lead in FY2002.

This investment is motivated both by current funding problems, which are creating a threat to U.S. pre-eminence in this critical field, as well as by the demand for new results and more trained specialists to work on what has been an explosion of new applications. Without substantial increases in federal funding, the mathematical sciences cannot continue to advance science and technology in the U.S. and the world.

The proposed investment will be directed primarily to: fundamental mathematics, connections to other sciences and engineering (through applied mathematics, statistics, and computational mathematics), and mathematical sciences education.

**Academic Research Funding in the Mathematical Sciences**

In 1997, the federal government invested roughly $200M in academic research in the mathematical sciences. Of this amount, the National Science Foundation's share was approximately two thirds; the Department of Defense (AFOSR, ARO, ONR, and DARPA) provided approximately 18%, and DOE 6%. NSF's share of university funding is larger because the other agencies, especially DOE, spend a substantial portion of their research funding within their own laboratories. During the four-year period from FY1997 to FY2000, the NSF budget in the mathematical sciences grew at an annual rate of about 3.3%—sufficient to keep pace with inflation but not enough to provide for any real growth in research. While NSF funds a majority of federal academic research in mathematics, other agencies play an important role in specific sub-areas. The Air Force Office of Scientific Research is the major source of support for mathematical control theory, and the Department of Energy's Science Office plays a major role in computational mathematics, to cite two examples.

Federal funding for basic research in the mathematical sciences from DoD and DOE has not increased since the mid-1980s. As these sources of academic funding have declined in real terms, the National Science Foundation has experienced increased pressure on its research budget.

**Role of the Mathematical Sciences in Science and Technology**

Basic research in the mathematical sciences is a fundamental enabler of developments in virtually all areas of science and technology. Mathematical modeling and computational simulation are essential tools for modern design and control, as well as for the prediction and understanding of phenomena across the sciences.

This role is well understood, but its impact is perhaps less well appreciated. Through the use of computational design and simulation, mathematical research of the past decades has, in large part, fueled recent increases in productivity. Mathematics has transformed industrial design. The design of modern commercial aircraft and high-performance military aircraft, for example, rely on mathematical tools developed in the last two decades. The Boeing 777 is a well-known example of a commercial aircraft designed through computational simulation. In military aircraft, the development and testing of stealth capability depend in large measure on the large-scale computational codes that simulate the reflection of electromagnetic waves from airplane surfaces. Mathematics has contributed to advances in nearly every niche of technology. Numerical weather prediction is an everyday tool that uses models of fluid flow coupled with modern computational algorithms. Even the technology for ensuring the reliability and safety of our nation's nuclear stockpile, as well as for reducing environmental hazards from waste, now depends on computational simulation.

Modern medicine and the life sciences increasingly rely on mathematics to simulate, visualize, and predict new structures, and then to organize and understand, quickly and cheaply, the data needed for the development and certification of more effective devices and pharmaceuticals.

Recent advances in information technology and the explosion of the Internet have depended on mathematical advances of the past few decades, and further progress will require continued new developments in mathematics and related disciplines. Web search engines use modern numerical algorithms. Dynamic routing algorithms developed by mathematicians over the past fifteen years are speeding up Internet routing. This last case is an example of how research in the mathematical sciences quickly finds its way into applications and the U.S. economy--such ideas led to the creation of a startup company (Akamai) that is now publicly traded, and is but one example among many.

Even the national economy and financial markets rely strongly on mathematical tools that were developed through U.S.-funded basic research. Mathematical tools, such as the Black-Scholes algorithm, are used to price securities and options and to regulate the markets. These powerful tools were developed through research in stochastic differential equations. Research in mathematical finance continues to be an important application area, and Wall Street remains a strong source of demand for graduates from the mathematical sciences.

Chart 1

As computers become more powerful, mathematics penetrates more deeply into every area of science and technology. Today, mathematics finds application directly and rapidly through computer codes. While advances in computers attract a lot of national attention, it should be recognized that mathematics is equally important for advancing capabilities, if not more so. As computer hardware improves, leading to a doubling of computer speeds every 1.5 years, improvements in mathematical algorithms yield similar improvements with similar speedups, as shown in Chart 1. More importantly, though, new mathematical developments make it possible to solve problems that could not be handled with older methods, even with hardware improvements.

**Indications of a Crisis**

Chart 2

There is reason for urgency. We face a serious decline in enrollments in undergraduate and graduate programs in the mathematical sciences. We have been surviving on talent from outside the U.S. but cannot rely on this source in the future. Chart 2 shows trends in graduate enrollments in the mathematical sciences.

What factors are contributing to this trend? Grant sizes are extremely small in the mathematical sciences. Furthermore, only 35% of active researchers in the mathematical sciences receive federal grant support, compared with roughly 67% of researchers in the biological and physical sciences. This imbalance may send signals to researchers, educators, and prospective students about how the discipline is valued. But more importantly, it results in a smaller pool of funds for training new students.

Chart 3

Chart 3 shows that the level of federal support for graduate students in the mathematical sciences is about a third of that for students in other major disciplines. Under these conditions, it is very difficult to attract and keep graduate students in the mathematical sciences.

**Status of Graduate Education in the Mathematical Sciences**

The following points highlight an alarming trend:

Between 1992 and 1999, full-time graduate enrollment dropped by 21% (by
27% for U.S. citizens).

In 1997, only 12% of full-time graduate students were supported by research
assistantships.

Between 1992 and 1999, numbers of upper-division mathematics majors dropped
by 23%.

The Senior Assessment Panel of the International Assessment of the U.S. Mathematical Sciences (The "Odom Report," National Academy of Sciences, March 1998) concluded that, "Based on present trends, it is unlikely that the U.S. will be able to maintain its world leadership in the mathematical sciences." It is essential that the U.S. maintain leadership in certain critical sub-fields in order to leverage advances and to transition those results into corresponding advances in sciences and technology critical to the U.S. The U.S. now relies on foreign talent—over 50% of U.S. graduate students in the mathematical sciences come from outside the U.S.—and while many of these students have chosen to remain in the U.S., this is not a sure way to develop future pre-eminence.

**Opportunities and an Investment Strategy**

Mathematics itself is not facing a decline in demand or opportunities. Quite the contrary—there is an explosive need for mathematical research.

Interdisciplinary collaborations have become an increasingly important aspect of research in the mathematical sciences, helping to steer research into important directions as well as to transition research results into useful applications (see the National Research Council Report "Strengthening the Linkages between the Sciences and the Mathematical Sciences," National Academy Press, 2000).

The interdisciplinary nature of the research necessitates that a certain portion of funding be for collaborative research among disciplines, but it also argues strongly for putting mathematics and the application disciplines on an "equal footing" with respect to federal funding. The mathematical research itself must be nurtured and renewed along with the applications.

One of the reasons for the utility of mathematics lies in its abstraction. Models for phenomena in one application area are often applicable in another. As a result, mathematical results can be transferred quickly. But mathematics has its own infrastructure to be developed, and to that end further fundamental research in mathematics itself is required. Interdisciplinary research lies on the border between mathematics and an application discipline. It can be driven by the pull of the application or by the push to drive new mathematical ideas into new understanding and technology in an application area.

We are living on past investments in basic research in the mathematical sciences.

Future advances are at risk.

**The Mathematical Sciences Investment Will Focus on Strategic Targets**:

**Individual Investigators **

*support researchers in the mathematical sciences at a level competitive
with other sciences; renew the creative base*

**Focused Research Teams **

*enable team approaches to critical problems and applications*

**Interdisciplinary Grants **

*nurture connections to new problem areas*

**Educational Awards**

*establish career paths in teaching, training and research*

**Research Centers and Institutes**

*broaden support across more application domains where persistent attacks
are needed*

**Benefits to the Nation**

Foster advances in fundamental mathematics with important long-term benefits
for science and technology.

Integrate state-of-the-art mathematical and statistical principles, tools
and concepts with all NSF research.

Expand interdisciplinary research partnerships with other sciences and
engineering disciplines.

Create a new generation of interdisciplinary researchers.

Achieve increased diversity of U.S. students to meet labor requirements
in academia, industry, and government.

Provide vigorous technical preparation of mathematics teachers.

The Society for Industrial and Applied Mathematics (SIAM) endorses the proposed increase for funding in the mathematical sciences as part of the overall NSF budget and urges enhanced federal funding for science.