Research institutes in the mathematical sciences have fostered and can continue to accelerate both the central role that the mathematical sciences are playing in our rapidly evolving society and the growth of core and applied mathematics and statistics to meet ever increasing demands for mathematical sciences knowledge. These institutes are catalysts stimulating far-reaching mathematical developments while serving the strategic and critical needs of science and society.

The mathematical sciences community's responses to the committee's call for comments clearly indicated that the existing broadly based mathematical research institutes are serving well, and should continue to fulfill, very important needs of both the mathematical community and its applications partners.

Considering new needs in a rapidly changing environment, this report presents sketches of two new types of research institutes proposed for the mathematical sciences. These new institutes would address needs of the mathematical sciences community as it strives to meet the challenges presented by other sciences and society. The sketches offered are intended only as guidelines. These two new types of research institutes would enable the mathematical sciences to increase their contributions to U.S science, engineering, and technology, and would strengthen the nation's position in science and technology.

The committee believes that the needs (identified in Chapter 3) that the two new types of institutes would be designed to address, and the contributions they would make in addressing those needs, constitute a strong justification for the new funding that would be necessary to establish them. It is hoped that a competition for proposals for such institutes would encourage many innovative approaches to their design.

The committee emphasizes that mathematical research institutes must be based on a strong and vigorous mathematical research community. With this in mind, *the committee strongly believes that it would not be in the best interest of either the mathematical sciences community or society as a whole to transfer funding from existing mathematical sciences individual (principal investigator) research grant programs to funding for existing or additional mathematical sciences institutes*.

At the same time, the committee notes that the modest increase by 53% in federal funding for U.S. mathematical sciences research during the past 15 years has not kept pace even with the growth of the U.S. mathematical research community and does not correct imbalances noted in the David I report.^{1} Changes in national and federal agency priorities have also occurred in that time frame, leading to decreases in support for curiosity-driven research. For example, sponsors such as the Air Force Office of Scientific Research, the Office of Naval Research, and the Army

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4
Recap and Closing Comments
Research institutes in the mathematical sciences have fostered and can continue to accelerate both the central role that the mathematical sciences are playing in our rapidly evolving society and the growth of core and applied mathematics and statistics to meet ever increasing demands for mathematical sciences knowledge. These institutes are catalysts stimulating far-reaching mathematical developments while serving the strategic and critical needs of science and society.
The mathematical sciences community's responses to the committee's call for comments clearly indicated that the existing broadly based mathematical research institutes are serving well, and should continue to fulfill, very important needs of both the mathematical community and its applications partners.
Considering new needs in a rapidly changing environment, this report presents sketches of two new types of research institutes proposed for the mathematical sciences. These new institutes would address needs of the mathematical sciences community as it strives to meet the challenges presented by other sciences and society. The sketches offered are intended only as guidelines. These two new types of research institutes would enable the mathematical sciences to increase their contributions to U.S science, engineering, and technology, and would strengthen the nation's position in science and technology.
The committee believes that the needs (identified in Chapter 3) that the two new types of institutes would be designed to address, and the contributions they would make in addressing those needs, constitute a strong justification for the new funding that would be necessary to establish them. It is hoped that a competition for proposals for such institutes would encourage many innovative approaches to their design.
The committee emphasizes that mathematical research institutes must be based on a strong and vigorous mathematical research community. With this in mind, the committee strongly believes that it would not be in the best interest of either the mathematical sciences community or society as a whole to transfer funding from existing mathematical sciences individual (principal investigator) research grant programs to funding for existing or additional mathematical sciences institutes.
At the same time, the committee notes that the modest increase by 53% in federal funding for U.S. mathematical sciences research during the past 15 years has not kept pace even with the growth of the U.S. mathematical research community and does not correct imbalances noted in the David I report.1 Changes in national and federal agency priorities have also occurred in that time frame, leading to decreases in support for curiosity-driven research. For example, sponsors such as the Air Force Office of Scientific Research, the Office of Naval Research, and the Army
1
In its 1984 report, Renewing U.S. Mathematics: Critical Resource for the Future (David I report; NRC, 1984), a committee of the National Research Council found that although the mathematical sciences in the United States were thriving intellectually, funding had deteriorated to the dangerously low level of $78 million in constant 1984 dollars. To enable the mathematical sciences to continue to play their synergistic role with the broader science and engineering community, which had expanded, the committee called for an increase in federal support to the level of $180 million (in 1984 dollars) over the 5 years from 1985 to 1989. However, that level of funding was never achieved. In 1998 the total federal support for mathematics was $180.8 million ($119.1 million in 1984 dollars), 57% of which was concentrated in the National Science Foundation (Thompson, 1998).

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Research Office have grown more mission oriented in recent years. This trend is cause for concern because many, and perhaps most, significant innovations in scientific or engineering fields began with one or two people, on a small scale, and in an unglamorous but curiosity-driven way. For example, it was not anticipated that purely mathematical, number-theoretic basic research on elliptic curves would become of key importance in secure encryption procedures that are now widely used in (to give just one context) banking.
The mathematical sciences are a comparatively low-cost, high-leverage contributor to the nation's research enterprise. In many instances, mathematical developments have been springboards for the launching of new industries, medical imaging and image compression being just two. The ever increasing role that the mathematical sciences play in science, technology, engineering, business, and industry warrants increased government funding for mathematical sciences programs and, in particular, new funding for new research institutes in the mathematical sciences. New funds should be made available so that serious consideration can be given to establishing research institutes for mathematical sciences in emerging fields and a research institute for experimental and electronic tools in the mathematical sciences. The establishment of these institutes would be a good investment involving minimal risk and offering a solid likelihood of great impact and benefit to the nation.