This chapter considers the impact of the current configuration of U.S. mathematical sciences research institutes in facilitating the development of the mathematical sciences and shaping the U.S. mathematical sciences community. It is based both on the collective experience of the committee members and on the inputs received through the committee's call for comments (see the appendix). Overall, the committee was favorably impressed by the breadth and depth of contributions of the many mathematical research institutes both within and outside the United States.

Perhaps their most important contribution has been that the U.S. mathematical sciences research institutes provide an especially favorable atmosphere for high-level mathematical sciences researchers to focus with considerable concentration on their investigations. In exceptional cases, a few of the world's best mathematicians have had a lifetime opportunity to achieve their goals in a mathematical institute. More typically, established mathematical scientists visiting these institutes have carried out important projects and developed new ideas during their stays.

The senior mathematical scientists working at institutes have helped set the direction of the mathematical sciences and emphasized the importance of current mathematical discoveries and developments. Institutes have served as a forum for the mathematical sciences through their postdoctoral programs, conference and lecture series, and pre-publications of new mathematical research results. Because short-term visitors from both academia and industry have the opportunity to pose problems and enter into collaborations with longer-term institute visitors, the effect on the mathematical sciences is far broader than might be suggested by the relatively small numbers of long-term visitors.

By sponsoring special year-long programs that highlight specific mathematical areas and applications, mathematical institutes have provided encouragement and support for targeted areas. Frequently, a particular field experiences considerable stimulation from such a program, and the resulting mathematical progress continues to fuel developments in succeeding years. New and important areas of research have emerged and evolved as a result. For example, the mathematics of materials science, the linkage of operator algebras and knot theory, and the applications of statistics to problems of transportation and drug design have all emerged from special-year programs at U.S. mathematical research institutes (see Box 2.1).

Most mathematical research institutes host mathematical scientists from many nations. The opportunity for mathematical scientists from different countries to work together has been a

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2
Current U.S. Research Institutes in the Mathematical Sciences—Impact and Continuing Need
Impact of Existing U.S. Research Institutes in the Mathematical Sciences
This chapter considers the impact of the current configuration of U.S. mathematical sciences research institutes in facilitating the development of the mathematical sciences and shaping the U.S. mathematical sciences community. It is based both on the collective experience of the committee members and on the inputs received through the committee's call for comments (see the appendix). Overall, the committee was favorably impressed by the breadth and depth of contributions of the many mathematical research institutes both within and outside the United States.
Impact on Research
Perhaps their most important contribution has been that the U.S. mathematical sciences research institutes provide an especially favorable atmosphere for high-level mathematical sciences researchers to focus with considerable concentration on their investigations. In exceptional cases, a few of the world's best mathematicians have had a lifetime opportunity to achieve their goals in a mathematical institute. More typically, established mathematical scientists visiting these institutes have carried out important projects and developed new ideas during their stays.
The senior mathematical scientists working at institutes have helped set the direction of the mathematical sciences and emphasized the importance of current mathematical discoveries and developments. Institutes have served as a forum for the mathematical sciences through their postdoctoral programs, conference and lecture series, and pre-publications of new mathematical research results. Because short-term visitors from both academia and industry have the opportunity to pose problems and enter into collaborations with longer-term institute visitors, the effect on the mathematical sciences is far broader than might be suggested by the relatively small numbers of long-term visitors.
By sponsoring special year-long programs that highlight specific mathematical areas and applications, mathematical institutes have provided encouragement and support for targeted areas. Frequently, a particular field experiences considerable stimulation from such a program, and the resulting mathematical progress continues to fuel developments in succeeding years. New and important areas of research have emerged and evolved as a result. For example, the mathematics of materials science, the linkage of operator algebras and knot theory, and the applications of statistics to problems of transportation and drug design have all emerged from special-year programs at U.S. mathematical research institutes (see Box 2.1).
Impact on Mathematical Quality and Culture
Most mathematical research institutes host mathematical scientists from many nations. The opportunity for mathematical scientists from different countries to work together has been a

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Box 2.1 Some Noteworthy Impacts of U.S. Mathematical Sciences Institutes
Advances in Materials Science
The Institute for Mathematics and Its Applications (IMA) recognized early that materials science, a critical technology area for the nation, also represented an important opportunity for the mathematical sciences. The IMA helped build a mathematical research community in this area more than five times the size of the community 10 years earlier, and research advances at the IMA were significant. Based on work done at the 1995–1996 IMA Year on Mathematics in Materials Science (which in turn built on a 1984–1985 IMA program focusing on continuum physics and partial differential equations), microstructure theories of martensitic materials were developed to the point that the behavior of new materials could be predicted. A particular advance was the new concept for a hypothetical material combining two types of transformations: a ferromagnetic transition and a martensitic transformation. This research led to a successfully implemented strategy for developing this class of materials. As a result, new alloys have been produced that exhibit this magnetostrictive effect to a degree some 50 times greater than what had been observed in the previous record holders, the so-called giant magnetostrictive materials. Today, nearly every area of active materials science research includes research mathematicians, and the mathematical sciences are playing a crucial role in several large federal thrusts in materials research. The IMA had a major influence in coalescing and nurturing this development.
Knots and Protein Folding
In the mid-1980s a deliberately planned juxtaposition of two research programs at the Mathematical Sciences Research Institute was the catalyst for interaction between Vaughan Jones, whose research concerned operator algebras, and researchers in low-dimensional topology. That interaction led Jones to notice that a sequence of algebraic relations discussed was similar to those that define a mathematical object called the braid group. Jones then found that this correspondence was more than just an analogy, and he obtained a general invariant for characterizing knots and links that was different from and more useful than a previously known classical one. This insight rapidly led to many advances and, ultimately, to a deep pairing of these ideas with another mathematical area, algebraic K-theory. Jones received a Fields Medal (the mathematical equivalent of a Nobel Prize, but which is awarded only once every 4 years) in part for this development.
Pharmaceutical Design
A new research approach to drug discovery was opened through sequential drug design strategies developed at the National Institute for Statistical Sciences. As a consequence, the high-throughput screening of hundreds of thousands of drug compounds is now feasible, in a unique combination of computational chemistry, computer science, and statistics.
considerable boon for the mathematical sciences both in the United States and abroad. Frequently, U.S. mathematical scientists form a large cohort in foreign institutes just as foreign mathematical scientists are frequent visitors to U.S. mathematical institutes. It is vital to preserve these opportunities for mixing different mathematical cultures. Fundamental mathematical research in the United States will continue to profit by having a significant proportion of the most active U.S. mathematical researchers exposed to the best experts worldwide.
Increasingly, mathematical institutes are providing a bridge linking academic mathematical scientists and industries keen on finding new mathematical tools for their work. For instance, the industrial postdoctoral research fellow positions developed by the Institute for Mathematics and Its Applications have given considerable encouragement to mathematical areas that are directly applicable to industry and have also heightened recognition in the marketplace of

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the relevance and importance of mathematical sciences research. Interactions between mathematical scientists and those industrial researchers are typically not one-way transfers of mathematical techniques. Many mathematical science ideas—for example, finite elements, posipolynomials, splines, and spectral partitioning—have arisen in some heuristic form during contacts between mathematical researchers and applications-oriented researchers with questions to be addressed. Mathematical research institutes have helped to foster this important interaction.
Vitality of the U.S. Mathematical Enterprise
Of course, mathematical research institutes not only contribute to the development of the mathematical sciences and their applications, but also enhance the well-being of the U.S. mathematical community. Senior mathematical scientists visiting a mathematical institute are provided with the opportunity to interact at some length with their peers, resulting in collaborative efforts among like-minded researchers and also leading to unexpected interplay between seemingly disparate areas. This cross-fertilization contributes to the coherence of the mathematical community as well as to the advancement of scientific knowledge (see Box 2.2).
Moreover, when junior mathematical scientists attend one of the mathematical research institutes, they often enjoy their first extended period of uninterrupted research. This period of relative calm has frequently made the critical difference in enabling young mathematical researchers to develop their most creative ideas. Mathematical institutes also provide younger mathematical researchers with opportunities to encounter for the first time other researchers working on similar problems. Frequently, younger researchers are captivated by new developments discussed at these institutes and may find that they can profitably shift the focus of their own research.
For postdoctoral fellows, an appointment at a mathematical research institute is an opportunity to deepen and broaden their doctoral education. The opportunity to associate with important figures in their area of research has often opened vistas to postdoctoral fellows that have led to dramatic advances in their research careers. The mentoring such postdoctoral fellows received from their PhD thesis advisors is often then carried on by other senior mathematicians.
Many others in the mathematical community derive considerable benefit from short visits to research institutes, typically for conferences and workshops. For mathematicians who are somewhat isolated, a visit to a mathematical institute can provide an especially important opportunity to connect with others working on similar problems, and to learn what areas of research are being studied most intensively at mathematical centers.
Benefits to Mathematical Education and Other Areas
The postdoctoral programs of mathematical institutes constitute their major educational component. However, in addition to holding workshops and conferences that educate all levels of the mathematical sciences community (including university faculty and researchers from government laboratories and industry), increasingly institutes organize research conferences that include special lectures, and sometimes entire courses, for graduate students. Mathematical institutes now sponsor summer programs (each focused on a specific research area) targeted to graduate students and summer programs for undergraduate students and pre-college mathematical sciences teachers. They also occasionally sponsor workshops devoted to mathematical education.
Beyond benefiting the mathematical sciences and the community of mathematical scientists, mathematical research institutes benefit industry and commerce through the aforementioned industrial postdoctoral programs and workshops on applications of the

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Box 2.2 Personal Experiences at U.S. Mathematical Institutes (extracted from the committee's call for comments)
[A] short visit at IMA (and MCIM) ... expose[d] me to the people and resources of IMA, which ... eventually ... [led] to several interesting collaborations (one extraordinarily successful).
The interactions with the IMA have been extremely beneficial to me and my company.
I spent about 25 days, in three installments, at the MSRI .... Two weeks were workshops, the rest just participation in the regular activities. It had a tremendous impact on me. In addition to being informed about recent progress and controversies in my subfield proper in one of the workshops, I learned about main current directions in the related field of algebraic geometry in the other workshop. I also had a chance to discuss with colleagues my then emerging interest in a new field ... [and received encouragement rather than discouragement]. Both helped me shape my program. Those discussions taught me about results and open problems ... that might be related to my new interests. In a number of cases those issues indeed proved to be related and had a major impact on the first paper I wrote on the subject. MSRI ... has enriched my research—or shall I say, my mathematical existence—in a way that no other institutional framework has ever done.
My first job after getting my PhD was at the IMA.... For me, it was exactly the right place to be. I found the mix of pure and applied topics exciting, and the opportunity to mix with the leaders of the field on a regular basis invaluable. I rank my stay at the IMA as the pivotal event in my career—I'm still working on some of the research projects I started in Minneapolis.
I visited MSRI for a semester. Without a doubt this was the most exciting, stimulating, challenging period I have experienced since my PhD.... No university in the world can offer the stimulation and challenge that simultaneous contact with so many top researchers offered.... One might think that a good conference offers much the same stimulation; but this is erroneous. I needed the continuing contact over a period of time for the issues to really become clear.
I spent 10 months at the IMA as the main principal organizer of a year's program. Besides postdocs who were at the IMA for a year, there were several people who came for about one quarter, some who came for two quarters, and a couple more who spent the whole academic year. Then of course there were people who came just for a workshop and others who were there for 2 to 4 weeks. There was considerable interaction among the visitors and the postdocs who were selected for the specific program. Many joint papers were written that would not have been written otherwise ... most of them were very good to excellent. The experience at the IMA was very valuable for the postdocs not only for the mathematics learned and created but also for the contacts they made. Daily interaction was the norm rather than the exception.
mathematical sciences to real-world problems. Scientists and engineers outside academia can visit these institutes as well as interact through participation in problem-solving forums. Moreover, basic scientific research in other scientific, technological, and engineering areas, especially mathematical physics, has traditionally been an important beneficiary of the programs and concerted efforts of mathematical research institutes. Indeed, some of today's institutes consist of a roughly equal mix of mathematical scientists and physicists.
Finally, some mathematical research institutes have commenced activities to make the results of research in the mathematical sciences more accessible to the general public. Their experimentation with making materials available on the Internet suggests that this aspect of their work might expand.

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The Continuing Value of Broadly Based Research Institutes in the Mathematical Sciences
The committee believes that broadly based mathematical research institutes (labeled as category 2 in Chapter 1) are of critical importance to the continuing success of mathematical sciences research in the United States. Such institutes are now indispensable, strategic components for the health, strength, and world preeminence (COSEPUP, 1997) of the nation's mathematical research enterprise. Among the specific needs addressed successfully by such mathematical research institutes are the following:
1.
Decisively advancing mathematical research, and ensuring that its progress is robust;
2.
Catalyzing group and team interactions focused on topics of noteworthy potential;
3.
Supplying high-quality outreach to and interaction with industry and the scientific community;
4.
Providing first-class postdoctoral programs in both core and interdisciplinary mathematics;
5.
Enabling renowned senior researchers to direct, influence, and mentor younger scientists at crucially beneficial points in those younger researchers' careers;
6.
Sharpening, both in core and applied areas, mathematical research's focus via quick-response workshops on key, cutting-edge issues and fast-breaking ''hot topics";
7.
Enriching and invigorating mathematical education at every level; and
8.
Being a hub for mathematical resources, archives, and tools.
Of course some university departments also can and do address at least some of these needs.
One of the main purposes of broadly based mathematical research institutes is to bring together internationally recognized scientists to work on fundamental problems and forge new directions for research. A successful program at such an institute can have an impact on an entire mathematical field by setting common standards and establishing research agendas, thereby communicating beyond the leading centers of research an understanding of what is difficult, what is key, and what is exciting. The broadly based institutes strongly emphasize innovation and provide a national forum for research on both core mathematics and mathematical sciences areas at the interface of fields beyond a discipline's traditional boundaries.
Mathematical scientists at all levels of experience who participate in broadly based institute programs return to their home institutions with a new perspective on important research problems and with contacts to a network of other researchers that can influence their future research collaborations. In particular, the 1-year institute programs serve as an important training ground for postdoctoral fellows. By providing mentors who are internationally recognized researchers, such programs give young researchers a chance to immerse themselves in their own and related areas to a depth unmatched at many universities. Institutes also help foster graduate student involvement in the mathematical research enterprise.
Some broadly based mathematical institute programs bring together researchers from industry, government, and academia in order to address in a concentrated time period more facets of mathematical problems from a given area than would be feasible at most individual universities. These initiatives build partnerships among the mathematical sciences, other scientific disciplines, and the industrial sector. They also foster creative research that can lead to innovative applications of the mathematical sciences in business and industry. The broadly based institutes are now also contributing to the enrichment of mathematical education by providing special events and resources for high school teachers. At the same time, their public outreach activities are helping to enhance the image of the mathematical sciences in society.
In short, broadly based mathematical research institutes play a major international role in the development of improved communication of major developments in the sciences. The proliferation of mathematics research institutes worldwide, many modeled on the existing facilities in the United States, testifies to the impact and success of these broadly based institutes.

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While many sorts of other mathematical institutes now exist, there nevertheless continues to be a vital need for broadly based mathematical research institutes in the United States along the lines of, but not necessarily identical to, the ones now in operation.