*Donald J. Lewis*

*Director, Division of Mathematical Sciences National Science Foundation*

Thank you for the invitation to participate in this workshop, "Actions for the Mathematical Sciences in the Changed Environment." I very much appreciate what the Board on Mathematical Sciences is seeking to achieve by this workshop and commend the Board for its efforts. I will address the issues implied by the title assigned to me by the organizers, but I want to do so in a somewhat broader context.

As you know there is a bipartisan agreement to balance the budget, and at the moment, defense and the entitlement programs appear to be politically untouchable and there is a mood to reduce, not increase, taxes. This leaves only the discretionary programs to Carry the burden of reaching a balanced budget. Both parties in their outreach projections have targeted the National Science Foundation with at most a 3% increase per year, probably the rate of inflation and maybe less. So at the very best, in constant dollars, NSF's funding will be flat—just as it has been for the last 5 to 6 years. This is optimistic, for as you probably know the American Association for the Advancement of Science is predicting a substantial drop in NSF's constant-dollar budget.

NSF Currently provides 52 to 55% of the federal funding for mathematical research and a considerably smaller percentage in the case of other sciences. These percentages could change radically in the next few years. All the Department of Defense agencies are reviewing their basic research portfolios, and it is rumored that there are pressures from the admirals and generals to reduce substantially or eliminate 6,100 budget accounts—the accounts that fund basic research. If this should occur, all the basic sciences will be hit severely, and the proposal pressure on NSF will be enormous in all the sciences.

The foundation is a very conservative organization and if you look back over its 45-year history you will find, with a few exceptions, that the percent of its budget going to any particular discipline has been pretty constant. One exception was a 15% increase in DMS's budget in response to David I (NRC, 1984); others were in response to the Mansfield amendment, and to decisions to build major facilities. As one famous chemist remarked, "God decreed in 1950 the distribution of resources amongst the sciences and no one has dared to go against the decree." So while I see a growing awareness within the foundation that the mathematical sciences are underfunded relative to their importance to science, I am not optimistic that there is currently the will or the capacity within the foundation to double DMS's budget, which would alleviate some of the pain now being experienced, or to triple it, which would put things in a real comfort zone. At least, there will not be such a will or capacity until the scientific community and especially the mathematicians convince the general American public that investments in science and mathematics will improve their lives. The public believes that the National Institutes of Health is working for the improvement of their health, and so NIH thrives even in a bad budget climate. We need to make the case to the corner grocer and the factory worker why their money should be invested in the mathematical sciences.

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Expectations and New Opportunities at the Division of Mathematical Sciences
Donald J. Lewis
Director, Division of Mathematical Sciences National Science Foundation
Thank you for the invitation to participate in this workshop, "Actions for the Mathematical Sciences in the Changed Environment." I very much appreciate what the Board on Mathematical Sciences is seeking to achieve by this workshop and commend the Board for its efforts. I will address the issues implied by the title assigned to me by the organizers, but I want to do so in a somewhat broader context.
As you know there is a bipartisan agreement to balance the budget, and at the moment, defense and the entitlement programs appear to be politically untouchable and there is a mood to reduce, not increase, taxes. This leaves only the discretionary programs to Carry the burden of reaching a balanced budget. Both parties in their outreach projections have targeted the National Science Foundation with at most a 3% increase per year, probably the rate of inflation and maybe less. So at the very best, in constant dollars, NSF's funding will be flat—just as it has been for the last 5 to 6 years. This is optimistic, for as you probably know the American Association for the Advancement of Science is predicting a substantial drop in NSF's constant-dollar budget.
NSF Currently provides 52 to 55% of the federal funding for mathematical research and a considerably smaller percentage in the case of other sciences. These percentages could change radically in the next few years. All the Department of Defense agencies are reviewing their basic research portfolios, and it is rumored that there are pressures from the admirals and generals to reduce substantially or eliminate 6,100 budget accounts—the accounts that fund basic research. If this should occur, all the basic sciences will be hit severely, and the proposal pressure on NSF will be enormous in all the sciences.
The foundation is a very conservative organization and if you look back over its 45-year history you will find, with a few exceptions, that the percent of its budget going to any particular discipline has been pretty constant. One exception was a 15% increase in DMS's budget in response to David I (NRC, 1984); others were in response to the Mansfield amendment, and to decisions to build major facilities. As one famous chemist remarked, "God decreed in 1950 the distribution of resources amongst the sciences and no one has dared to go against the decree." So while I see a growing awareness within the foundation that the mathematical sciences are underfunded relative to their importance to science, I am not optimistic that there is currently the will or the capacity within the foundation to double DMS's budget, which would alleviate some of the pain now being experienced, or to triple it, which would put things in a real comfort zone. At least, there will not be such a will or capacity until the scientific community and especially the mathematicians convince the general American public that investments in science and mathematics will improve their lives. The public believes that the National Institutes of Health is working for the improvement of their health, and so NIH thrives even in a bad budget climate. We need to make the case to the corner grocer and the factory worker why their money should be invested in the mathematical sciences.

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I do not wish to convey that we cannot make DMS's budget grow; we can and did increase it this year. But the growth will need to be incremental and will come because we have been innovative and are demonstrating careful management of the funds entrusted to us. The mathematical community will need to identify opportunities and help sell them. It probably will need to help identify the more mature areas, and those that have the greatest promise, and be prepared for DMS to do some shifting of funds. As a community, we will need to determine how best to use the limited funds provided to maintain and advance the discipline. This may mean arriving at alternative funding patterns. If we are not prepared to make hard choices, how can we expect others to do so? When the community is willing to make choices and move forward, suggesting areas of opportunity and responding to innovative modes of funding—as, for example, it did for the Group Infrastructure Grants (GIG) program—the foundation does respond with increased funding. The mathematical community must understand the changing environment in which we live, and then be agile in taking advantage of opportunities presented. The purpose of this meeting is to begin the process of taking charge of our destiny.
You should be aware that Congress is now requiring greater accountability of the agencies it funds. The NSF needs to come up with a set of goals and metrics to measure achievement of these goals. Crude counting, whether of students produced, papers published, and so on, will just distort the purpose of the foundation. For that reason it is seeking to have qualitative, rather than quantitative, measurements.
The foundation was founded to promote the progress of science, to advance the national health, prosperity, and welfare of U.S. citizens, and to secure the national defense. The national populace is the foundation's constituency, not the academic research community. Thus it is not sufficient for the NSF to ensure the welfare and progress of academic researchers, confident that their brilliance and hard work will take care of the needs of the nation. It needs to lead the community of researchers and educators in the most beneficial directions for the nation. That being the case, in a qualitative manner we will need to address how our funding has been beneficial to the nation. NSF is funding basic research, and so the time lines for this assessment cannot be one or two years, but since many lines of inquiry will not have a payoff, ever, it will not be convincing to Congress if every investment takes in excess of 50 years before there is any indication that the investment has had a payoff. As Eugene Wigner so ably asserted and demonstrated, mathematics is inordinately effective in its contributions to science—and, as we now know, to management and industry. In the years ahead we mathematicians will need to document this time and time again, and we will need to assist in a faster transfer of knowledge.
As I see it, the mathematics community is going to have to get into the business of telling its story and describing the impact it has had on the general populace, not only to Congress but, to the extent possible, to the person in the street. For far too long, we have been content to communicate individually with a small, select group of colleagues, making little effort to communicate even with the mathematical sciences community as a whole, let alone with our science colleagues. To the educated public we appear to be a small sect, inwardly looking, muttering in an incomprehensible language. If this is the case, how do we demonstrate our contributions to our nation's citizenry?
Our fellow scientists need our help. They need to reduce experimental trials, and they need models to simulate experiments and suggest the most appropriate areas of inquiry. They need the mathematician to help not only in modeling and simulation but also in reformulating their questions. Formulating precise questions is our forte. We have become so inner driven that we have forgotten our ancestry; we forget how much of mathematics was stimulated by physics.

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Today every area of science from the physical, to the biological, to the social, and to the managerial is becoming highly mathematized. Researchers in these areas need our help and, in turn, we will discover new areas of challenging mathematics. As I wander about the foundation, I find division directors excited about the idea of mathematicians collaborating with their researchers—excited not just in words, but willing to put up money to back up their words. If more mathematicians were to become active in multidisciplinary research, we could more quickly transfer mathematical discoveries and speed up the time line from mathematical discoveries to contributions to our nation's populace. Let me add that the mathematical expertise for multidisciplinary research should go beyond that of the analysts and computational mathematicians. All the subdisciplines of mathematics have a role to play and need to get involved.
Currently the easiest way to increase the DMS budget is via multidisciplinary research. Many fields of science have progressed to the point that it is easier to get new results by combining ideas from several disciplines, and frequently such results are closer to being useful for the well-being of society. Also, with minimal increases, if funding goes to a multidisciplinary area, then two or more disciplines benefit—thus providing the director with increased brownie points. Areas currently under discussion for possible new funding that would involve mathematics are communications, mining of massive data sets, and machinery for predictability. Each of these should involve several programs within mathematics. We also expect to see growth in materials science, mathematical biology, and computational science.
But the returns to mathematicians from entering into multidisciplinary research are far greater than increased funding and better assessment reports. There are the intellectual opportunities of opening new areas of basic mathematical research. The next half-century could be the golden age of mathematics both as an enabling science and as the queen of intellectual achievements. The National Research Council will soon undertake an in-depth study of the interplay between mathematics and science, looking at how mathematics enables and serves science and, in turn, finds new challenges. Recent interchanges have led to the development of the new field of quantum geometry. There is no one who would not consider it as belonging to core mathematics, and yet it will have deep implications in physics and cosmology. We believe that as mathematicians become more open to multidisciplinary discourse they will not only be good handmaidens but also will find exciting new areas of mathematical research. The opportunities are there. Will we seize them or squander them?
The integration of science and education is a theme we hear of daily within the foundation. It has become the perception of Congress and many citizens that the foundation's funding patterns have compromised the mission of the research universities, causing a bifurcation of their duties and of their staff into two groups—researchers who function as in a research institute, on the other hand, and a lower class that transmits knowledge, often in rote fashion, to the undergraduates. Further, there is the perception that our faculties have failed to convey the excitement and intellectual stimulation of inquiry, and have failed to provide students with hands-on experience in scientific inquiry—skills needed not only by those who become working scientists, but also by other workers and by citizens. This criticism of the foundation is not new. In 1965, after an extensive series of hearings by the House Science and Astronautics Committee, it was reported that serious scrutiny was needed regarding the balance between the foundation's support of teaching and education and its support of research. The foundation and the National Science Board were challenged to take a more proactive leadership role in setting science policy on such matters. The current foundation leadership is seeking to inspire and motivate the research

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community to address the issue of its role in ensuring that undergraduates learn the benefits and comprehend the method of scientific inquiry. Whether it is accurate or not, the perception that research universities are not functioning properly is undermining university funding by state legislatures and putting many at financial risk. The problem thus extends beyond the foundation and needs to be addressed in order for universities to survive.
The mathematics community has a better record than most of the other scientific disciplines regarding concern for undergraduates. But this record is spotty; not all departments have been involved. Too often our instruction has been too algorithmic, too formalistic, and unchallenging, and has required little intellectual involvement by students. Too often in the past, and even now, we would be hard-pressed to prove that we have taught students to think as mathematicians. Too often we have acquiesced to the use of large lectures and poorly prepared lecturers and temporary faculty. All too often we have been content to be the cash cow for our colleges.
As we know from the University of Rochester experience, failure to look after the undergraduates can undermine the viability of an excellent research group.1 As you surely know, Rochester's math department was (and is) not the only math department undergoing close scrutiny. When attention was paid to the undergraduates, the departments under scrutiny not only were kept intact but also frequently were given additional funds.
As mathematics faculty we need to be aware of the accounting changes rapidly spreading over the country. With these accounting systems, each unit sits on its own bottom. Tuition income and indirect costs need to balance salaries, supplies, equipment, and, in some cases, libraries and faculty maintenance and heating costs. Other units are free to provide their students with mathematical instruction in any way that meets their needs. Hence it is tempting for engineering colleges and others to build a contingent of adjunct lecturers to provide mathematical instruction, cutting the cost below what a mathematics department might charge under these new accounting procedures. If we do not provide top-quality instruction, we will lose much of our service instruction. Considering the ratio of credit hours earned by in-service teaching compared to that for majors and graduate students, it is clear that service teaching has been an important source of income. Can we afford to lose it?
At present the NSF provides very little funding for mathematics graduate students (about $10 million including fringe indirect costs). This is low compared to the other sciences. But it is nigh impossible to make the case for increased funding of graduate students in mathematics when 15% of the new doctorates are unemployed, and perhaps as many are underemployed, and the time to degree keeps lengthening. In contrast to other disciplines, we have created a culture where academic employment is the only honorable employment. Yet we see time and time again that industry wants, needs, and values mathematicians. If we are concerned with the nation's well-being, don't we need to address how to provide industry with the mathematicians it needs and some of its citizens with the skills to contribute?
A year ago the Directorate of Mathematics and Physical Sciences had a workshop on graduate education and postdoctoral training (NSF, 1996). Mathematicians need to consider the resulting report and respond to it or, alternatively, decide on better principles. I expect MPS, within the month, to fund several innovative graduate math programs as demonstration projects, and more next year, if they are forthcoming from the community.
1
See the paper by Stillinger for a brief description of this experience.

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One problem the community needs to address is its current dependence on foreign graduate students. There is clear evidence that as the Pacific Rim continues to develop its science investment, many of those educated in the United States will return home. This could deplete our current ranks of top-level young researchers and will certainly lead to overseas economic challenges. Has this dependence really been in the nation's best interests? We once produced excellent researchers who were U.S. citizens. Why now the dependence on foreign students? Have we failed at the undergraduate level? Have we taken the easy way of preferring the more strongly prepared student? Can we defend our actions to the American public in terms they understand?
As mathematicians we envy the way astronomers have developed support from the general public for their research, which we view as being just as esoteric as anything mathematicians do. We hear that they succeed because one-third of the citizens once owned a telescope, even if it did no more than show faint details of the moon's surface. We hear that many college students study introductory astronomy to satisfy distribution requirements and so have had some exposure. We fail to remember that a far larger percentage take calculus, and that when colleges introduced quantitative reasoning requirements, we let other disciplines respond or we offered college algebra. The newspapers run mathematical puzzles each week, which suggests that there are readers who find such things interesting. So, there is at least a latent interest in mathematics in quite a good percentage of the reading public.
Years ago Dick Otter and I offered a course in number theory to freshman humanities students in which the students computed, conjectured, and often made proofs. They experienced what it was to be a mathematician. These students consistently asked for the second semester course, and in the senior poetry publication they made numerous references to number theory. Every time a readable book like Chaos (Gleick, 1987) appears, it makes a reasonable climb on the bestseller's list. Have we totally missed opportunities to build a supportive group for mathematics among the general public? Can we learn from the astronomers? Do we need to train and encourage those who can communicate about mathematics to the general public?
I have raised the issues of education, communicating with the general public, especially the educated public, and multidisciplinary research. I have not said much about inner-directed research because I believe we all agree on its importance. I do feel that each of the activities mentioned will flourish only if all the others do. Individually, we probably cannot be involved in all these activities at the same time, and some individuals may participate in only one or two in their lifetime. But if all are necessary for mathematics to flourish, then as a community we must see that each activity thrives, and honor those who contribute to any one of them we cannot have one activity being viewed as more superior.
Funding modes need the community's attention. Currently, DMS is overloaded with proposals; it funds fewer than a third of those submitted, and funds them very inadequately—all this despite the fact that many mathematicians do not even submit proposals because of the conviction that to do so is pointless. The program officers have no time to visit and find out what researchers are doing. As a community we should be looking for alternative, efficiently managed modes of support that would advance our discipline. Would departmental grants serve that goal? They would be a more efficient mode, and they would allow us to consider both the education (graduate and undergraduate) and the research effort of a department as well as expository and informal communications efforts. Your thoughts on this approach could be informative.
I look forward to your discussions and final report, and hope they are directed to maintaining and enhancing the discipline we so love.

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References
Gleick, J., 1987, Chaos: Making a New Science, Viking, New York.
National Research Council (NRC), 1984, Renewing U.S. Mathematics: Critical Resource for the Future ("David I" report), National Academy Press, Washington, D.C.
National Science Foundation (NSF), 1996, Graduate Education and Postdoctoral Training in the Mathematical and Physical Sciences, Workshop Report, June 5-6, 1995, document NSF 96-21 , National Science Foundation, Arlington, Va. (To order, see the summary on NSF's home page on the World Wide Web at http://www.nsf.gov:80/mps/workshop.htm, or request a copy via e-mail to sspencer@nsf.gov.)