The system of education evinced by mathematical sciences doctoral and postdoctoral programs in the United States has deep roots in the European system. The European emphasis on research, and, in particular, on fundamental research, has guided the development of the American system during the past half century. A brief overview of the history of the mathematical sciences in the United States suggests why this is the case.

The statistical and historical data presented in this chapter and on occasion in the rest of this report are taken from the annual AMS-MAA surveys in the *Notices of the American Mathematical Society* (1980–1991), *The Mathematical Sciences: A Report* (NRC, 1968), *A Century of Mathematics in America, Part I* (Duren, 1988), *Science and Engineering Doctorates: 1960–86* (NSF, 1988), and *A Challenge of Numbers* (NRC, 1990b).

During the 19th century, mathematics consisted of pure mathematics, some mathematical physics (the applied mathematics of those days), and statistics. Statistics had a separate professional identity, fostered by the American Statistical Association, which was founded in 1839, and thus was the earliest domestic professional society in the mathematical sciences. However, statistics as an academic discipline was considered to be an integral part of mathematics. Europe was the center of mathematical research, and American mathematics relied mainly on European universities to train American doctoral students and on people trained abroad to staff American colleges and universities. A base for future development was built with the establishment in 1876 of Johns Hopkins University, the first American research university, the founding of the American Mathematical Society in 1888, and the development of research programs at the University of Chicago toward the turn of the century. Respected journals were established to disseminate and stimulate mathematical research, among them the *American Journal of Mathematics,* in 1878, the *Annals of Mathematics,* in 1884, and the *Transactions of the American Mathematical Society,* in 1900.

During the period 1910 through 1930, the American mathematical community was small but active. Strong programs grew at Harvard and Princeton Universities, joining that at the University of Chicago. Mathematicians played a role in World War I primarily by calculating trajectory and range tables. Concern for mathematics education was growing, as attested to by the founding of the Mathematical Association of America in 1915.

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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States
2
HISTORICAL PERSPECTIVE
The system of education evinced by mathematical sciences doctoral and postdoctoral programs in the United States has deep roots in the European system. The European emphasis on research, and, in particular, on fundamental research, has guided the development of the American system during the past half century. A brief overview of the history of the mathematical sciences in the United States suggests why this is the case.
The statistical and historical data presented in this chapter and on occasion in the rest of this report are taken from the annual AMS-MAA surveys in the Notices of the American Mathematical Society (1980–1991), The Mathematical Sciences: A Report (NRC, 1968), A Century of Mathematics in America, Part I (Duren, 1988), Science and Engineering Doctorates: 1960–86 (NSF, 1988), and A Challenge of Numbers (NRC, 1990b).
THE EARLY YEARS
During the 19th century, mathematics consisted of pure mathematics, some mathematical physics (the applied mathematics of those days), and statistics. Statistics had a separate professional identity, fostered by the American Statistical Association, which was founded in 1839, and thus was the earliest domestic professional society in the mathematical sciences. However, statistics as an academic discipline was considered to be an integral part of mathematics. Europe was the center of mathematical research, and American mathematics relied mainly on European universities to train American doctoral students and on people trained abroad to staff American colleges and universities. A base for future development was built with the establishment in 1876 of Johns Hopkins University, the first American research university, the founding of the American Mathematical Society in 1888, and the development of research programs at the University of Chicago toward the turn of the century. Respected journals were established to disseminate and stimulate mathematical research, among them the American Journal of Mathematics, in 1878, the Annals of Mathematics, in 1884, and the Transactions of the American Mathematical Society, in 1900.
During the period 1910 through 1930, the American mathematical community was small but active. Strong programs grew at Harvard and Princeton Universities, joining that at the University of Chicago. Mathematicians played a role in World War I primarily by calculating trajectory and range tables. Concern for mathematics education was growing, as attested to by the founding of the Mathematical Association of America in 1915.

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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States
With the coming of modern manufacturing methods following World War I, statistics grew into a separate discipline of the mathematical sciences. Statistical quality control and sampling schemes, two of the major developments of the 1920s, had far-reaching consequences for industrial development prior to World War II. However, the training ground for the statistician was still the mathematics graduate programs.
In the late 1920s, the American mathematical community grew rapidly, aided by new support from foundations and a National Research Council fellowship program funded by the Rockefeller Foundation. The top research universities established research instructorships in mathematics, a precursor to the postdoctoral positions of a later era. These instructorships were nonrenewable term appointments for two to four years with slightly reduced teaching loads. They were funded by the universities and increased the opportunity for postdoctoral experience at research universities.
For mathematics as for other sectors of society, the Depression brought an increase in unemployment and a decrease in salaries. The job market remained difficult during this period, and many new job seekers found only temporary employment or positions at the pre-college level. In spite of the prevailing poor economic conditions, the 1930s were a period of growth for the mathematical sciences community: the number of doctorates awarded increased from 351 in 1920–1929 to 780 in 1930–1939. PhD production during the 1930s was fairly constant, with some 80 degrees awarded annually. About 15% of the degrees in mathematics came from three departments (at the University of Chicago, Harvard University, and Princeton University) that were rated as “distinguished” and a further 50% from a group of about 15 “strong” departments. Opportunities for postdoctoral education were increased with the establishment of the Institute for Advanced Study in 1932. The Institute for Mathematical Statistics, a professional society for mathematical statistics, was organized in 1935.
In the United States through the 1930s, teaching was emphasized, both in the way departments conducted their business and in the way graduate students were educated. The heavy teaching load of U.S. university and college faculty allowed little time for research. In much of Europe, however, the emphasis was on research, and faculty had a normal teaching load approximately half that of their American counterparts.
In the mid-to late 1930s, the political conditions in Nazi Germany induced a number of mathematicians at universities in German-controlled areas to emigrate to the United States. World-renowned mathematicians such as Richard Courant, Hermann Weyl, and Hans Rademacher were among the emigrés. The U.S. mathematical community was infused with mathematical talent that would have an effect for generations to come. The additional strain on the U.S. job market for mathematicians was noticeable but not severe. By the end of 1939, 51 mathematicians had left their posts at German-speaking universities and come to the United States. Some were hired directly by the Institute for Advanced Study and a few universities, while others were placed in temporary positions. Although

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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States
efforts were made to avoid placing refugees in regular positions and using university funds for their support, some domestic faculty members still reacted negatively at a time when funds were limited, teaching loads were increasing, and native-born PhDs were without jobs.
THE ERA OF GROWTH
World War II provided new opportunities for mathematicians, including the newly immigrated mathematicians. World War II brought technology to weaponry. However, very few mathematicians—American or foreign-born—had the applied skills needed for the tasks at hand. Mathematics was forced to broaden its perspective to meet war needs, and many pure mathematicians learned to do applied work. Statistics increased in importance with the growing demand for quality control, sequential analysis, and analytical/statistical methods for solving dynamical problems such as bombing patterns. The new disciplines of operations research and computer science were born in the war effort.
By the end of World War II, the number of mathematicians who had come to the United States from countries affected by the war totaled only 120 to 150. These immigrant mathematicians and World War II profoundly changed the culture of the American mathematical community. The immigrant mathematicians added to the research and scholarship in the United States and became a driving force in changing the emphasis at many institutions from teaching to research. The war made new areas of research important and brought new sources of support, greatly strengthening the ties between the government and the mathematical sciences. Government support continued after the war through the Office of Naval Research (ONR), founded in 1947, and the National Science Foundation (NSF), founded in 1950. The new system of grants for summer research, graduate students, and conferences, as well as of peer review for the awarding of such grants, gradually changed the atmosphere in U.S. colleges and universities. Research became an integral part of the university structure, and institutions across the country sought to hire mathematicians capable of doing research and obtaining grants.
By 1951, the number of PhDs awarded annually was already over 200 per year. The horizons of the mathematical sciences were being widened. Applied mathematics, scientific computing, and operations research became recognized disciplines of the mathematical sciences. The Society for Industrial and Applied Mathematics was founded in 1951, the Operations Research Society of America in 1952.
By 1961, the number of PhDs awarded annually was well over 300 per year. The mathematics departments at Chicago, Harvard, and Princeton still had distinguished programs but now were joined by departments at Columbia University, the University of Michigan, Massachusetts Institute of Technology, Stanford University, the University of

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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States
California, Berkeley, and Yale University, which had moved up from the “strong” category. Other programs replenished the ranks of the strong departments, bringing the number of recognized programs to about 25. Strong departments of statistics also emerged during this period, among them the departments at the University of North California, Iowa State University, and the University of California, Berkeley.
The challenges facing the United States in the 1960s—to be first on the moon and to develop the technology base for economic and military security—strongly influenced both research and PhD production in the mathematical sciences. During this period, the number of doctorates conferred annually increased from 332 in the 1960–1961 academic year to 1070 in 1968–1969, and the number of doctoral programs went from around 200 to nearly 325. Throughout most of the 1960s, employment opportunities were unlimited for new PhDs. This situation was fueled by the growth in the number of research departments, increased enrollment in undergraduate mathematical sciences courses, and a new demand for mathematicians in government and industry. This was also a period of rapid growth for statistics. The number of doctoral statistics departments rose from 8 in 1950 to 17 in 1960 to 34 in 1970. The number of doctoral degrees awarded in statistics, including degrees awarded by mathematics, biology, and social science departments, increased from 110 in 1962 to 324 in 1972.
THE ERA OF CONTRACTION
The 1970s, however, was a time of great difficulty for many doctoral and postdoctoral programs in mathematics. During the 1970s, federal funding and employment opportunities decreased and the service role of mathematical sciences departments increased. Enrollment in undergraduate mathematical sciences courses continued to increase while the number of faculty positions remained level or decreased slightly. The mathematical sciences were increasingly viewed by many university and college administrators as a service discipline. Class sizes grew to accommodate increasing undergraduate enrollment, contributing to problems in collegiate mathematics. At many institutions, more of the undergraduate teaching load was shifted to graduate teaching assistants. This close identification of mathematics departments with teaching had serious implications as administrators looked for savings by promoting larger classes.
While undergraduate enrollment was increasing, graduate enrollment was decreasing. Graduate enrollment in the 155 PhD-granting mathematics departments declined by 17% from the fall of 1969 to the fall of 1974, while the decrease for the top 65 departments was 25%. The number of first-year graduate students decreased by nearly 50% at the top 65 departments. During the same period, federal support for graduate studies suffered a sharp decrease. Annual doctoral production reached a maximum of 1281 in the 1971–1972 academic year and then began to decline. The traditional mathematics programs reacted by looking inward and restricting access to the profession. Students

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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States
were discouraged from pursuing doctoral studies in the mathematics. As a result PhD production in the mathematical sciences dropped further and took longer to recover than in any of the other sciences or in engineering. PhD production in the mathematical sciences continued to decline through the 1970s. By the 1979–1980 academic year, the number of new PhDs conferred had dropped to 745. However, the decline was not uniform over all areas of the mathematical sciences. The more applied areas of the mathematical sciences reacted by creating non-academic opportunities for their new PhDs to make up for the loss of academic positions. The number of doctorates awarded in pure areas decreased by 55%, while those in applied mathematics, statistics, and operations research remained roughly constant or increased slightly. This shift in emphasis from pure to applied areas was reflected in the strength of statistics during this period of decline for mathematics. The number of autonomous statistics departments continued to increase, reaching 65 by 1987. In this year, there was a total of 164 degree programs in statistics, including 47 degree programs in mathematics and mathematical sciences departments.
By the 1970s, computer science had emerged as a separate discipline with an identity distinct from that of the mathematical sciences. The growth of computer science and its establishment as a separate discipline drew students and resources from the mathematical sciences. Mathematics departments, which originally had had to teach computer science, were often left with a relative decrease in resources and a relative increase in service teaching when independent computer science departments were established.
During the late 1970s, concerns began to surface about not only the decrease in the number of new PhDs but also the decrease in the percentage of U.S. citizens among new PhDs and about underrepresentation of women and ethnic minorities. The percentage of mathematical sciences doctorates awarded to U.S. citizens decreased from 82% in the 1970–1971 academic year to 74% in the 1978–1979 academic year. Growing concerns for increasing the participation of underrepresented minorities and women in the mathematical sciences were made evident by the founding of the National Association of Mathematicians in 1969 and the founding of the Association for Women in Mathematics in 1971.
Instead of increasing the employment opportunities for mathematical sciences PhDs by making mathematicians more employable in nontraditional institutions, most of the community worked to restrict the number of PhDs to fit the small traditional marketplace. When stability in the employment market returned at the start of the 1980s, many mathematical scientists realized that severe problems had been created by this inward-looking attitude. Despite success in both research and advanced study, the mathematical sciences in the United States were now unable to attract sufficiently many domestic graduate students for renewal. In addition, a serious imbalance had developed in federal funding of the mathematical sciences as compared with other sciences.
One of the first reports to call attention to the imbalance in funding was the report Renewing U.S. Mathematics: Critical Resource for the Future (NRC, 1984), which was

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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States
influential in bringing about a modest increase in federal support for the mathematical sciences. By 1987, however, there was still nothing close to a balance with other fields. In 1987, 75% and 56% of the R&D faculty in physics and chemistry, respectively, received federal support, whereas only 37% of the mathematical sciences R&D faculty received federal support. In that same year, 51% and 49% of the graduate students in physics and chemistry, respectively, received federal support, whereas only 18% of the mathematical sciences graduate students received federal support. A reassessment, Renewing U.S. Mathematics: A Plan for the 1990s (NRC, 1990c), summarized the effects of the 1984 report and argued strongly for the continued need for increasing federal funding.
THE RECENT PAST
In contrast to the preceding three decades of rapid change, the 1980s was a period of stability for the mathematical sciences. In spite of some competition with computer science for students, the number of PhDs awarded annually in the mathematical sciences was roughly constant, decreasing from 839 in the 1980–1981 academic year to a minimum of 726 in the 1984–1985 academic year and then increasing to 1061 in the 1990–1991 academic year. But the trend toward a higher percentage of non-U.S. citizens receiving degrees accelerated sharply. In 1980–1981, 68% of the doctorates granted by U.S. institutions in the mathematical sciences went to U.S. citizens, whereas in 1990–1991, only 43% of the doctorates awarded went to U.S. citizens. Increasing undergraduate enrollments, a booming economy, and retirements provided job opportunities for almost all new PhDs in the decade.