By Mark L. Green and Scott T. Weidman
How healthy are the mathematical sciences today? What will the discipline look like in the coming years? How should the mathematical enterprise adjust? These questions are addressed in The Mathematical Sciences in 2025,* a recent report from the Board on Mathematical Sciences and Their Applications of the U.S. National Academies.
The study that led to the report was initiated and supported by the National Science Foundation’s Division of Mathematical Sciences. BMSA assembled a broad, blue-ribbon committee to carry out the study, with members from core and applied mathematics, statistics, computational biology, computer science, finance, theoretical physics, industry, and university administration. The study was chaired by Thomas Everhart, president emeritus of the California Institute of Technology, with one of us (MLG) as vice-chair.
The study revealed three major themes:
- The mathematical sciences enterprise is qualitatively different from the one that prevailed during the second half of the 20th century, and a different model is emerging—one of a discipline with a much broader reach and greater potential impact. The mathematical sciences are an increasingly integral and essential component of research in a growing array of areas in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and many other fields. All these areas are crucial to economic growth, national competitiveness, and national security.
- Reflecting this expansion, the culture of the mathematical sciences needs to evolve and a serious rethinking of postsecondary mathematics education is needed.
- Stresses that are beginning to affect all of higher education are likely to impact the mathematical sciences especially severely, and our community must be proactive in addressing these challenges.
The pervasive use of computing, the creation of massive datasets, and their widespread availability through the Internet have placed the mathematical sciences in a pivotal position. As a result, the study committee learned, the mathematical sciences reach into many other disciplines, and mathematically sophisticated research is being produced by a distributed community that extends well beyond the scope of most academic departments of mathematics or statistics.
Even as the mathematical sciences are broadening their reach, they are enjoying a period of striking accomplishments. As noted in the report,
The opening years of the twenty-first century have been remarkable ones for the mathematical sciences. The list of exciting accomplishments includes . . . proofs of the long-standing Poincaré conjecture and the “fundamental lemma”; progress in quantifying the uncertainties in complex models; new methods for modeling and analyzing complex systems such as social networks and for extracting knowledge from massive amounts of data from biology, astronomy, the Internet, and elsewhere; and the development of compressed sensing.
Against this backdrop of accomplishments and broadening, the discipline has retained its unity and coherence, with ideas flowing freely between different areas and between the core and applications. The core of the subject is flourishing, as longstanding problems are solved and a steady supply of innovative ideas are pursued for internal reasons, but with benefits percolating ever more rapidly into other subjects. The health of the core is essential to the welfare of the mathematical sciences. Much of 21st-century science and engineering will be built on a mathematical foundation, and that foundation must continue to flourish, evolve, and expand. Many mathematical scientists remain unaware of the expanding role of their field, however, and this limits the community’s ability to fully realize the benefits of the expansion.
One conclusion of the study is that the dramatic expansion in the role of the mathematical sciences “has not been matched by a comparable expansion in federal funding, either in amount or in the diversity of sources.”
The number of students entering mathematics-intensive disciplines is inadequate to meet the opportunities of the future. The mathematical sciences community has a critical role to play in educating a broad range of students: those within the mathematical sciences; those pursuing degrees in science, medicine, engineering, business, and social science; and those already in the workforce who need additional quantitative skills. There is a need for multiple entry points and pathways, and for new courses, majors, programs, and educational partnerships with those in other disciplines, both inside and outside academe. Students would benefit from encounters with a variety of modes of mathematical and statistical thinking—including core mathematics, modeling and simulation, algorithms, probabilistic methods, and statistical inference—and should learn to deal with problems that are not precisely formulated. The widespread need for graduates with math-intensive career skills provides an opportunity for the mathematical sciences.
The traditional lecture–homework–exam format also needs re-examination. A large and growing body of research indicates that diversification of teaching methods can result in substantial improvements in STEM education. As part of this effort, it is essential that educators motivate students at all levels by explaining how the mathematical topics they are teaching are used and identifying careers that make use of them. Modest steps in this direction could lead to greater success in attracting and retaining students in mathematical sciences courses.
Cost pressures on universities and the advent of online education have the potential to elicit rapid change. The current business model of mathematical sciences departments, with its heavy reliance on lower-division service courses, may prove unsustainable. The mathematical sciences community should be proactive and engaged in discussions of institutional changes.
The market for mathematical talent is now global, the study found, and the U.S. is in danger of losing its global pre-eminence in the discipline; a brain drain from the U.S. is now a real threat. The report also recommends that every academic department in the mathematical sciences explicitly include recruitment and retention of women and underrepresented groups in the responsibilities of the faculty members in charge of the undergraduate program, graduate program, and faculty hiring and promotion, and it provides an extensive list of best practices.
The study committee recognized that the size of the pipeline of students preparing for careers that rely on the mathematical sciences is fundamentally limited by the quality of K–12 mathematics and statistics education. Although the study did not encompass K–12 education, this is an issue that should concern all mathematical scientists, especially in their role in the education of K–12 teachers.
A PDF of The Mathematical Sciences in 2025 can be downloaded for free. The study also produced an earlier report, Fueling Innovation and Discovery: The Mathematical Sciences in the 21st Century, which illustrates how the discipline is providing clear benefits for diverse areas of science and engineering, for industry and technology, for innovation and economic competitiveness, and for national security; a free PDF of that report can be downloaded here.
*National Research Council, National Academies Press, Washington, DC, 2013. Portions of this article replicate or paraphrase the wording in the report.
Mark Green, a founding director of the Institute of Pure and Applied Mathematics, is a professor emeritus of mathematics at the University of California, Los Angeles. Scott Weidman is director of the Board on Mathematical Sciences and Their Applications.