40 Things to Know about The Mathematical Sciences in 2025

1. 40 Things to Know aboutThe Mathematical Sciences in 2025 Mark L. Green, UCLA A National Research Council/National Academies Report for the National Science Foundation

2. 1. It’s the First Decadal Study ofUS Math in 15 years • 1984 David Committee report: Renewing Mathematics: Critical Resource for the Future, • 1990 David II committee report: Renewing U.S. Mathematics: A Plan for the 1990’s, • 1998 Odom Report of the Senior Assessment Panel for the International Assessment of the U.S. Mathematical Sciences.

3. 2. NSF Sponsored the Study • Suggested by Tony Chan when he was Assistant Director for MPS at NSF • Commissioned by Peter March when he was Division Director of DMS • Study done by the National Research Council/National Academies under the supervision of the Board on Mathematical Sciences and their Applications (BMSA)

4. 3. The Committee wasn’t only Mathematical Scientists Core Math: LUIS A. CAFFARELLI, MARK L. GREEN (VC), DAVID EISENBUD, PETER W. JONES (applied also), JU-LEE KIM, JOHN W. MORGAN, YUVAL PERES (also industry) Applied Math: EMMANUEL J. CANDES (Stat also), PHILLIP COLELLA (also computational science), Statistics: JAMES O. BERGER, JUN LIU (Computational Biology also) Computer Science: YANN LeCUN, EVA TARDOS, MARGARET H. WRIGHT(also applied math) Electrical Engineering: THOMAS E. EVERHART (C) Finance: TANYA S. BEDER Theoretical Physics: JUAN MALDACENA Education/Math/CS: JOE B. WYATT

5. 4. There are Two Publications Vignettes The Full Report

6. 5. The Vignettes had a professional writer, deal with Math inthe real world • Compressed Sensing • Eigenvectors: from Math to an IPO • Simulating Supernovas • Bayesian Inference • Diffusion Tensor Imaging • Fast Multipole • Cellular Automata • Graph Spectra • Bioinformatics • Geometry and Physics • Statistical Physics • Patents

7. 6. There is cool stuff in the Full Report that I won’t get to talk about Ch 1: Examples of how the mathematical sciences impact everyday life Ch 2: A sampling of recent advances Ch 3 and Appendix D: Examples of the uses of mathematical sciences in other disciplines, including evidence from decadal studies from other subjects Ch 5: A list of best practices for the inclusion of women and other underrepresented groups Appendix C: Basic data about the mathematical sciences

8. VITALITY

9. 7. Research in the Mathematical Sciences is on a Roll The vitality of the U.S. mathematical sciences enterprise is excellent. The discipline has consistently been making major advances in research, both in fundamental theory and in high-impact applications. The discipline is displaying great unity and coherence as bridges are increasingly built between subfields of research…The discipline’s vitality is providing clear benefits to most areas of science and engineering and to the nation.

10. 8. The Role of the Mathematical Sciences has Expanded—a Lot This major expansion in the uses of the mathematical sciences has been paralleled by a broadening in the range of mathematical science ideas and techniques being used. Much of 21st century science and engineering is going to be built on a mathematical science foundation, and that foundation must continue to evolve and expand.

11. 9. The Core is Essential Support for basic science is always fragile, and this may be especially true of the core mathematical sciences. In order for the whole mathematical sciences enterprise to flourish long-term, the core must flourish. This requires investment by universities and by the government in the core of the subject. These investments are repaid not immediately and directly in applications but rather over the long term as the subject grows and retains its vitality. From this ever-increasing store of fundamental theoretical knowledge many innovative future applications will be drawn. To give short shrift to maintaining this store would shortchange the country.

12. 10. The “Unreasonable Effectiveness of Mathematics” • Prime Numbers-> Secure Internet Commerce • Operators on Hilbert Space-> Quantum Mechanics • Quaternions->Satellite Tracking, Video Games • Eigenvectors-> Google’s PageRank • Stochastic Processes-> Black-Scholes • Integral Geometry-> MRI and PET scans • Connections-> Gauge Fields

13. 11. The Mathematical Sciences are being used everywhere Finding: Mathematical sciences work is becoming an increasingly integral and essential component of a growing array of areas of investigation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and much more. This work involves the integration of mathematics, statistics, and computation in the broadest sense, and the interplay of these areas with areas of potential application; the mathematical sciences are best conceived of as including all these components. These activities are crucial to economic growth, national competitiveness, and national security. This Finding has ramifications for both the nature and scale of funding of the mathematical sciences and for education in the mathematical sciences.

14. CONNECTIONS

15. 12. National studies and priorities give evidence for the importance of Math • A New Biology for the 21st Century • NAE Grand Challenges • High Performance Computing in Astrophysics • DoD 2012 Priorities for National Security • OSTP Big Data Initiative • Cross-Agency Priority Goal on Cybersecurity • Almost every decadal study in science…

16. 13. An Expanded EnterpriseNeeds More Funding • While the expansion of the mathematical sciences and their ever-wider reach is all to the good, the committee is concerned about the adequacy of current federal funding for the discipline in light of this broadening. • Conclusion:The dramatic expansion in the role of the mathematical sciences over the past 15 years has not been matched by a comparable expansion in federal funding, either in the total amount or in the diversity of sources. The discipline—especially the core areas—is still heavily dependent on the National Science Foundation.

17. 14. The Entire EcosystemMust be Funded DMS is faced with an innate conflict: As the primary funding unit charged with maintaining the health of the mathematical sciences, it is naturally driven to aid the expansions discussed in Chapter 3 [Connections]; yet it is also the largest of a very few sources whose mission includes supporting the foundations of the discipline, and thus it plays an essential role with respect to those foundations. … There are challenges inherent in supporting a broad, loosely knit community while maintaining its coherence, and the adequacy and balance of funding is a foremost concern. As noted in Chapter 3, funding of excellence wherever it is found should still be the top priority.

18. 15. What Math do R&D Managers Want (1996) • Modeling and Simulation • Mathematical Formulation of Problems • Algorithm and Software Development • Problem-Solving • Statistical Analysis • Verifying Correctness • Analysis of Accuracy and Reliability

19. 16. Connections are Important;Non-Math Researchers need us • The complexity of phenomena … are pushing frontiers in the mathematical sciences and challenging those who could have previously learned the necessary skills • As this complexity increases, we are finding more and more occasions where specialized mathematical and statistical experience is required or would be beneficial

20. 17. The Cost of Missed Connections is High When researchers do not have the best tools or collaborators from the mathematical sciences available… • Instruments do not achieve maximum resolution • Information in data is not fully utilized • Experiments are not designed optimally • Genes are not found, patterns are not recognized, unifying principles are missed

21. 18. The Mathematical Sciences are a Unified Whole The committee members—like many others who have examined the mathematical sciences—believe that it is critical to consider the mathematical sciences as a unified whole. Distinctions between “core” and “applied” mathematics increasingly appear artificial; in particular, it is difficult today to find an area of mathematics that does not have relevance to applications. It is true that some mathematical scientists primarily prove theorems, while others primarily create and solve models, and professional reward systems need to take that into account. But any given individual might move between these modes of research, and many areas of specialization can and do include both kinds of work.

22. TRENDS IN THE MATHEMATICAL SCIENCES

23. 19. Two Major Drivers are Computing and Data Two major drivers of the increased reach of the mathematical sciences are the ubiquity of computational simulations—which build on concepts and tools from the mathematical sciences—and exponential increases in the amount of data available for many enterprises. The Internet, which makes these large quantities of data readily available, has magnified the impact of these drivers.

24. 20. Big Data has Many Faces • Statistics • Adaptive and streaming algorithms • Image processing and analysis/shape analysis/text mining • Search algorithms • Inverse problems • Dimensionality reduction • Network science • Encryption and privacy • Large-scale Simulations

25. PEOPLE IN THE MATHEMATICAL SCIENCES

26. 21. The Nation Needs More People with Math Skills • McKinsey report (2011) estimates USbusinesses will need an additional 140,000-190,000 people by 2018 with “deep analytical talent” and a high level of quantitative skills and anticipates a shortage • “Engage to Excel” has the goal of an additional 1,000,000 STEM graduates over 10 years

27. 22. The Changing Role of Math Impacts What Students Need The expansion of research opportunities in the mathematical sciences necessitates changes in the way students are prepared.

28. 23. Math Should be a GatewayNot a Barrier • Motivation: Motivate math by how it is used • Incorporate multiple modes of mathematical thinking • New entry-points and new pathways • Partner with other disciplines to create a compelling menu of lower-division courses • Diversify teaching methods, engage with online education • A community-wide effort to bring successful experiments to scale

29. 24. We’re Responsible for Multiple Student Populations who Need Math • The mathematical sciences community has a critical role in educating a broad range of students. Some will exhibit a special talent in mathematics from a young age, but there are many more whose interest in the mathematical sciences arises later and perhaps through nontraditional pathways, and these latter students constitute a valuable pool of potential majors and graduate students. A third cadre consists of students from other STEM disciplines who need strong mathematical sciences education. All three pools of students need expert guidance and mentoring from successful mathematical scientists, and their needs are not identical. The mathematical sciences must successfully attract and serve all three of these cadres. • It is important to enable fluidity of entry into the mathematical sciences and the subjects they support to take account of changes in student interest over the college years.

30. 25. New Majors, New Programs, New Pathways are Needed Mathematical sciences curricula need attention. The educational offerings of typical departments in the mathematical sciences have not kept pace with the large and rapid changes in how the mathematical sciences are used in science, engineering, medicine, finance, social science, and society at large. This diversification entails a need for new courses, new majors, new programs, and new educational partnerships with those in other disciplines, both inside and outside universities. New educational pathways for training in the mathematical sciences need to be created—for students in mathematical sciences departments, for those pursuing degrees in science, medicine, engineering, business, and social science, and for those already in the workforce needing additional quantitative skills.

31. 26. A Community-Wide Effort at Change is Needed Most mathematics departments still tend to use calculus as the gateway to higher-level coursework, and that is not appropriate for many students. Although there is a very long history of discussion about this issue, the need for a serious reexamination is real, driven by changes in how the mathematical sciences are being used. Different pathways are needed for students who may go on to work in bioinformatics, ecology, medicine, computing, and so on. It is not enough to rearrange existing courses to create alternative curricula; a redesigned offering of courses and majors is needed. Although there are promising experiments, a community-wide effort is needed in the mathematical sciences to make its undergraduate courses more compelling to students and better aligned with the needs of user departments.

32. 27. Motivate Mathematics by How It Is Used • Research shows this is key for K-12 students • Addresses the 27% with high math skills and low STEM interest • Impacts the dropoff in STEM majors over college years • Faculty and grad students need to know how math is used • K-12 teacher training needs to incorporate it

33. 28. Students Should See Different Modes of Thinking in the Mathematical Sciences • Formal manipulation • Logical reasoning, proof • Modeling and simulation • Algorithms • Probabilistic and statistical thinking Goals: Expose students at all levels to a variety of modes of thinking. Foster the ability to deal with problems that are not precisely formulated.

34. 29.Diversify Teaching Methods The traditional lecture-homework-exam format that often prevails in lower-division mathematics courses would benefit from a reexamination. A large and growing body of research indicates that STEM education can be substantially improved through a diversification of teaching methods. Change is unquestionably coming to lower-division undergraduate mathematics, and it is incumbent on the mathematical sciences community to ensure that it is at the center of these changes and not at the periphery.

35. 30. Those Already in the Workforce Will Need to Update their Skills New credentials may be needed, such as professional master’s degrees for those about to enter the workforce or already in it. The trend toward periodic acquisition of new job skills by those already in the workforce provides an opportunity for the mathematical sciences to serve new needs.

36. 31. Forge Lower-Division Partnerships With Other Disciplines • The needs of 21st century students call for a truly compelling menu of creatively taught lower-division courses in the mathematical sciences. • The mathematical sciences have a critical role in educating a broad range of students, including from other STEM disciplines. • Partnerships with mathematics-intensive disciplines in designing such courses are eminently worth pursuing.

37. 32. Successful Changes Need to be Brought to Scale • The stick: Change is coming no matter what, business model is unsustainable • The carrot: Putting mathematics education on a sustainable and exciting course that will serve the country well for the next 15 years • Steps: Mobilize the stakeholders, provide tools for a community wide effort, involve those with experience in bringing successful reform efforts to scale

38. 33. Change isn’t Easy • Involves culture change • Faculty need to venture out of their comfort zone • Occurs at a time of intense cost pressures and of major changes in how universities and colleges operate • Represents a large scale, rapid change Nevertheless, a once-in-a-generation opportunity for positive change

39. 34. PCAST is a Wake-Up Call The PCAST report should be viewed as a wake-up call for the mathematical sciences community. While there have been numerous promising experiments within the community for addressing the issues it raises—especially noteworthy has been the tremendous expansion in opportunities for undergraduate research in the mathematical sciences—at this point a community-wide effort is called for. The professional societies should work cooperatively to spark this. Change is unquestionably coming to lower- division undergraduate mathematics, and it is incumbent upon the mathematical sciences community to ensure that it is at the center of these changes and not at the periphery.

40. 35. A Deep Rethinking is Needed Recommendation:Mathematics and statistics departments, in concert with their university administrations, should engage in a deep rethinking of the different types of students they are attracting and wish to attract, and must identify the top priorities for educating these students. This should be done for bachelor’s, master’s, and Ph.D.-level curricula. In some cases, this rethinking should be carried out in consultation with faculty from other relevant disciplines.

41. 36. Make Recruitment/Retention of Women & Underrepresented Groups an Explicit Responsibility Recommendation: Every academic department in the mathematical sciences should explicitly incorporate recruitment and retention of women and underrepresented groups into the responsibilities of the faculty members in charge of the undergraduate program, graduate program, and faculty hiring and promotion. Resources need to be provided to enable departments to monitor and adapt successful recruiting and mentoring programs that have been pioneered at many schools and to find and correct any disincentives that may exist in the department.

42. 37. Math Needs to Reach Out to the General Public Recommendation:More professional mathematical scientists should become involved in explaining the nature of the mathematical sciences enterprise and its extraordinary impact on society. Academic departments should find ways to reward such work. Professional societies should expand existing efforts and work with funding entities to create an organizational structure whose goal is to publicize advances in the mathematical sciences.

43. THE CHANGING ACADEMIC CONTEXT

44. 38. The Business Model for Math Depts is in Trouble • Math graduate students are supported primarily by TA-ships; what will happen if many of these disappear? • The size of mathematical science departments is largely justified/paid for by service teaching; will there be a collapse of ecological niches for research mathematicians and statisticians? • This will hit early career people especially hard; what will this do to the pipeline?

45. 39. Mathematical Scientists Must Get Engaged in Shaping Efforts in Online Education While online education in the mathematical sciences is a work in progress, effective ways to deliver this material at a level of quality comparable to large university lecture classes most likely will be found. It is strongly in the interests of mathematical scientists to be involved in initiatives for online education, which will otherwise happen in a less-than-optimal way.

46. 40. The Time to do Something is Now Recommendation:Academic departments in mathematics and statistics should begin the process of rethinking and adapting their programs in order to keep pace with the evolving academic environment and to be sure they have a seat at the table as online content and other innovations in the delivery of mathematical science coursework are created. The professional societies have important roles to play in mobilizing the community in these matters, through mechanisms such as opinion articles, online discussion groups, policy monitoring, and conferences.

47. THANK YOU!