40 Things to Know about
The Mathematical Sciences in
2025
Mark L. Green, UCLA
A National Research Council/National
Academies Report for the National
Science Foundation
1. It’s the First Decadal Study of
US 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.
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)
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
4. There are Two Publications
Vignettes
The Full Report
5. The Vignettes had a professional
writer, deal with Math in
the real world
• Compressed Sensing
• Eigenvectors: from
Math to an IPO
• Simulating
Supernovas
• Bayesian Inference
• Diffusion Tensor
Imaging
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Fast Multipole
Cellular Automata
Graph Spectra
Bioinformatics
Geometry and Physics
Statistical Physics
Patents
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
VITALITY
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 highimpact 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.
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.
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.
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
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.
CONNECTIONS
12. National studies and priorities
give evidence for the importance
of Math
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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…
13. An Expanded Enterprise
Needs 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.
14. The Entire Ecosystem
Must 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.
15. What Math do R&D Managers
Want (1996)
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Modeling and Simulation
Mathematical Formulation of Problems
Algorithm and Software Development
Problem-Solving
Statistical Analysis
Verifying Correctness
Analysis of Accuracy and Reliability
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
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
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.
TRENDS IN THE
MATHEMATICAL
SCIENCES
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.
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
PEOPLE IN THE
MATHEMATICAL
SCIENCES
21. The Nation Needs More People
with Math Skills
• McKinsey report (2011) estimates US
businesses will need an additional 140,000190,000 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
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.
23. Math Should be a Gateway
Not 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
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.
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.
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.
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
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.
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.
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.
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.
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
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
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.
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.
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.
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.
THE CHANGING
ACADEMIC CONTEXT
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?
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-thanoptimal way.
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.
THANK YOU!