40 Things to Know about The Mathematical Sciences in 2025

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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

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1. Its 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 1990s,
• 1998 Odom Report of the Senior Assessment Panel
  for the International Assessment of the U.S.
  Mathematical Sciences.

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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)

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3. The Committee wasnt 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

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4. There are Two Publications
Vignettes
The Full Report
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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

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6. There is cool stuff in the Full Report that I
wont 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

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• VITALITY

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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
  researchThe disciplines vitality is providing
  clear benefits to most areas of science and
  engineering and to the nation.

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8. The Role of the Mathematical Sciences has
Expandeda 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.

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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.

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10. The Unreasonable Effectiveness of
Mathematics

• Prime Numbers-gt Secure Internet Commerce
• Operators on Hilbert Space-gt Quantum Mechanics
• Quaternions-gtSatellite Tracking, Video Games
• Eigenvectors-gt Googles PageRank
• Stochastic Processes-gt Black-Scholes
• Integral Geometry-gt MRI and PET scans
• Connections-gt Gauge Fields

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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.

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• CONNECTIONS

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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

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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
  disciplineespecially the core areasis still
  heavily dependent on the National Science
  Foundation.

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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.

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15. What Math do RD 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

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16. Connections are ImportantNon-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

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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

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18. The Mathematical Sciences are a Unified Whole

• The committee memberslike many others who have
  examined the mathematical sciencesbelieve 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.

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• TRENDS IN THE MATHEMATICAL SCIENCES

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19. Two Major Drivers are Computing and Data

• Two major drivers of the increased reach of the
  mathematical sciences are the ubiquity of
  computational simulationswhich build on concepts
  and tools from the mathematical sciencesand
  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.

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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

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• PEOPLE IN THE MATHEMATICAL SCIENCES

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21. The Nation Needs More People with Math Skills

• McKinsey report (2011) estimates US businesses
  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

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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.

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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

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24. Were 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.

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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 createdfor 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.

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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.

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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

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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.

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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.

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30. Those Already in the Workforce Will Need to
Update their Skills

• New credentials may be needed, such as
  professional masters 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.

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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.

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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

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33. Change isnt 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

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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 raisesespecially noteworthy has
  been the tremendous expansion in opportunities
  for undergraduate research in the mathematical
  sciencesat 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.

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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 bachelors,
  masters, and Ph.D.-level curricula. In some
  cases, this rethinking should be carried out in
  consultation with faculty from other relevant
  disciplines.

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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.

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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.

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• THE CHANGING ACADEMIC CONTEXT

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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?

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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.

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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.

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• THANK YOU!