The Mathematical Sciences
in 2025

Committee on the Mathematical Sciences in 2025

Board on Mathematical Sciences and Their Applications

Division on Engineering and Physical Sciences

NATIONAL RESEARCH COUNCIL
OF THE NATIONAL ACADEMIES

THE NATIONAL ACADEMIES PRESS

Washington, D.C.

www.nap.edu



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The Mathematical Sciences in 2025 Committee on the Mathematical Sciences in 2025 Board on Mathematical Sciences and Their Applications Division on Engineering and Physical Sciences

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THE NATIONAL ACADEMIES PRESS  500 Fifth Street, NW  Washington, DC 20001 NOTICE: The project that is the subject of this report was approved by the Govern- ing Board of the National Research Council, whose members are drawn from the councils of the National Academy of Sciences, the National Academy of Engineer- ing, and the Institute of Medicine. The members of the committee responsible for the report were chosen for their special competences and with regard for appropri- ate balance. This project was supported by the National Science Foundation under grant number DMS-0911899. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the organizations or agencies that provided support for the project. International Standard Book Number-13: 978-0-309-28457-8 International Standard Book Number-10: 0-309-28457-0 Library of Congress Control Number: 2013933839 Additional copies of this report are available from the National Academies Press, 500 Fifth Street, NW, Keck 360, Washington, DC 20001; (800) 624-6242 or (202) 334-3313; http://www.nap.edu. Suggested citation: National Research Council. 2013. The Mathematical Sciences in 2025. Washington, D.C.: The National Academies Press. Copyright 2013 by the National Academy of Sciences. All rights reserved. Printed in the United States of America

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The National Academy of Sciences is a private, nonprofit, self-perpetuating society of distinguished scholars engaged in scientific and engineering research, dedicated to the furtherance of science and technology and to their use for the general welfare. Upon the authority of the charter granted to it by the Congress in 1863, the Acad- emy has a mandate that requires it to advise the federal government on scientific and technical matters. Dr. Ralph J. Cicerone is president of the National Academy of Sciences. The National Academy of Engineering was established in 1964, under the charter of the National Academy of Sciences, as a parallel organization of outstanding e ­ ngineers. It is autonomous in its administration and in the selection of its members, sharing with the National Academy of Sciences the responsibility for advising the federal government. The National Academy of Engineering also sponsors engineer- ing programs aimed at meeting national needs, encourages education and research, and recognizes the superior achievements of engineers. Dr. Charles M. Vest is presi- dent of the National Academy of Engineering. The Institute of Medicine was established in 1970 by the National Academy of Sciences to secure the services of eminent members of appropriate professions in the examination of policy matters pertaining to the health of the public. The Insti- tute acts under the responsibility given to the National Academy of Sciences by its congressional charter to be an adviser to the federal government and, upon its own initiative, to identify issues of medical care, research, and education. Dr. Harvey V. Fineberg is president of the Institute of Medicine. The National Research Council was organized by the National Academy of Sci- ences in 1916 to associate the broad community of science and technology with the Academy’s purposes of furthering knowledge and advising the federal government. Functioning in accordance with general policies determined by the Academy, the Council has become the principal operating agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the government, the public, and the scientific and engineering communities. The C ­ ouncil is administered jointly by both Academies and the Institute of Medicine. Dr. Ralph J. Cicerone and Dr. Charles M. Vest are chair and vice chair, respectively, of the National Research Council. www.national-academies.org

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COMMITTEE ON THE MATHEMATICAL SCIENCES IN 2025 THOMAS E. EVERHART, California Institute of Technology, Chair MARK L. GREEN, University of California, Los Angeles, Vice-chair TANYA S. BEDER, SBCC Group, Inc. JAMES O. BERGER, Duke University LUIS A. CAFFARELLI, University of Texas at Austin EMMANUEL J. CANDES, Stanford University PHILLIP COLELLA, E.O. Lawrence Berkeley National Laboratory DAVID EISENBUD, University of California, Berkeley PETER W. JONES, Yale University JU-LEE KIM, Massachusetts Institute of Technology YANN LeCUN, New York University JUN LIU, Harvard University JUAN MALDACENA, Institute for Advanced Study JOHN W. MORGAN, Stony Brook University YUVAL PERES, Microsoft Research EVA TARDOS, Cornell University MARGARET H. WRIGHT, New York University JOE B. WYATT, Vanderbilt University Staff SCOTT WEIDMAN, Study Director THOMAS ARRISON, Senior Program Officer MICHELLE SCHWALBE, Program Officer BARBARA WRIGHT, Administrative Assistant v

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BOARD ON MATHEMATICAL SCIENCES AND THEIR APPLICATIONS DONALD G. SAARI, University of California, Irvine, Chair GERALD G. BROWN, U.S. Naval Postgraduate School LOUIS ANTHONY COX, JR., Cox Associates BRENDA L. DIETRICH, IBM T.J. Watson Research Center CONSTANTINE GATSONIS, Brown University DARRYLL HENDRICKS, UBS Investment Bank ANDREW LO, Massachusetts Institute of Technology DAVID MAIER, Portland State University JAMES C. McWILLIAMS, University of California, Los Angeles JUAN MEZA, University of California, Merced JOHN W. MORGAN, Stony Brook University VIJAYAN N. NAIR, University of Michigan CLAUDIA NEUHAUSER, University of Minnesota, Rochester J. TINSLEY ODEN, University of Texas at Austin FRED ROBERTS, Rutgers University J.B. SILVERS, Case Western Reserve University CARL SIMON, University of Michigan EVA TARDOS, Cornell University KAREN VOGTMANN, Cornell University BIN YU, University of California, Berkeley Staff SCOTT WEIDMAN, Director NEAL GLASSMAN, Senior Program Officer MICHELLE SCHWALBE, Program Officer BARBARA WRIGHT, Administrative Assistant BETH DOLAN, Financial Associate vi

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Preface When I was asked to chair a committee of mathematical scientists charged with examining the field now with an eye toward how it needs to evolve to produce the best value for the country by 2025, I demurred because I am not a mathematical scientist. The counter was that therefore I would not be biased, could be objective to prevent possible internal politics from “capturing” the report, and would be continuing a tradition of having such committees chaired by nonexperts. The assignment was educational in many ways. The committee was extraordinary in its makeup, with experts from the core of mathematics as well as from departments of statistics and com- puter science, from both academia and industry. My eyes were opened to the power of the mathematical sciences today, not only as an intellectual undertaking in their own right but also as the increasingly modern founda- tion for much of science, engineering, medicine, economics, and business. The increasingly important challenges of deriving knowledge from huge amounts of data, whether numerical or experimental, of simulating com- plex phenomena accurately, and of dealing with uncertainty intelligently are some of the areas where mathematical scientists have important contribu- tions to make going forward—and the members of this committee know it. They have demonstrated a great capacity to envision an emerging era in which the mathematical sciences underpin much of twenty-first century science, engineering, medicine, industry, and national security. I hope that this report persuades many others to embrace that vision. While all members of the committee contributed to this report, vice- chair Mark Green, from the University of California at Los Angeles, and vii

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viii PREFACE NRC staff, headed by Scott Weidman, worked tirelessly to provide much of the writing and data that give the report its coherence, organization, and credibility. I especially thank them, for myself and for the rest of the com- mittee, for their essential contributions. Thomas E. Everhart, Chair

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Acknowledgments This report has been reviewed in draft form by individuals chosen for their diverse perspectives and technical expertise, in accordance with procedures approved by the National Research Council’s Report Review Committee. The purpose of this independent review is to provide candid and critical comments that will assist the institution in making its published report as sound as possible and to ensure that the report meets institutional standards for objectivity, evidence, and responsiveness to the study charge. The review comments and draft manuscript remain confidential to protect the integrity of the deliberative process. We wish to thank the following individuals for their review of this report: Emery Brown, Massachusetts General Hospital and Massachusetts Institute of Technology Anna Gilbert, University of Michigan Leslie Greengard, New York University Yu-Chi Ho, Harvard University Stephen Robinson, University of Wisconsin Kenneth Ribet, University of California, Berkeley Terence Tao, University of California, Los Angeles Yannis Yortsos, University of Southern California Bin Yu, University of California, Berkeley Robert Zimmer, University of Chicago Although the reviewers listed above have provided many constructive comments and suggestions, they were not asked to endorse the conclusions ix

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x ACKNOWLEDGMENTS or recommendations nor did they see the final draft of the report before its release. The review of this report was overseen by Lawrence D. Brown of the University of Pennsylvania and C. Judson King of the University of California, Berkeley. Appointed by the National Research Council, they were responsible for making certain that an independent examination of this report was carried out in accordance with institutional procedures and that all review comments were carefully considered. Responsibility for the final content of this report rests entirely with the authoring committee and the institution. The committee also acknowledges the valuable contribution of the following individuals, who provided input at the meetings on which this report is based or by other means: Theodore T. Allen, Ohio State University Yali Amit, University of Chicago Nafees Bin Zafar, DreamWorks Animation Emery Brown, Massachusetts General Hospital Robert Bryant, Mathematical Sciences Research Institute Philip Bucksbaum, Stanford University Russel Caflisch, University of California, Los Angeles James Carlson, Clay Mathematics Institute William Cleveland, Purdue University Ronald Coifman, Yale University Peter Constantin, University of Chicago James Crowley, Society for Industrial and Applied Mathematics Brenda Dietrich, IBM T.J. Watson Research Center David Donoho, Stanford University Cynthia Dwork, Microsoft Research Lawrence Ein, University of Illinois at Chicago Charles Fefferman, Princeton University Robert Fefferman, University of Chicago John S. Gardenier, Centers for Disease Control and Prevention (ret.) Scott Guthery, Docent Press Alfred Hales, Institute for Defense Analyses’ Center for Communications Research, La Jolla Kathryn B. Hall, Hewlett Packard James J. Higgins, Kansas State University Shi Jin, University of Wisconsin C. Judson King, University of California, Berkeley William E. Kirwan, University System of Maryland Bryna Kra, Northwestern University Deborah Lockhart, National Science Foundation Dana Mackenzie, mathematics writer

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ACKNOWLEDGMENTS xi Wen Masters, Office of Naval Research Donald McClure, American Mathematical Society Jill Mesirov, Broad Institute Diane K. Michelson, International Sematech Manufacturing Initiative Assaf Naor, New York University Deborah Nolan, University of California, Berkeley Martin Nowak, Harvard University Sastry Pantula, National Science Foundation Colette Patt, University of California, Berkeley Walter Polansky, Department of Energy Adrian Raftery, University of Washington Samuel Rankin, American Mathematical Society Nancy Reid, University of Toronto Fadil Santosa, University of Minnesota Terence Sejnowski, University of California, San Diego Harry Shum, Microsoft Corporation James Simons, Renaissance Technologies Douglas Simpson, University of Illinois at Urbana-Champaign Hal Stern, University of California, Irvine Tina Straley, Mathematical Association of America Terence Tao, University of California, Los Angeles Richard Taylor, Harvard University Charles Toll, National Security Agency Kam Tsui, University of Wisconsin Gunther Uhlmann, University of Washington Ron Wasserstein, American Statistical Association S.-T. Yau, Harvard University Bin Yu, University of California, Berkeley Robert Zimmer, University of Chicago

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Contents SUMMARY 1 1 INTRODUCTION 15 Study Overview, 15 Nature of the Mathematical Sciences, 19 Everyone Should Care About the Mathematical Sciences, 21 Structure of the Report, 25 2 VITALITY OF THE MATHEMATICAL SCIENCES 27 The Topology of Three-Dimensional Spaces, 28 Uncertainty Quantification, 32 The Mathematical Sciences and Social Networks, 33 The Protein-Folding Problem, 35 The Fundamental Lemma, 37 Primes in Arithmetic Progression, 39 Hierarchical Modeling, 40 Algorithms and Complexity, 42 Inverse Problems: Visibility and Invisibility, 44 The Interplay of Geometry and Theoretical Physics, 46 New Frontiers in Statistical Inference, 50 Economics and Business: Mechanism Design, 53 Mathematical Sciences and Medicine, 54 Compressed Sensing, 55 xiii

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xiv CONTENTS 3 CONNECTIONS BETWEEN THE MATHEMATICAL SCIENCES AND OTHER FIELDS 58 Introduction, 58 Broadening the Definition of the Mathematical Sciences, 59 Implications of the Broadening of the Mathematical Sciences, 66 Two Major Drivers of Expansion: Computation and Big Data, 71 Contributions of Mathematical Sciences to Science and Engineering, 82 Contributions of Mathematical Sciences to Industry, 82 Contributions of Mathematical Sciences to National Security, 87 4 IMPORTANT TRENDS IN THE MATHEMATICAL SCIENCES 93 Increasing Importance of Connections for Mathematical Sciences Research, 93 Innovation in Modes for Scholarly Interactions and Professional Growth, 99 The Mathematical Sciences Should More Thoroughly Embrace Computing, 106 Funding Implications of Increasing Connectivity of the Mathematical Sciences, 107 A Vision for 2025, 110 5 PEOPLE IN THE MATHEMATICAL SCIENCES ENTERPRISE 116 Introduction, 116 Changing Demand for the Mathematical Sciences, 117 The Typical Educational Path in the Mathematical Sciences Needs Adjustments, 124 Attracting More Women and Underrepresented Minorities to the Mathematical Sciences, 128 The Critical Role of K-12 Mathematics and Statistics Education, 137 Enrichment for Precollege Students with Clear Talent in Mathematics and Statistics, 140 6 THE CHANGING ACADEMIC CONTEXT 145 APPENDIXES A Past Strategic Studies 155 B Meeting Agendas and Other Inputs to the Study 159 C Basic Data about the U.S. Mathematical Sciences 163

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CONTENTS xv D Examples of the Mathematical Sciences Support of Science and Engineering 175 E Illustrative Programs Aimed at Increasing Participation in the Mathematical Sciences by Women and Underrepresented Minorities 190 F Biographical Sketches of Committee Members and Staff 197

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