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Fueling Innovation and Discovery
The Mathematical Sciences
in the 21st Century
Committee on the Mathematical Sciences in 2025
Board on Mathematical Sciences and Their Applications
Division on Engineering and Physical Sciences
THE NATIONAL ACADEMIES PRESS
Washington, D.C.
www.nap.edu
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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 Academy 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 engineers. 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 engineering programs aimed at meeting national
needs, encourages education and research, and recognizes the superior achievements of
engineers. Dr. Charles M. Vest is president 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 Institute 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 Sciences 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 Council 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
FUELING
innovation and discovery
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COMMITTEE ON THE MATHEMATICAL BOARD ON MATHEMATICAL SCIENCES AND
SCIENCES IN 2025 THEIR APPLICATIONS
THOMAS E. EVERHART, Chair, California Institute of C. DAVID LEVERMORE, Chair, University of Maryland
Technology TANYA STYBLO BEDER, SBCC Group, Inc.
MARK L. GREEN, Vice Chair, University of California, PATRICIA FLATLEY BRENNAN, University of Wisconsin
Los Angeles GERALD G. BROWN, Naval Postgraduate School
TANYA STYBLO BEDER, SBCC Group, Inc. L. ANTHONY COX, JR., Cox Associates, Inc.
JAMES O. BERGER, Duke University BRENDA DIETRICH, IBM Thomas J. Watson Research
LUIS A. CAFFARELLI, University of Texas at Austin Center
EMMANUEL J. CANDES, Stanford University CONSTANTINE GATSONIS, Brown University
PHILLIP COLELLA, Lawrence Berkeley National Laboratory DARRYLL HENDRICKS, UBS Investment Bank
DAVID EISENBUD, University of California, Berkeley KENNETH L. JUDD, Hoover Institution
PETER WILCOX JONES, Yale University DAVID MAIER, Portland State University
JU-LEE KIM, Massachusetts Institute of Technology JAMES C. McWILLIAMS, University of California,
YANN LeCUN, New York University Los Angeles
JUN LIU, Harvard University JUAN C. MEZA, University of California, Merced
JUAN MALDACENA, Institute for Advanced Study JOHN W. MORGAN, Stony Brook University
JOHN W. MORGAN, Stony Brook University VIJAYAN N. NAIR, University of Michigan
YUVAL PERES, Microsoft Research CLAUDIA NEUHAUSER, University of Minnesota,
EVA TARDOS, Cornell University Rochester
MARGARET H. WRIGHT, New York University J. TINSLEY ODEN, University of Texas at Austin
JOE B. WYATT, Vanderbilt University DONALD G. SAARI, University of California, Irvine
J.B. SILVERS, Case Western Reserve University
STAFF GEORGE SUGIHARA, University of California, San Diego
EVA TARDOS, Cornell University
SCOTT WEIDMAN, Director, Board on Mathematical KAREN L. VOGTMANN, Cornell University
Sciences and Their Applications BIN YU, University of California, Berkeley
DANA MACKENZIE, Mathematics Writer
TOM ARRISON, Senior Staff Officer STAFF
MICHELLE SCHWALBE, Associate Program Officer
BARBARA WRIGHT, Administrative Assistant SCOTT WEIDMAN, Director
NEAL GLASSMAN, Senior Program Officer
MICHELLE SCHWALBE, Associate Program Officer
BARBARA WRIGHT, Administrative Assistant
BETH DOLAN, Financial Manager
in the 21st Century
The Mathematical Sciences
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Fueling Innovation and Discovery
The Mathematical Sciences in the 21st Century
Launched in 2010 with funding from the National Science Foundation, the National Academies’ study The Mathematical
Sciences in 2025 (MathSci 2025) is a forward-looking assessment of the current state of the mathematical sciences in the
United States. The final report of the MathSci 2025 project will be released later in 2012. More information about
the project is available at www.nas.edu/mathsci2025.
This publication, Fueling Innovation and Discovery: The Mathematical Sciences in the 21st Century, is a separate product
being released in advance of the final report. It is based on the committee’s identification of recent advances in the
mathematical sciences or advances enabled by mathematical sciences research, drawn from the committee’s as-
sessment of the vitality of the discipline. This report is geared toward general readers who would like to know more
about ongoing advances in the mathematical sciences and how these advances are changing our understanding of
the world, creating new technologies, and transforming industries.
In selecting the topics for this report, the committee aimed to cover a range of mathematical sciences subfields and
areas of impact, choosing topics where information was accessible and where developments could be described in a
few pages. While the committee believes that all the topics covered are important and interesting, this publication is
not intended to be a comprehensive selection of the most important developments in the mathematical sciences.
The committee worked primarily with mathematics writer Dana Mackenzie to prepare this report. It greatly
appreciates his insights and hard work. During late 2010 and 2011, appropriate topics were identified, experts
consulted, drafts prepared and revised, and accompanying images compiled. This report contains no committee
conclusions or recommendations.
This report has been reviewed in draft form by individuals chosen for their diverse perspectives and technical exper-
tise, in accordance with procedures approved by the National Academies’ Report Review Committee. The review of
this report was overseen by Samuel Fuller, Analog Devices, Inc. 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 quality and objectivity. The review comments and draft
manuscript remain confidential to protect the integrity of the process.
We wish to thank the following individuals for their review of this report:
JOHN BRADY, California Institute of Technology,
JAMES CARLSON, Clay Mathematics Institute,
ANNA GILBERT, University of Michigan,
MARVIN GOLDSTEIN, NASA Glenn Research Center,
RONALD GRAHAM, University of California, San Diego,
JON KETTENRING, Telcordia Technologies, Inc. (retired),
AROGYASWAMI PAULRAJ, Stanford University,
STEVEN STROGATZ, Cornell University,
LARRY WASSERMAN, Carnegie Mellon University,
EUGENE WONG, University of California, Berkeley, and
BIN YU, University of California, Berkeley.
Although the reviewers listed above have provided many constructive comments and suggestions, they were not
asked to endorse the content of the report, nor did they see the final draft before its release. Responsibility for the
final content of this report rests entirely with the committee and the institution.
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CONTENTS
1 Introduction
3 Compressed Sensing: Through the Kaleidoscope
7 Eigenvectors: From the Mathematical Sciences to . . . an IPO
11 Mathematical Simulations: When the Lab Isn’t Big Enough
18 Mathematical Sciences Inside . . . Tsunamis
20 Bayesian Inference: Not an Enigma Anymore
24 Diffusion Tensor Imaging: A New View of the Brain
29 Fast Multipole Method: A Long-Term Payoff
34 Mathematical Sciences Inside . . . the Battlefield
36 Cellular Automata: Sublimely Complex
40 Graph Spectra: Sparsest Cuts in Minimum Time
44 Bioinformatics: Interpreting the Human Genome
49 Geometry and Physics: Endlessly Intertwined
53 Probability and Statistical Physics: Connecting Microscopic
and Macroscopic
56 Mathematical Sciences Inside . . . Inventions
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