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

NATIONAL RESEARCH COUNCIL
                                 OF THE NATIONAL ACADEMIES

THE NATIONAL ACADEMIES PRESS
Washington, D.C.
www.nap.edu



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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 i

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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 ii

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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 iii

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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. iv

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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 v

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