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of the( N E W S E R I E S )

BULLETIN

Articles Susan Friedlander Chief Editor

Book Reviews Peter Kuchment

Editors

VOLUME 53 NUMBER 4 OCTOBER 2016

Available electronically at www.ams.org/bull/

Bulletin (New Series) of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research writ-ten in a way that gives insight to mathematicians who may not be experts in the particulartopic. The Bulletin publishes, by invitation only, reviews of selected books in mathematicsand short articles in the “Mathematical Perspectives” section.

The book review appearing on pp. 701–705 of this volume is reprinted with permission.Copyright c©2015 Society for Industrial and Applied Mathematics. All rights reserved.

Submission information. See Information for Authors at the end of this issue.Publication on the AMS website. Articles are published on the AMS website

individually after proof is returned from authors and before appearing in an issue.Subscription information. Bulletin (New Series) of the American Mathematical

Society is published quarterly. It is available electronically from www.ams.org/journals/.

For paper delivery, subscription prices for Volume 53 (2016) are US$574 list, US$459.20institutional member, US$344.40 individual member, US$516.60 corporate member. Thesubscription price for members is included in the annual dues. Subscription renewalsare subject to late fees. See www.ams.org/journal-faq for more journal subscriptioninformation. If ordering the paper version, add US$5 for delivery within the UnitedStates; US$8 for delivery outside the United States.

Back number information. For back issues see www.ams.org/backvols.Subscriptions and orders should be addressed to the American Mathematical Society,

P.O. Box 845904, Boston, MA 02284-5904 USA. All orders must be accompanied by pay-ment. Other correspondence should be addressed to the American Mathematical Society,201 Charles Street, Providence, RI 02904-2294 USA.

Copying and reprinting. Individual readers of this publication, and nonprofit li-braries acting for them, are permitted to make fair use of the material, such as to copyan article for use in teaching or research. Permission is granted to quote brief passagesfrom this publication in reviews, provided the customary acknowledgment of the sourceis given. Republication, systematic copying, or multiple reproduction of any material inthis publication is permitted only under license from the American Mathematical Soci-ety. Permissions to reuse portions of AMS publication content are now being handledby Copyright Clearance Center’s RightsLink�service. For more information, please visithttp : //www.ams.org/rightslink.

Translation rights and licensed reprint requests should be sent to [email protected].

Excluded from these provisions is material for which the author holds copyright. Insuch cases, requests for permission to reuse or reprint should be addressed directly to theauthor(s). Copyright ownership is indicated in the notice in the lower right-hand cornerof the first page of each article.

Bulletin (New Series) of the American Mathematical Society (ISSN 0273-0979 (print); ISSN 1088-9485 (online)) is published quarterly by the American Mathematical Society at 201 Charles Street,Providence, RI 02904-2294 USA. Periodicals postage is paid at Providence, Rhode Island, and addi-tional mailing offices. POSTMASTER: Send address changes to Bulletin (New Series) of the AmericanMathematical Society, American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294USA.

c© 2016 American Mathematical Society. All rights reserved.

This journal is indexed in Mathematical Reviews, Science Citation Index�,Science Citation IndexTM–Expanded, ISI Alerting ServicesSM, CompuMath

Citation Index�, and Current Contents�/Physical, Chemical &Earth Sciences. This journal is archived in Portico and in CLOCKSS.

Printed in the United States of America.∞© The paper used in this journal is acid-free and falls withinthe guidelines established to ensure permanence and durability.

10 9 8 7 6 5 4 3 2 1 21 20 19 18 17 16

BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY

CONTENTS

Vol. 53, No. 4 October 2016

ARTICLES

Michael Holst, Olivier Sarbach, Manuel Tiglio, and MicheleVallisneri, The emergence of gravitational wave science: 100 years ofdevelopment of mathematical theory, detectors, numerical algorithms,and data analysis tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

Michael Aschbacher and Bob Oliver, Fusion systems . . . . . . . . . . . . . . . . . 555

Sourav Chatterjee, An introduction to large deviations for random graphs 617

Skip Garibaldi, E8, the most exceptional group . . . . . . . . . . . . . . . . . . . . . . . . . 643

MATHEMATICAL PERSPECTIVES

Gerald L. Alexanderson, contributor Leonard F. Klosinski, Aboutthe cover: Ramanujan in bronze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673

Selected Mathematical Reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

BOOK REVIEWS

Andrew Hassell (Reviewer), Hangzhou lectures on eigenfunctions of theLaplacian, by Christopher D. Sogge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

Lloyd N. Trefethen (Reviewer), The Princeton companion to appliedmathematics, edited by Nicholas J. Higham . . . . . . . . . . . . . . . . . . . . . . . . . . 701

http://www.ams.org/bull/2016-53-04/S0273-0979-2016-01544-8/








http://www.ams.org/bull/2016-53-04/S0273-0979-2016-01545-X/



INDEX TO VOLUME 53 (2016)

bulletin articles

Aganagic, Mina. String theory and math: Why this marriage may last. Mathematics and dual-

ities of quantum physics, 93Aschbacher, Michael, and Bob Oliver. Fusion systems, 555

Belolipetsky, Mikhail. Arithmetic hyperbolic reflection groups, 437

Chatterjee, Sourav. An introduction to large deviations for random graphs, 617Fløystad, Gunnar, Jason McCullough, and Irena Peeva. Three themes of syzygies, 415

Garibaldi, Skip. E8, the most exceptional group, 643

Holst, Michael, Olivier Sarbach, Manuel Tiglio, and Michele Vallisneri. The emergence of gravita-tional wave science: 100 years of development of mathematical theory, detectors, numerical

algorithms, and data analysis tools, 513

Kuchment, Peter. An overview of periodic elliptic operators, 343McCullough, Jason. See Fløystad, Gunnar

Nahmod, Andrea R. The nonlinear Schrodinger equation on tori: Integrating harmonic analysis,geometry, and probability, 57

Okounkov, Andrei. Limit shapes, real and imagined, 187

Oliver, Bob. See Aschbacher, MichaelPeeva, Irena. See Fløystad, Gunnar

Penner, R. C. Moduli spaces and macromolecules, 217

Sarbach, Olivier. See Holst, MichaelTiglio, Manuel. See Holst, Michael

Vallisneri, Michele. See Holst, Michael

Weinstein, Jared. Reciprocity laws and Galois representations: recent breakthroughs, 1Wright, Alex. From rational billiards to dynamics on moduli spaces, 41

Zakharevich, Inna. Perspectives on scissors congruence, 269

mathematical perspectives

Alexanderson, Gerald L., and Leonard F. Klosinski. About the cover: Gauss and the Disquisi-

tiones Arithmeticæ, 117. About the cover: The Newton–Leibniz controversy, 295

Alexanderson, Gerald L., contributor Leonard F. Klosinski. About the cover: Ramanujan inbronze, 673

Klosinski, Leonard F. See Alexanderson, Gerald L.

Oswald, Nicola M. R., and Jorn Steuding. About the cover: Zeta-functions associated with qua-dratic forms in Adolf Hurwitz’s estate, 477

Steuding, Jorn. See Oswald, Nicola M. R.

book reviews

Bates, Daniel J., Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler. Nu-

merically solving polynomial systems with Bertini, reviewed by Henry Schenck, 179Brudnyi, Alexander, and Yuri Brudnyi. Methods of geometric analysis in extension and trace

problems. Volume 1, reviewed by Garving K. Luli, 143. Methods of geometric analysis in extension and trace problems. Volume 2, reviewed by

Garving K. Luli, 143Brudnyi, Yuri. See Brudnyi, Alexander

Grebert, B., and T. Kappeler. The defocusing NLS equation and its normal form, reviewed byDario Bambusi, 337

Hales, Thomas C. Dense sphere packings: a blueprint for formal proofs, reviewed by Jeffrey C.

Lagarias, 159Hauenstein, Jonathan D. See Bates, Daniel J.

Higham, Nicholas J., editor. The Princeton companion to applied mathematics, reviewed byLloyd N. Trefethen, 701

Iwaniec, H. Lectures on the Riemann zeta function, reviewed by Brian Conrey, 507

Jean, Frederic. Control of nonholonomic systems: from sub-Riemannian geometry to motionplanning, reviewed by I. Zelenko, 151

Kappeler, T. See Grebert, B.

http://www.ams.org/journal-getitem?pii=S0273-0979-2015-01517-X


http://www.ams.org/journal-getitem?pii=S0273-0979-2016-01538-2























INDEX TO VOLUME 53 (2016)

Lo Bello, Anthony. Origins of mathematical words: a comprehensive dictionary of Latin, Greek,

and Arabic roots, reviewed by John Stillwell, 331

Maggi, Francesco. Sets of finite perimeter and geometric variational problems. An introductionto geometric measure theory, reviewed by Alex Iosevich, 167

Marsh, Robert J. Lecture notes on cluster algebras, reviewed by Ralf Schiffler, 325

Mazur, Joseph. Enlightening symbols: a short history of mathematical notation and its hiddenpowers, reviewed by John Stillwell, 331

Ostrovskii, M. I. Metric embeddings: bilipschitz and coarse embeddings into Banach spaces,

reviewed by Florent P. Baudier and William B. Johnson, 495Sogge, Christopher D. Hangzhou lectures on eigenfunctions of the Laplacian, reviewed by An-

drew Hassell, 693

Sommese, Andrew J. See Bates, Daniel J.Talagrand, M. Upper and lower bounds for stochastic processes, reviewed by Antonio Auffinger,

173Wampler, Charles W. See Bates, Daniel J.

Weiß, Christian. Twisted Teichmuller curves, reviewed by Thomas A. Schmidt, 313

BULLETIN( N E W S E R I E S ) OF THE

A M E R I C A N M A T H E M A T I C A L S O C I E T Y

PROVIDENCE, RHODE ISLAND USA

ISSN 0273-0979

VOLUME 53 2016

Available electronically at www.ams.org/bull/

EDITORSSusan Friedlander, Chief EditorPeter Kuchment, Book Reviews

ASSOCIATE EDITORS

Emmanuel CandesDavid Cox Richard T. DurrettMark EmbreeDaniel S. FreedEdward Frenkel Irene M. GambaMark GoreskyAndrew J. GranvilleRobert M. Guralnick Craig L. Huneke

Lisa Jeffrey Bryna Kra Steven G. KrantzBarry MazurKen OnoPhilip E. ProtterAndrew Ranicki Israel Michael Sigal Christoph M. ThieleYuri TschinkelMichael Wolf

CONSULTANTS TO THE EDITORSGerald L. AlexandersonEdward G. Dunne

Bulletin (New Series) of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research writ-ten in a way that gives insight to mathematicians who may not be experts in the particulartopic. The Bulletin publishes, by invitation only, reviews of selected books in mathematicsand short articles in the “Mathematical Perspectives” section.

Submission information. See Information for Authors at the end of this issue.Publication on the AMS website. Articles are published on the AMS website

individually after proof is returned from authors and before appearing in an issue.Subscription information. Bulletin (New Series) of the American Mathematical

Society is published quarterly. It is available electronically from www.ams.org/journals/.

For paper delivery, subscription prices for Volume 53 (2016) are US$574 list, US$459.20institutional member, US$344.40 individual member, US$516.60 corporate member. Thesubscription price for members is included in the annual dues. Subscription renewalsare subject to late fees. See www.ams.org/journal-faq for more journal subscriptioninformation. If ordering the paper version, add US$5 for delivery within the UnitedStates; US$8 for delivery outside the United States.

Back number information. For back issues see www.ams.org/backvols.Subscriptions and orders should be addressed to the American Mathematical Society,

P.O. Box 845904, Boston, MA 02284-5904 USA. All orders must be accompanied by pay-ment. Other correspondence should be addressed to the American Mathematical Society,201 Charles Street, Providence, RI 02904-2294 USA.

Copying and reprinting. Individual readers of this publication, and nonprofit li-braries acting for them, are permitted to make fair use of the material, such as to copyan article for use in teaching or research. Permission is granted to quote brief passagesfrom this publication in reviews, provided the customary acknowledgment of the sourceis given. Republication, systematic copying, or multiple reproduction of any material inthis publication is permitted only under license from the American Mathematical Soci-ety. Permissions to reuse portions of AMS publication content are now being handledby Copyright Clearance Center’s RightsLink�service. For more information, please visithttp : //www.ams.org/rightslink.

Translation rights and licensed reprint requests should be sent to [email protected].

Excluded from these provisions is material for which the author holds copyright. Insuch cases, requests for permission to reuse or reprint should be addressed directly to theauthor(s). Copyright ownership is indicated in the notice in the lower right-hand cornerof the first page of each article.

Bulletin (New Series) of the American Mathematical Society (ISSN 0273-0979 (print); ISSN 1088-9485 (online)) is published quarterly by the American Mathematical Society at 201 Charles Street,Providence, RI 02904-2294 USA. Periodicals postage is paid at Providence, Rhode Island, and addi-tional mailing offices. POSTMASTER: Send address changes to Bulletin (New Series) of the AmericanMathematical Society, American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294USA.

c© 2016 American Mathematical Society. All rights reserved.

This journal is indexed in Mathematical Reviews, Science Citation Index�,Science Citation IndexTM–Expanded, ISI Alerting ServicesSM, CompuMath

Citation Index�, and Current Contents�/Physical, Chemical &Earth Sciences. This journal is archived in Portico and in CLOCKSS.

Printed in the United States of America.∞© The paper used in this journal is acid-free and falls withinthe guidelines established to ensure permanence and durability.

10 9 8 7 6 5 4 3 2 1 21 20 19 18 17 16


CONTENTS

Vol. 53, No. 1 January 2016

Articles

Jared Weinstein, Reciprocity laws and Galois representations: recentbreakthroughs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Alex Wright, From rational billiards to dynamics on moduli spaces . . . . . . 41

Andrea R. Nahmod, The nonlinear Schrodinger equation on tori:Integrating harmonic analysis, geometry, and probability . . . . . . . . . . . . . 57

Mina Aganagic, String theory and math: Why this marriage may last.Mathematics and dualities of quantum physics . . . . . . . . . . . . . . . . . . . . . . . . 93

Mathematical Perspectives

Gerald L. Alexanderson and Leonard F. Klosinski, About the cover:Gauss and the Disquisitiones Arithmeticæ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Selected Mathematical Reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Book Reviews

Garving K. Luli (Reviewer), Methods of geometric analysis in extensionand trace problems. Volume 1, by Alexander Brudnyi and Yuri Brudnyi;Methods of geometric analysis in extension and trace problems. Volume2, by Alexander Brudnyi and Yuri Brudnyi . . . . . . . . . . . . . . . . . . . . . . . . . . 143

I. Zelenko (Reviewer), Control of nonholonomic systems: from sub-Riemannian geometry to motion planning, by Frederic Jean . . . . . . . . . . 151

Jeffrey C. Lagarias (Reviewer), Dense sphere packings: a blueprint forformal proofs, by Thomas C. Hales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Alex Iosevich (Reviewer), Sets of finite perimeter and geometric variationalproblems. An introduction to geometric measure theory, by FrancescoMaggi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Antonio Auffinger (Reviewer), Upper and lower bounds for stochasticprocesses, by M. Talagrand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Henry Schenck (Reviewer), Numerically solving polynomial systems withBertini, by Daniel J. Bates, Jonathan D. Hauenstein, Andrew J.Sommese, and Charles W. Wampler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Vol. 53, No. 2 April 2016

Articles

Andrei Okounkov, Limit shapes, real and imagined . . . . . . . . . . . . . . . . . . . . . . 187

R. C. Penner, Moduli spaces and macromolecules . . . . . . . . . . . . . . . . . . . . . . . . 217

Inna Zakharevich, Perspectives on scissors congruence . . . . . . . . . . . . . . . . . . . 269





























Gerald L. Alexanderson and Leonard F. Klosinski, About the cover:The Newton–Leibniz controversy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295


Book Reviews

Thomas A. Schmidt (Reviewer), Twisted Teichmuller curves, by ChristianWeiß . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Ralf Schiffler (Reviewer), Lecture notes on cluster algebras, by Robert J.Marsh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

John Stillwell (Reviewer), Origins of mathematical words: a comprehensivedictionary of Latin, Greek, and Arabic roots, by Anthony Lo Bello;Enlightening symbols: a short history of mathematical notation and itshidden powers, by Joseph Mazur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Dario Bambusi (Reviewer), The defocusing NLS equation and its normalform, by B. Grebert and T. Kappeler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Vol. 53, No. 3 July 2016

Articles

Peter Kuchment, An overview of periodic elliptic operators . . . . . . . . . . . . . . 343

Gunnar Fløystad, Jason McCullough, and Irena Peeva, Three themesof syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

Mikhail Belolipetsky, Arithmetic hyperbolic reflection groups . . . . . . . . . . . 437


Nicola M. R. Oswald and Jorn Steuding, About the cover: Zeta-functions associated with quadratic forms in Adolf Hurwitz’s estate . . 477


Book Reviews

Florent P. Baudier and William B. Johnson (Reviewers), Metricembeddings: bilipschitz and coarse embeddings into Banach spaces, byM. I. Ostrovskii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

Brian Conrey (Reviewer), Lectures on the Riemann zeta function, by H.Iwaniec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507























Articles

Michael Holst, Olivier Sarbach, Manuel Tiglio, and MicheleVallisneri, The emergence of gravitational wave science: 100 yearsof development of mathematical theory, detectors, numerical algorithms,and data analysis tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513



Skip Garibaldi, E8, the most exceptional group . . . . . . . . . . . . . . . . . . . . . . . . . . 643


Gerald L. Alexanderson, contributor Leonard F. Klosinski, About thecover: Ramanujan in bronze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673


Book Reviews

Andrew Hassell (Reviewer), Hangzhou lectures on eigenfunctions of theLaplacian, by Christopher D. Sogge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

Lloyd N. Trefethen (Reviewer), The Princeton companion to appliedmathematics, edited by Nicholas J. Higham . . . . . . . . . . . . . . . . . . . . . . . . . . 701












Editorial Board for Articles

Emmanuel CandesRichard T. DurrettDaniel S. FreedEdward FrenkelSusan Friedlander, ChairIrene M. GambaMark GoreskyAndrew J. Granville

Robert M. GuralnickBryna R. KraBarry C. MazurAndrew RanickiChristoph M. ThieleYuri TschinkelMichael Wolf

Editorial Board for Book Reviews

David CoxMark EmbreeCraig L. HunekeLisa JeffreySteven G. Krantz

Peter Kuchment, ChairKen OnoPhilip E. ProtterIsrael Michael Sigal

Consultants to the Editors

Gerald L. AlexandersonEdward Dunne

Chief Editor: Susan Friedlander

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CONTENTS


ARTICLES

Michael Holst, Olivier Sarbach, Manuel Tiglio, and Michele

Vallisneri, The emergence of gravitational wave science: 100 years of

development of mathematical theory, detectors, numerical algorithms,

and data analysis tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513



Skip Garibaldi, E8, the most exceptional group . . . . . . . . . . . . . . . . . . . . . . . . . 643

MATHEMATICAL PERSPECTIVES

Gerald L. Alexanderson, contributor Leonard F. Klosinski, About

the cover: Ramanujan in bronze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673

Selected Mathematical Reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

BOOK REVIEWS

Andrew Hassell (Reviewer), Hangzhou lectures on eigenfunctions of the

Laplacian, by Christopher D. Sogge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

Lloyd N. Trefethen (Reviewer), The Princeton companion to applied

mathematics, edited by Nicholas J. Higham . . . . . . . . . . . . . . . . . . . . . . . . . . 701

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erican Mathem

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enter 35 M

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