168 pages on 40 lb Velocity Spine: 3/16" Print Cover on 9pt CarolinaISSN 0002-9920 Volume 57, Number 9 Notices of the American Mathematical Society Volume 57, Number 9, Pages 10731240, October 2010of the American Mathematical SocietyTrim: 8.25" x 10.75"October 2010Topological Methods for Nonlinear Oscillationspage 1080Tilings, Scaling Functions, and a Markov Processpage 1094Reminiscences of Grothendieck and His Schoolpage 1106Speaking with the Natives: Reectionson Mathematical Communicationpage 1121New Orleans Meetingpage 1192 What if Newton saw an apple as just an apple?Take a closer look at TI-Nspire software and get started for free. Visit education.ti.com/us/learnandearn Mac is a registered trademark of Apple, Inc.2010 Texas Instruments AD10100Would we have his laws of gravity? Thats the power of seeing things in diferent ways. When it comes to TI technology, look beyond graphing calculators.Think TI-Nspire Computer Software for PC or Mac. Software thats easy to use. 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About the Cover: New Orleans Vieux Carr(see page 1151) ,!4%8 ,!4%8,!4%8c=LL vcn ===Lic=+iceHow to appIy Procedures of appIications avaiIabIe atwww.sciencesmaths-paris.frand, for PGSM, atwww.sciencesmaths-paris.fr/pgsmrcLc=+ic eciccce M=+icm=+icLce cc ==nie - www.ecicccem=+ie-==nie.vn=, nLc =icnnc c+ M=nic cLnic re& ==nie ccccx ce1cL. : + & ee c - r=x : + & ee crcLccne ecic+ivc ==n+cne ==n+cneThe Foundation Sciences Mathmatiques de Paris is pIeased to offer the foI-Iowing opportunities to outstanding scientists and exceIIent students:c=ccmic c=n&c-&c& D The Foundation Prize 2011D The Research Chair 2011D Fifteen postdoctoraI positionsD PGSM: Ten master course schoIarships (new!)Algebra and Algebraic GeometryCollege Algebra Henry Burchard FineAMS Chelsea Publishing, Volume 354; 1961; 631 pp.; hardcover; ISBN: 978-0-8218-3863-1; List US$72; Sale Price US$25.20; Order code: CHEL/354.HSelected Works of Phillip A. Griffiths with Commentary Enrico Arbarello, Robert L. Bryant, C. 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Go to www.ams.org/bookstore to view the entire list of titles on sale.AMERICAN MATHEMATICAL SOCIETYFeatures Communications 1116 Alice Turner Schafer (19152009): RemembrancesGeorgia Benkart, Bhama Srinivasan, Mary Gray, Ellen Maycock, Linda Rothschild, edited by Anne Leggett 1125 WHAT IS... a Linear Algebraic Group?Skip Garibaldi 1127 Doceamus: The Doctor Is InRobert Borrelli 1132 Meta-Morphism: From Graduate Student to Networked MathematicianAndrew Schultz 1137 Lovsz Receives Kyoto Prize Commentary 1079 Opinion: Agenda for a Mathematical RenaissanceTeodora-Liliana R adulescu and Vicentiu R adulescu 1129 The Cult of Statistical SignificanceA Book ReviewReviewed by Olle HggstrmNoticesof the American Mathematical SocietyThe feature articles this month exhibit the dynamic and diversity of modern mathematics. The article by Chris Byrnes considers applications of modern topology to questions of physics. The article by Dick Gundy shows how wavelets can be used to understand probability theory. And the article by Jerry Folland considers questions of communication that will resonate with us all. The interview with Luc Illusie aboutAlexander Grothendieck will give us all pause for thought.Steven G. KrantzEditor 1080 Topological Methods for Nonlinear Oscillations Christopher I. Byrnes 1094 Tilings, Scaling Functions, and a Markov Process Richard F. Gundy 1106 Reminiscences of Grothendieck andHis School Luc Illusie, with Alexander Beilinson, Spencer Bloch, Vladimir Drinfeld, et al. 1121 Speaking with the Natives: Reflections on Mathematical Communication Gerald B. FollandOctober 20101106 11291094 1116From the AMS SecretaryAMS Officers and Committee Members . . . . . . . . . . . . . . . . . . . 1152Statistics on Women Compiled by the AMS . . . . . . . . . . . . . . . 1163EDITOR: Steven G. KrantzASSOCIATE EDITORS: Krishnaswami Alladi, David Bailey, Jonathan Borwein, Susanne C. Brenner, Bill Casselman (Graphics Editor), Robert J. Daverman, Susan Friedlander, Robion Kirby, Rafe Mazzeo, Harold Parks, Lisette de Pillis, Peter Sarnak, Mark Saul, Edward Spitznagel, John SwallowSENIOR WRITER and DEPUTY EDITOR: Allyn JacksonMANAGING EDITOR: Sandra FrostCONTRIBUTING WRITER: Elaine KehoeCONTRIBUTING EDITOR: Randi D. RudenEDITORIAL ASSISTANT: David M. CollinsPRODUCTION ASSISTANT: Muriel ToupinPRODUCTION: Kyle Antonevich, Anna Hattoy, Teresa Levy, Mary Medeiros, Stephen Moye, Erin Murphy, Lori Nero, Karen Ouellette, Donna Salter, Deborah Smith, Peter Sykes, Patricia ZinniADVERTISING SALES: Anne NewcombSUBSCRIPTION INFORMATION: Subscription prices for Volume 57 (2010) are US$488 list; US$390 insti-tutional member; US$293 individual member. (The subscription price for members is included in the annual dues.) A late charge of 10% of the subscrip-tion price will be imposed upon orders received from nonmembers after January 1 of the subscription year. 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For more information, see the section Reference and Book List.NOTICES ON THE AMS WEBSITE: Supported by the AMS membership, most of this publication is freely available electronically through the AMS website, the Societys resource for delivering electronic prod-ucts and services. Use the URL http://www.ams.org/notices/ to access the Notices on the website.[Notices of the American Mathematical Society (ISSN 0002-9920) is published monthly except bimonthly in June/July by the American Mathemati cal Society at 201 Charles Street, Providence, RI 02904-2294 USA, GST No. 12189 2046 RT****. Periodicals postage paid at Providence, RI, and additional mailing offices. POSTMASTER: Send address change notices to Notices of the American Mathematical Society, P.O. Box 6248, Providence, RI 02940-6248 USA.] Publication here of the Societys street address and the other information in brackets above is a technical requirement of the U.S. Postal Service. Tel: 401-455-4000, email: notices@ams.org. 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The paper used in this journal is acid-free and falls within the guidelines established to ensure permanence and durability.DepartmentsAbout the Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1151Mathematics People . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138Nagel and Wainger Receive 2007-2008 Bergman Prize, AMS Menger Awards at the 2010 ISEF, Gupta and Grattan-Guinness Awarded May Prize, Buchweitz Receives Humboldt Research Award, Prizes of the Royal Society, SIAM Prizes Awarded, Prizes of the London Mathematical Society, Prizes of the Canadian Mathematical Society, 2010 International Mathematical Olympiad, SIAM Fellows Elected.Mathematics Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144NSF Postdoctoral Research Fellowships, NSF Conferences and Workshops in the Mathematical Sciences, NSF Research Networks in the Mathematical Sciences, Visiting Positions at CIRM, PIMS Call for Proposals, News from IPAM.For Your Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146Report: Considering the Future of K12 STEM Curricula and Instructional Design, Correction.Inside the AMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146National Who Wants to Be a Mathematician, Deaths of AMS Members.Reference and Book List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147 Mathematics Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164New Publications Offered by the AMS . . . . . . . . . . . . . . . . . . . . . . 1167Classified Advertisements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174Mathematical Sciences Employment Center inNew Orleans, LA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182AMS Short Courses in New Orleans, LA . . . . . . . . . . . . . . . . . . . . 1185Meetings and Conferences of the AMS . . . . . . . . . . . . . . . . . . . . . 1054New Orleans Meeting Timetable . . . . . . . . . . . . . . . . . . . . . . . . . . 1226Meetings and Conferences Table of Contents . . . . . . . . . . . . . . . 1239Noticesof the American Mathematical SocietyOpinions expressed in signed Notices articles are those of the authors and do not necessarily reflect opinions of the editors or policies of the American Mathematical Society.I thank Randi D. Ruden for her splendid editorial work, and for helping to assemble this issue. She is essential to everything that I do.Steven G. KrantzEditorGOOD MATH.ITS WEBASSIGN, ONLY BETTER.Only WebAssign is supported by every publisher and supports every course in the curriculum from developmental math to calculus and beyond. Thats good. Now weve added the latest homework tools to bring you the ultimate learning environment for math. Thats better. Welcome to a better WebAssign. There are new tools including number line, that lets students plot data graphically, then automatically assesses their work. There are more books, over 170 titles from every major publisher. And theres more server capacity with an expanded support team just waiting to serve you and over half a million loyal WebAssign users each term. Its the culmination of a team-wide effort to build the best WebAssign ever. The ingenuity of WebAssign for math. Its available now and its free to faculty at WebAssign.net. Now what could be better than that?Visit www.webassign.net to sign up for your free faculty account today.Students use number line to plot data graphically with automatic assessment.Terence Tao Professor of Mathematics, University of California, Los AngelesThe American Mathematical Society presentsThe AMS Einstein Public Lecture in MathematicsThe Cosmic Distance LadderHow do we know the distances from the earth to the sun and moon, from the sun to the other planets, and from the sun to other stars and distant galaxies? Clearly we cannot measure these directly. Nevertheless there are many indirect methods of measurement, combined with basic mathematics, which can give quite convincing and accurate results without the need for advanced technology (for instance, even the ancient Greeks could compute the distances from the earth to the sun and moon to moderate accuracy). These methods rely on climbing a cosmic distance ladder, using measurements of nearby distances to deduce estimates on distances slightly farther away. In this lecture, Tao will discuss several of the rungs in this ladder.Saturday, October 9, 2010 6:15 p.m. with a reception to follow Schoenberg Hall on the UCLA campusSponsored by the American Mathematical Society Hosted by the UCLA Department of Mathematics This event is part of the AMS 2010 Fall Western Sectional Meeting, October 9-10www.ams.org/meetings/einstein-lect.html Mars: NASA/JPL-Caltech/University of ArizonaSaturn: NASA/ESA/Erich Karkoschka (University of Arizona)Jupiter: NASA/JPLGalaxy-1: NASA/JPL-Caltech/STSclGalaxy-2 (background): MPIA/NASA/Calar Alto ObservatoryOCTOBER 2010 NOTICES OF THE AMS 1079OpinionAgenda for a Mathematical RenaissanceMost of us have wondered at some point in our careers how to motivate the interest of students in mathematics and encourage young talents to learn mathematics. In 1975 Paul Halmos [3] said: The best way to learn is to do; the worst way to teach is to talk. The best way to teach is to make students ask, and do. Dont preach factsstimulate acts. Halmos was a wise and respected scholar and there is general agreement that what he said is correct and important, but change is so difficult that perhaps we need more than words; we need a new agenda.One may begin with a recent analysis developed for The Wall Street Journal [4] that evaluates 200 professions. Ac-cording to the study mathematician is the top job in the U.S., placing first in terms of good environment, income, employ-ment outlook, physical demands, and low stress. Actuary and statistician, two related professions, rank second and third, respectively. The sociological analysis provides answers to the young students question Why do mathematics? Math-ematicians are in demand in terms of job prospects. Also, should anyone wonder if interest in mathematics has slowed down, the study shows that the role of mathematicians, both pure and applied, in the development of our society is as important as ever.Science and technology developed in an impressive way before, during, and immediately after World War II, which at-tracted post-war students to careers in research. This trend received a powerful stimulus in the 1960s and 1970s with the advent of the space age: launch of the Soviet satellite Sputnik (1957), Gagarins first human travel into space (1961), and the first manned mission to land on the Moon (Neil A. Armstrong and Edwin E. Aldrin, Apollo 11, 1969). The impact of these advances was huge. Science was recognized for the political and economic power it could generate. In those golden years, research became as important to society as it was fascinating to practitioners.Decades later, societys interest in many areas of research has diminished in most countries around the world. With exceptions in biomedical areas, genetics, or software engineer-ing, the importance of mathematics and science in society is less often recognized. Mathematics and the sciences are no longer perceived as offering desirable career opportuni-ties. Both in developed and developing countries, brilliant mathematics students who could have chosen careers in mathematics are not doing so.It is strange that students interest is low at a time when career opportunities for professional mathematicians are greater than ever [2]. This is true both for the applied fields where demand for mathematicians will continue to grow rap-idly in the next decades and for traditional areas that are rich with new developmentssee the seven Millennium Problems, cf. [1]. Yet today we notice a fundamental lack of appreciation for the richness and relevance of mathematics itself.It is possible for mathematics and mathematicians to regain social stature. The scientific enterprise can function at full potential if there is a fast flow of knowledge between the creators and users of mathematics. This is something mathematics education can and should facilitate, especially since mathematics is currently so active and vital both in research and applications.The culture of this millennium shows itself to be highly in-teractive and collaborative. It is an opportunity for mathemati-cians to work with scientists in other fields and also to reach out to the community at large. Mathematicians are uniquely qualified to articulate the value of mathematics in catalyz-ing major advances in science, health, business, economics, biomedical engineering, genetics, software engineering, and, more generally, in proving the patterns and the truths of the universe in which we live.The trend toward interactivity is an important feature of the sciences in our time. Unfortunately, some institutions have been slow to adapt to this reality. Mathematics loses a lot when it is isolated or fragmented according to various paradigms. Universities around the world, as well as many in-dustries and government agencies, will benefit from removing barriers to collaboration. In particular, powerful and diverse interactions between academic and industrial mathematicians should be enhanced. While the primary missions of academia and industries are different, the two cultures have much to learn from one another.In short, while mathematician is a top job in the U.S. today, it is no longer possible for a mathematician to remain aloof from the passing needs of the world or to continue work-ing in an ivory tower. As funds get scarce, the future of our profession is at stake. It is time for mathematicians to bring the vitality and usefulness of modern mathematics to the classrooms, to demonstrate its social impact, and to support this centurys mathematical renaissance.References[1] http://www.claymath.org/millennium.[2] P. A. Griffiths, Mathematics at the turn of the millennium, Current Science Online 77 (1999), 750758.[3] P. Halmos, The problem of learning to teach, Amer. Math. Monthly 82 (1975), 466476.[4] S. E. Needleman, Doing the math to find the good jobs. Mathematicians land top spot in new ranking of best and worst occupations in the U.S., The Wall Street Journal, Jan. 26, 2009, p. D2 (http://online.wsj.com/article_email/SB123119236117055127-lMyQjAxMDI5MzAxNjEwOTYyWj.html).[5] A. N. Whitehead, The Aims of Education and Other Essays, MacMillan Co, 1929 (reprinted in Education in the Age of Sci-ence, edited by Brand Blanshard, New York, Basic Books, 1959).Teodora-Liliana RadulescuDepartment of Mathematics, Fratii Buzesti National Collegeteodoraradulescu@yahoo.comVicentiu RadulescuInstitute of Mathematics Simion Stoilow of the Romanian Academyvicentiu.radulescu@imar.rovicentiu.radulescu@math.cnrs.frTopological Methods forNonlinear OscillationsChristopher I. ByrnesIntroductionPeriodic phenomena play a pervasive role in natu-ral and in man-made systems. They are exhibited,for example, in simple mathematical models ofthe solar system and in the observed circadianrhythms by which basic biological functions areregulated. Electronic devices producing stable pe-riodic signals underlie both the electrication ofthe world and wireless communications. My inter-est in periodic orbits was heightened by researchinto the existence of oscillations in nonlinear feed-back systems. While these kinds of applicationsare illustrated in Examples 2.2 and 4.2, a more de-tailed expedition into this important applicationarea is omitted here for the sake of space andfocus.Periodic orbits have played a prominent role inthe mathematics of dynamical systems and its ap-plications to science andengineering for centuries,due to both the importance of periodic phenom-ena and the formidable intellectual challengesinvolved in detecting or predicting periodicity.As a rst step toward addressing this challenge,Poincar developed his method of sections, begin-ning with the observation that, if a periodic orbit for a smooth vector eld X exists, if x0 and ifH is a hyperplane complementary to the tangentline Tx0() to x0 at , then on a suciently smallneighborhood S H of x0 one can dene, by theimplicit function theorem, a (least) positive timetx > 0 so that for each x M the solution to theChristopher Byrnes, former dean of the School of Engi-neering and Applied Science at Washington Universityin St. Louis, was a distinguished visiting professor in op-timization and systems theory at the Royal Institute ofTechnology in Stockholm when he died unexpectedly inFebruary 2010.dierential equation dened by X with initial con-dition x returns to H. In particular, one can denea smooth Poincar, or rst-return, map P on S,which sends the initial condition x to the solutionof the dierential equation at time tx. Moreover,the dynamics of the iterates of P on S are thenintimately related to the dynamics of X, near ,in positive time. Conversely, if a local sectionS transverse to X exists for which there exists aPoincar map P, the existence of periodic pointsfor P implies the existence of periodic orbits forX, allowing for the use of powerful topologicalxed point and periodic point theorems in thestudy of nonlinear oscillations. The importance ofPoincars method of sections led G. D. Birkhoto develop two sets of necessary and sucientconditions [1] for the existence of a section for adierential equation evolving in Rn. One of thesewas formulated in terms of what Birkho calledan angular variable, and the other involved what,in modern terminology, would be called an angu-lar one-form. Both concepts are reviewed in thisarticle.The existence of a section is, of course, bothone of the standard paradigms for the existenceof nonlinear oscillations and one of the grandtautologies of nonlinear dynamics, since to knowwhether S is section for X is to know a lotabout the long-time behavior of the trajectories ofthe corresponding dierential equationin whichcase one might already know whether there areperiodic orbits. Nonetheless, this paradigm hasactually been used with great success in appli-cations, most notably beginning with Birkhosproof of Poincars Last Theorem, which arosein the restricted three-body problem in celestialmechanics. An easier paradigm is provided by the1080 Notices of the AMS Volume 57, Number 9principle of the torus, which has been widely usedin applications to biology, chemistry, dynamics,engineering, and physics [13]. In this literature, ifDNdenotes the closed unit disc in RN, a subman-ifold M Rnthat is dieomorphic to Dn1 S1is called a toroidal region, and the principle ofthe torus asserts that, if a smooth vector eld Xleaves a toroidal region positively invariant andhas a section S that is dieomorphic to Dn1, thenX has a periodic orbit in M by Brouwers xedpoint theorem. Of course, among the limiting fea-tures of the principle of the torus is the need notonly to nd a section but also to have the abilityto characterize familiar topological spaces such astoroidal regions, disks, and spheres. Fortunately,remarkable advances in dynamics and topologysince Poincars time now allow us to eectivelyaddress both of these technical issues.Among the tools from dynamics that play arole in the results described in this paper arethe properties of nonlinear dynamical systemsthat dissipate energy. This has been developedin two separate schools, one pioneered by Lia-punov and the other beginning with Levinson andsignicantly developed by Hale, Ladyzhenskaya,Sell, and others. One of the main results of thelatter school concerns the existence of Liapunovstable global attractors for dissipative systems.The topological methods described here are alsoglobal, allowing one to bring techniques such asthe classical combination of cobordism and ho-motopy theory, as described in [14], to bear onthe study of nonlinear oscillations. The early useof topological methods in the study of nonlineardynamics dates back to the work of Poincar,Birkho, Lefschetz, Morse, Krasnoselskii, Smale,and many others. The results described here alsorely on global topological methods developed byF. W. Wilson Jr. in his study of the topology ofLiapunov functions for global attractors. For thecase of periodic orbits, Wilsons results form thestarting point for the derivation of necessary con-ditions, derived in [5], for the existence of anasymptotically stable periodic orbit for a smoothvector eld dened on an orientable manifold. Theproof uses a very general cobordism theorem ofBarden, Mazur, and Stallings [10] in dimensionsbigger than 5. In lower dimensions, crucial use isalso made of the solution of the Poincar Con-jecture in dimensions 3 and 4 by Perelman andFreedman and a result of Kirby and Siebenmannon smoothings of 5-manifolds. The remarkablefact that the necessary conditions are sucientfor the existence of a periodic orbit follows froman explicit cobordism argument [5] involving theperiod maps of one-forms, introduced by Abel.In this paper, I give a brief overview of theproofs of these results and then describe howto combine them to derive a new sucient con-dition, which replaces a topological assumptionwith an assumption that the dynamical systemdissipates energy. These sucient conditions areeasier to use in practice and are illustrated inseveral ways, including examples taken from pop-ulation dynamics and feedback control systems.This exposition concludes with an existence the-orem that is valid for a much more general classof smooth manifolds but that requires a more re-strictive hypothesis. Among several applications,this result is illustrated in the case of the exis-tence of periodic orbits for smooth vector elds oncompact 3-manifolds, with or without boundary.In closing, it is a pleasure to acknowledgevaluable advice from Roger Brockett, Tom Farrell,Dave Gilliam, Moe Hirsch, John Morgan, Ron Stern,and Shmuel Weinberger.Stability of Equilibria, Periodic Orbits, andCompact AttractorsIn this section, some basic results about asymp-totic stability of compact sets that are invariantwithrespect toasmoothvector eldX arereviewedand illustrated for a feedback design problem pre-sented in Example 2.2. Except for the last twosections of this survey, I will only need theseresults for vector elds dened on Rn, on thetoroidal cylinder, RnS1, or on the solid torus,DnS1. In this section, I will conne the discussionto the case of vector elds on the toroidal cylinder,which in this section is denoted by M. In Sections3 and 4, vector elds on solid tori are studiedin more detail. It should be noted, however, thatthese results, suitably formulated, do hold forsmooth paracompact manifolds, with or withoutboundary, and with careful modication they alsohold in innite dimensions [9].Any point in M has coordinates (x, ), wherex Rnand S1, and therefore any smoothvector eld X on M has the formX =

f1(x, )f2(x, )where f1 takes values in Rnand f2 takes valuesin R. The vector space of smooth vector elds onM is denoted by Vect(M). In particular, a smoothvector eld denes, and is dened by, an ordinarydierential equation (ODE) x = f1(x, ) (2.1) = f2(x, ) (2.2)to which the local existence, uniqueness, andsmoothness theorem for solutions to ODEs ap-plies, since small variations in an initial conditionz0 = (x0, 0) take place in anopen subset of RnR.(t, z0), dened for suciently small t, will de-note the solution initialized at z0 at time t0 = 0.In this paper, only vector elds for which (t, z0)October 2010 Notices of the AMS 1081is dened, for each z0 M and for all t 0, areconsidered. Any such X denes a semiow(2.3) : [0, ) M M.When t is xed, it is often convenient to use thenotation t(z) := (t, z). In particular, denes asemigroup of smooth embeddings t of M.Anequilibriumfor X is a point z0 M satisfyingX(z0) = 0 or, equivalently, (t, z0) = z0 for allt 0. A solution curve of (2.3) initialized at anonequilibrium point z M is periodic providedt(z) = z for some t > 0. The minimumtime T > 0such that T(z) = z is its period and the set ofpoints in M transcribed by a periodic solution iscalled a periodic orbit. A subset I of M is positivelyinvariant for a vector eld X if (t, z) I for eachz I and every t 0. I is invariant if (t, z) I foreach z I and every t R. Equilibria and periodicorbits are invariant sets. A compact invariantset K is a maximal compact invariant set for thesemigroup (2.3) provided every compact invariantset of (2.3) is contained in K.For any B M, the -limit set of B is dened[9] as(B) = {z B| for zj B and tj +,with j +, (tj, zj) z}.(2.4)For B = {z}, this coincides with the -limit set(z) introduced by Birkho in [1]. The -limitsets, (B) and (z), are dened as in (2.4) withthe sequence of times tj tending to . Following[9], a closed set A M is said to attract a closedset B M provided the distance(2.5) (t(B), A) := supzB infyA d(t(z), y)betweenthe sets t(B) and Atends to 0 as t +,where d is any complete metric on M.Denition 2.1 ([9]). A compact invariant subset Kis said to(1) be stable provided that for every neighbor-hood V of K, there exists a neighborhoodVof K, satisfying t(V) V, for all t 0;(2) attract points locally if there exists a neigh-borhood W of K such that K attracts eachpoint in W;(3) be asymptotically stable if K is stable andattracts points locally.Remark 2.1. If K is a compact invariant set, thenotion of attracting a point or attracting a com-pact set is independent of the choice of metric, asit should be. Moreover, since K is compact, condi-tion (3) is equivalent to the existence of a positivelyinvariant neighborhood K L for which K attractsL [9, Lemma 3.3.1]. The largest open set, D, at-tracted by K is called the domain of attraction ofthe attractor K.Denition 2.2. If a compact subset K M satisesconditions (1) and attracts every point of M, thenK is called a global attractor.Compact attractors exist in many situations inwhich the dynamical system dissipates energy, anotion that can be mathematized in several ways.There are two formulations of dissipativity thatare very useful. In reverse chronological order, onehas its roots in the work by Levinson on the forcedvan der Pol oscillator and is developed in [9] forthe case of Banach spaces. In this exposition, ittakes the following form.Denition 2.3. X Vect(M) is point-dissipativeprovided there exists a compact set K M thatattracts all points in M.Remark 2.2. For any > 0, the -neighborhood,B = B(K), of a global attractor K is a relativelycompact absorbing set; i.e., every trajectory even-tually enters and remains in B. A system is point-dissipative if, and only if, there exists a relativelycompact absorbing set.Point-dissipative systems on Rnare also some-times referred to as being ultimately boundedsystems, and their origin lies in classical nonlinearanalysis.Example 2.1. Consider a Cperiodically time-varying ordinary dierential equation(2.6) x = f (x, t), f (x, t +T) = f (x, t)evolving on Rn. Historically, a central questionconcerning periodic systems is whether thereexists an initial condition (x0, 0) generating aperiodic solution having period T. Such solu-tions are called harmonic solutions. Followingthe pioneering work of Levinson on dissipativeforced systems in the plane, V. A. Pliss formulatedthe following general denition for periodicallytime-varying systems:Denition 2.4 ([16]). The periodic dierentialequation (2.6) is dissipative provided there existsan R > 0 such that(2.7) limtx(t; x0, t0) < R.In particular, the ball B(0, R) of radius R about0 Rnis an absorbing set for the time-varyingsystem (2.7). As noted in [16], the system (2.6)denes a time-invariant vector eld X on thetoroidal cylinder M via x = f (x, ) (2.8) = 1. (2.9)To say that (2.6) is dissipative on Rnis to say that(2.8) is point-dissipative on M. For a dissipativeperiodic system, one can dene a smooth Poincarmap P : RnRndened via(2.10) P(x0) = (T; x0, 0).An important consequence [16] of (2.7) is thatthere exists a closed ball 0 B Rnand an r N1082 Notices of the AMS Volume 57, Number 9such that if x0 B, then x(t; x0, 0) B for t rT;i.e.,(2.11) Pr: B B.ApplyingBrouwers xedpoint theoremtoPr, Pliss[16] showed the existence of a forced oscillation.In fact, Browders xed point theorem asserts thatany mapsatisfying (2.11) must have a xedpoint inB so that there always exists a harmonic solution.The theory of dissipative systems has beenstudied by many mathematicians. Indeed, dissi-pative systems play a central role in the workof Krasnoselskii, Hale, Ladyzhenskaya, Sell, andothers for both nite- and innite-dimensionalsystems. In the present context, the principalresult is the following.Theorem 2.1. If X Vect(M) is point-dissipative,then there exists a compact attractor Afor X on M.Ais the maximal compact attractor and satisesA= {z M : {(t, z) : < t < }is relatively compact}.(2.12)In particular, if B M is relatively compact, then(B) A. Moreover, Ais connected.Remark 2.3. For an ODE evolving on Rn, any tra-jectory with initial condition z such that {(t, z) : < t < } is bounded is called Lagrange stable.Example 2.2. The problem considered in this ex-ample is referred to as a set-point control prob-lem in the systems and control literature and iswidely used in engineering applications in whichcertain physical variables need to be maintainedasymptotically close to a desired constant. Impor-tant examples include controlling the temperatureand air quality in buildings, for which heating, ven-tilation, and air conditioning consume the largestportion of energy costs, and controlling the alti-tude or airspeed of aircraft, a problem of greatimportance for air trac control. In more detail,consider the system(2.13) x1 = gx1+u1, x2 = gx2+u2, x3 = x1u2x2u1x3,which models [2] the control of a rotor by an ACmotor, with (x1, x2) being the components of themagnetic eld, u1, u2 being the current throughthe armature coils, g the resistance in the coils,x3 modeling the angular velocity of the rotor, and the coecient of friction. The control objectivestudied in [2] was to design a Ccontrol law(2.14) u1 = u1(x1, x2, x3, d), u2 = u2(x1, x2, x3, d)so that the system (2.13)(2.14) has the propertythat, in steady-state, limtx3(t) = d, for a de-sired constant rate of rotation d > 0. In the engi-neering literature, a system of this form, in whichexplicit control laws are fed back into a controlsystem, such as (2.13), is called a closed-loop sys-tem, and the control laws are referred to as feed-back laws.A natural starting point is to determine nec-essary conditions on the control laws in order tohave the closed-loop system(2.13)(2.14) be point-dissipative on some positively invariant open setD R3and solve the set-point control problemfor all initial conditions x D. Point-dissipativitywould imply the existence of a global compact at-tractor A D, while solving the set-point controlproblem would consist of ensuring that x3|A =d. Since A is invariant, x3|A = 0, which impliesx1 x2x2 x1 = d or(2.15) = dx21+x22> 0, for xA,where (r, ) denotes polar coordinates in the(x1, x2)-plane. In particular, the magnetic eldmust rotate in steady-state, as it should in orderto generate torque. In fact, for conventional ACmotors, the rotational rate of the magnetic eld ofthe AC motor should be constant, and imposingthe condition = f > 0 yields several additionalconclusions. For example, since any trajectoryon A will be a closed curve in the ane plane,x3 = d, having constant amplitude A =

d/f ,the global compact attractor A must consist ofthis single periodic orbit. If one assumes that thecontrol laws (2.14) are dened on all of R3andthat the rotational rate for the magnetic eld ofthe AC motor is constant for all initial conditionsin D, one is led to the further constraint(2.16) x1u2x2u1 = f (x21+x22)on (2.14), which yieldsu1 = x1f x2+x1h(x1, x2)+H1(x1, x2, x3)(x3d),u2 = x2+f x2+x2h(x1, x2)+H2(x1, x2, x3)(x3d),for some R. Setting h = 0, each of thesecontrol laws produces a closed-loop system hav-ing a periodic orbit with period T = 2/ onx3 = d, evolving as a classical harmonic motion, x1 = f x2, x2 = f x1, on the circle (x21+x22) = d/f .In [2], Brockett shows that the feedback lawu1 = gx1f x2+(d x3)x1(2.17)u2 = gx2+f x1+(d x3)x2(2.18)where f , > 0 solves the set-point control prob-lem, inducing an asymptotically stable periodic or-bit on D = R3 X3, where X3 is the x3-axis.In fact, this system is point-dissipative on D R2S1, with as its compact global attractor. Thisis easiest to see using Liapunov methods, whichare described below.A very powerful way to formulate dissipationof energy near an equilibrium was developed byLiapunov in his 1892 thesis and has since beenextended to uniformly attractive closed invariantOctober 2010 Notices of the AMS 1083sets. The main results for compact invariant setssuce for point-dissipative systems.Denition 2.5. Suppose X leaves an open subsetD M positively invariant and that K D is acompact invariant subset. A Liapunov function Vfor X on the pair (D, K) is a Cfunction V : D Rthat satises(1) V|K = 0 and V(z) > 0 for z K,(2) V < 0 on DK, and(3) V tends to a constant value (possibly )on the boundary, D, of D.Theorem 2.2. Suppose X leaves an open subsetD M positively invariant and that K D is acompact invariant subset. If a function V exists sat-isfying conditions (1) and (3) of Denition 2.5 andif V 0 on D, then K is stable. If V also satisescondition (2), then K is a global compact attractoron D.Example 2.3 (Example 2.2 (bis)). Consider theclosed-loop vector eld X Vect(R3) obtainedby implementing the feedback law (2.17) in thesystem (2.13)(2.19) x1 =f x2+(d x3)x1, x2 =f x1+(d x3)x2, x3 =f (x21+x22) x3.As noted above, (2.19) has a periodic solution with initial condition x(0) = (

d/f , 0, d)T. Fol-lowing [2], consider the function V : D [0, )dened byV(x1, x2, x3) = (d x3)2+f (x21+x22)d ln(x21+x22) +d(ln(d/f ) 1).(2.20)The function V satises conditions (1) and (3) ofDenition 2.5 for K = . Moreover,(2.21) V(x1, x2, x3) = 2(d x3)2 0,along trajectories of X, so that is stable. Since Vis nonincreasing along trajectories, it follows thatfor any x D, (x0) is an invariant subset ofV1(0)a very useful result known as LaSallesinvariance principle. To say V = 0 is to say thatx3 = d and, as shown in Example 2.2, the only in-variant set on which x3 = d is . This proves that is a global attractor on D. A typical sublevel setof V together with a trajectory converging to isdepicted in Figure 1.In fact, F. W. Wilson Jr. has shown that Liapunovfunctions for compact attractors always exist. Inthis setting, his result takes the following form.Theorem 2.3. Suppose that X Vect(M). A nec-essary condition for a compact subset K M to bea global compact attractor on an open positivelyinvariant domain D M is that there exist a Lya-punov function V for X on the pair (D, K).! 1012! 101200.511.5x1x2x3Figure 1. The sublevel set V1[0, 1] V1[0, 1] V1[0, 1] and thetrajectory of XXX for initial condition (1, .75, 1.5). (1, .75, 1.5). (1, .75, 1.5).Remark 2.4. Wilson [19] also studied the topol-ogy of Liapunov functions in the case that K is asmooth submanifold. For example, if K S1, as inthe case of an asymptotically stable periodic orbit,then the domain of attraction D of K always sat-ises D Rn1 S1, in harmony with Examples2.22.3 and Figure 1, for which D = R3 X3 R2S1.Angular Variables and Angular One-FormsThe purpose of this section is to recast Birkhosseminal ideas on the existence of sections forsmooth dynamical systems in modern terms andto delineate the extent to which these ideas areapplicable. For the closed-loop system (2.19), animportant role in the analysis of the nonlinearoscillator was played by the variable , measuringthe rotation of the magnetic eld. More formally,for the Liapunov function V dened in (2.20) andfor a xed choice of c > 0, denote the sublevel setV1[0, c] by M and consider the function(3.1) J : M S1, J(r, , x3) = ,which satises J(x) = dJ(x), X(x) = f > 0. Thesign of J is irrelevant; only the fact that J issign denite is important. The existence of suchan angular variable also arises in the two-bodyproblem with a central force eld, since conserva-tion of angular momentum implies that = c, forc a constant and for (r, ) in an invariant plane ofmotion.The importance of angular variables in thetheory of nonlinear oscillations was elucidatedby G. D. Birkho in his 1927 book on nonlineardynamics. In [1, pp.143145] Birkho derived twosets of necessary and sucient conditions for theexistence of what he called a surface of sectionfor an arbitrary dierential equation evolving inRn. For Birkho, a smoothsectionfor X Vect(Rn)is a hypersurface S M in some region M Rnso that, for each x M, the ow line (trajectory)through x intersects S transversely at a (least)1084 Notices of the AMS Volume 57, Number 9forwardtime tx > 0and, replacing t by t, at a leasttime sx > 0 in reverse time. In this case, he denesthe map (x) = 2sx/(sx+ tx) and notes that increases along every streamline (trajectory ofX) by 2 between successive intersections withS. Abstracting from this construction, he calledany map satisfying ddt((x(t))) > 0 for alltrajectories x(t) M an angular variable for X.Moreover, if is an angular variable, he observesthat S = 1(0) is a surface of section for X on M.Of course, an angular variable is actually amultivalued function, but it can be made single-valued as a map with values in S1. For example,Birkhos construction leads to the map(3.2) J(x) = exp(2isx/(sx+tx)).Birkhos second set of conditions is the existenceof smooth functions ai such thatni=1aiXi > 0 and aixj= ajxi, for i, j = 1, . . . , n,where the Xi are the coordinates of X, abstractingthe observation that J = ni=1JxiXi is actually awell-dened, real-valued function on M.More precisely, suppose M Rnconsists of anopen subset together with a smooth boundary.In the language of one-forms, one may dene=ni=1aidxi and note that Birkhos conditionsassert that is closed, i.e., d = 0, and that thenatural pairing of the one-form and the vectoreld X satises(3.3) , X =ni=1aiXi > 0.It is important to note that (3.3) can be checkedpointwise (in particular, without explicit knowl-edge of trajectories) just as in the applicationsof Liapunov functions. In this spirit, one mayalso formulate the existence of an angular vari-able without reference to the trajectories x(t). IfVect+(M) denotes the set of vector elds thatpoint inward on the boundary of M, then M ispositively invariant under any X Vect+(M).Denition 3.1 ([5]). Suppose X Vect+(M). Wesay that a map J : M S1is an angular variablefor X if it satises(3.4) J = dJ, X > 0everywhere on M. If is a closed one-form on M,then is an angular one-form for X provided that(3.3) holds everywhere on M.For example, if M is the solid torus, Dn1S1Rn, the smooth boundary of M is Sn2S1, whereSrdenotes the r-sphere. In this case, every closedone-formcan be written as =n1i=1 aidxi+and.In fact, there exists c R such that(3.5) = cd +dffor a smooth function f : M R.This is most easily seenusing the basic theory offundamental groups, e.g., the fundamental group1(M) of M satises 1(M) Z with a generatorgiven by a simple closed path transversing{0} S1in either direction. Choose a xed basepoint x0 M and an arbitrary upper limit x Mfor the integral(3.6)

xx0.Of course, this integral may depend on a choiceof path joining x0 to x. Suppose that 1, 2 aretwo such paths and consider the closed path constructed by traversing 1 and then 2 inthe opposite direction. To say that

xx0 is pathindependent is to say that

= 0 for everyclosed path . If c =

, then

( cd) = 0and, since every closed path is homotopic to amultiple of ,

(cd) = 0 for all closed paths . Therefore

xx0( cd) = f (x) is independentof path and = cd +df .If c in (3.5) is an integer, then is calledan integral closed one-form, and every such one-form denes a smooth period map J : M S1in the following manner. As noted above, (3.6) isonly dened up to periods of , i.e., up to theelements of the subgroup() =

: [] 1(M)

= (c) Z R.If c 0, then () is an innite cyclic subgroup,and therefore the period map J of denes asmooth surjection from M to a circle(3.7) J : M S1,dened via(3.8) J(x) =

xx0

R mod (c).Moreover, J satises(3.9) J = dJ, X > 0if and only if is an angular one-form for X.Remark 3.1. If is an angular one-form, then itcan be shown that c 0. Moreover, the normaliza-tion /|c| is an integral angular one-form, wherethe constant in (3.5) is 1. Therefore, an angu-lar variable exists. Conversely, since S1 R/Z,a smooth map J : M S1can be regarded asa multivalued map J : M R where the valueJ(x) is determined only up to an integer constant.Nonetheless, = dJ is well-dened as a one-formon M, since the derivative of a constant is zero.Moreover, if J is an angular variable for X, then, X = dJ, X > 0, so that is an angularone-form. It is convenient to employ a synthesisof these two approaches.October 2010 Notices of the AMS 1085Remark 3.2. The calculations made above, such as(3.5), for the solid torus also hold, without changeof notation, for the toroidal cylinder RnS1.Periodic Orbits on Solid Tori and ToroidalCylindersWhile Birkho was not specic about the ambientsubmanifold M Rnin which a surface of sectionmight exist for X or what that might imply forthe topology of M, his statements clearly excludethe choice M = Rnand specically include sur-faces of section having a boundary. Moreover, itis necessary for his construction that M be aninvariant set. The development of angular vari-ables in [5] includes the case of a smooth manifoldwith boundary that is only required to be posi-tively invariant and allows for a characterizationof M topologically when there exists a locallyasymptotically stable orbit.Theorem 4.1 ([5]). Suppose that n > 1. If is an asymptotically stable periodic orbit ofX Vect(Rn), then there exists a smooth, pos-itively invariant n-dimensional submanifold M Rn, homeomorphic to a solid n-torus,on which X has an angular variable J : M S1.In fact, M is dieomorphic to Dn1 S1, exceptperhaps when n = 4.Remark 4.1. The idea underlying the proof is tobuild a positively invariant solid torus using a Li-apunov function V on the domain of stability Dof , generalizing what was observed about thesystem (2.19). Indeed, for c suciently small,(4.1) Mc = V1[0, c]is a positively invariant compact subset, con-sisting of an open subset and with a smoothboundary. More succinctly, Mc is a compactorientable smooth manifold with boundary. More-over, for c suciently small, Mc clearly admitsan angular variable, since does. The rest ofthe proof uses several key ingredients, startingwith the fact that, according to Remark 2.4, thedomain of attraction, D, for is dieomorphicto Rn1 S1. It can also be shown that the in-clusion Mc D induces an isomorphism ofhomotopy groups. In other words, the inclusionis a homotopy equivalence. On the other hand theprojection p2 : Rn1 S1 S1onto the secondfactor, p2(x, ) = , is also a homotopy equiva-lence, since Rnis contractible. Consequently, Mcis homotopy equivalent to S1.For n = 2, by the classication of surfacesit then follows that Mc A, the standard two-dimensional annulus. For n = 3, the solutionby Perelman of the classical Poincar conjecture[15] implies that, up to dieomorphism, D2 S1is the only 3-dimensional compact, orientablemanifold that is homotopy equivalent to S1. Sim-ilarly, for n = 4, the solution by Freedman ofthe 4-dimensional Poincar conjecture [7] impliesthat, up to homeomorphism, D3 S1is the only4-dimensional compact, orientable manifold thatis homotopy equivalent to S1. In higher dimen-sions, there are innitely many smooth compactorientable manifolds homotopy equivalent to S1,but Mc is special. Adapting some constructions ofWilson [19], one can show that Mc is homotopyequivalent to Sn1S1. This fact, along with somehomotopy theory, allows one to use Freedmansproof of the 4-dimensional Poincar conjecture[7] to show that, for n = 5, Mc is homeomorphic toD4 S1, while a fundamental result due to Kirbyand Siebenmann [11] implies that Mc is dieomor-phic to D4S1. An application of the theorem ofBarden, Mazur, and Stallings [10] completes theproof [5] of Theorem 4.1 for n 6.Remark 4.2. It is unknown how many dieren-tiable structures on D3S1may exist.In fact, the necessary conditions for the exis-tence of an asymptotically stable periodic orbitare also sucient for the existence of a periodicorbit.Theorem 4.2 ([5]). If M Rnis a smooth subman-ifold which is dieomorphic to Dn1S1, then anyX Vect(Rn) leaving M positively invariant andhaving an angular variable J : M S1has a pe-riodic orbit in M. Moreover, the homotopy class ofthis periodic solution generates 1(M).The idea behind the proof is to rst use alevel set S = J1(), for any regular value of an angular variable J, as a section for Xand to next prove that, after modifying J ifnecessary, S Dn1. In particular, one can applyBrouwers xedpoint theoremto the Poincar mapP : S S.Briey, since J is an angular variable, then dJ == cd +df according to (3.5) and, according toRemark 3.1, c 0 . Without loss of generality, onecan assume c = 1 and can embed in the familyof one-forms = d + df , 0 1, whichdenes a homotopy(4.2) J : M [0, 1] S1, J(x, ) =

xx0between the period mappings J0 = J(, 0) andJ1 = J(, 0) = J and therefore a deformation ofJ10 () Dn1into J11 () S. The remainder ofthe proof in [5] uses the fruitful relationship be-tween homotopy and cobordism, as describedin [14], to show that this deformation is adieomorphism.Remark 4.3. In [5], Theorems 4.1 and 4.2 areproven for the more general case in which Rnisreplaced by an arbitrary orientable paracompactmanifold N of dimension n > 1.1086 Notices of the AMS Volume 57, Number 9One corollary of Theorem 4.1 and the proofof Theorem 4.2 is that, except perhaps whenn = 4, the seemingly stringent hypotheses of theprinciple of the torus are actually necessary forthe existence of a locally asymptotically stableperiodic orbit. More importantly, a combinationof the proofs of Theorems 4.1 and 4.2 yieldsan amplication of Theorem 4.2 that combinestopology and dynamics in a form that is easier toapply in practice.Theorem 4.3. Suppose that N Rn S1. IfX Vect(N) is point-dissipative and has an an-gular variable J, then X has a periodic orbit .Moreover, the homotopy class determined by generates 1(N).This result was originally proved [3, Theorem2.1] for the class of dissipative periodic systemsdiscussed in Example 2.1, but the proof extends tothe general case. The key idea is to use a Liapunovfunction V for the global attractor Afor X to builda positively invariant torus for X as in Remark 4.1and then apply Theorem 4.2. Since A is not ingeneral smooth, this argument does require a bitmore work than the proof of Theorem 4.1.Theorem 4.3 generalizes Example 2.1 and givesa new proof of Browders theorem on the ex-istence of harmonic oscillations for dissipativeperiodic systems (2.6) as well. Indeed, any dis-sipative system (2.6) was noted to be equivalentto the point-dissipative system (2.8) evolving onthe toroidal cylinder N = Rn S1. Since (2.8)has as an angular variable, a periodic solution{(t, x0, 0)} N exists. That {(t, x0, 0)} is har-monic follows from the fact that the homotopyclass of (, x0, 0) generates 1(N) Z.As the next example shows, Theorem 4.3also applies directly to the May-Leonard equa-tions, modeling the population dynamics of threecompeting species with immigration [6].Example 4.1. The May-Leonard model for threecompeting species with immigration ( > 0),N1 = N1(1 N1N2N3) + (4.3)N2 = N2(1 N1N2N3) + (4.4)N3 = N3(1 N1N2N3) +, (4.5)where 0 < < 1 < , + > 2, leaves thepositive orthantP+= {(N1, N2, N3) : Ni > 0, i = 1, 2, 3}positively invariant. Let X Vect(P+) denote thevector eld dened by this dierential equation.The function V(N1, N2, N3) = N1 + N2 +N3 is positive on P+and has derivative V =LXV(N1, N2, N3) = N1+N2+N3(N21+N22+N23)(+)(N1N2+N2+N3+N1N3), which is negativefor (N1, N2, N3) suciently large in norm, so thatB(0, R) P+is an absorbing set, for the ball ofsome radius R. Since > 0, the vector eld X|P+points inward, and therefore there is a smaller,relatively compact absorbing set in P+. Hence, byRemark 2.2, X is point-dissipative.Following [6], there is a unique equilibrium((), (), ()) = (1 + (1 + 4)1/2)/2 P+,where = 1 + + . Immigration stabilizesthe population around this equilibrium if >2( 3)/( +3)2, but for(4.6) < 2( 3)/( +3)2the equilibriumis unstable with a one-dimensionalstable manifold Ws(0) = {(N, N, N)}, where Ni =N > 0 for i = 1, 2, 3.In this case, M = P+Ws R2S1is positivelyinvariant. Moreover, the one-form(4.7)= (N1dN2N2dN1)+(N2dN3N3dN2)+(N3dN1N1dN3)N21+N22+N23(N1N2+N2N3+N1N3)is an angular one-form for X on M. Therefore, byTheorem 4.3, there exists a periodic orbit P+whenever (4.6) is satised, as is illustrated in Fig-ure 2.Figure 2. A periodic trajectory for = 1.5 = 1.5 = 1.5, = .75 = .75 = .75.Remark 4.4. The existence of periodic orbits forthe May-Leonard equations when (4.6) is satisedis well known, and some of our calculations wereinspired by the analysis of these equations in [6],although our use of angular one-forms and dissi-pativity is new and more streamlined. Indeed, thetreatment in [6] proves the existence of a periodicsolution by checking some comparatively very re-strictive hypotheses in an existence theorem dueto Grasman [6]. In fact, Grasmans theorem is acorollary of Theorem 4.3.Example 4.2. (Voltage Controlled Oscillators withNonlinear Loop Filters.) A phase-locked loop (PLL)is a basic electronic component used in wirelesscommunication networks for the transmission ofstable periodic signals. A PLL consists of threecomponents: a phase detector (PD), a voltage-controlled oscillator (VCO), and a (low-pass) loopOctober 2010 Notices of the AMS 1087lter (LF), each of which can be described in termsof a mathematical model. For example, in a verysimple, commercially available form, the LF hasthe form y = y +u, where u, y R are the inputand output of a one-dimensional system, the VCOis an integrator, = y and the closed-loop systemproduced by a PD that compares the phase resultsin the feedback control u = asin(), where > a > 0. In this case the region y > a is pos-itively invariant, and using Poincar-Bendixsontheory for the pseudo-polar coordinates (y, ) inthis region shows that the interconnected feed-back system results in a sustained, or self-excited,oscillation in the steady-state response of y(t).In this example, I consider the 3-dimensional,nonlinear LF dened by x1 = 2x1x1ex2+x2(4.8) x2 = x13x2x32+y (4.9) y = u (4.10)with the same VCO and feedback law as above, re-sulting in the interconnected feedback system onM = R3S1dened by x1 = 2x1x1ex2+x2(4.11) x2 = x13x2x32+y (4.12) y = y ++asin() (4.13) = y (4.14)where > a > 0. When = 0, this system is un-coupled, consisting of a two-dimensional globallyasymptotically stable systemon R2and the classicvoltage-controlled oscillator. In fact, using Theo-rem4.3, one can showthat there exists a sustainedoscillation for any 0. First, note that since > a N = {(x1, x2, y, ) : y > 0} is a positivelyinvariant submanifold which is dieomorphic toR3S1. Moreover, is an angular variable for X onN since y > 0. Finally, using the energy function(4.15) V(x1, x2, y) = x21+x22+(y )2,it is straightforward to see that the system ispoint-dissipative on N, provided 0. Therefore,by Theorem 4.3 there exists a periodic orbit, as isillustrated in Figure 3.The Existence of Periodic Orbits for VectorFields on Closed Three ManifoldsIn this and the next section, I will presume famil-iarity with the concept of a smooth manifold. Anexcellent introduction, and invitation, to the sub-ject is the book [14]. As a prelude to investigatinghowgenerally the sucient conditions in Theorem4.2 might hold, consider the case of X Vect(M),where M is a compact orientable 3-manifold. Forexample, an irrational constant vector eld onthe 3-torus, T3, is nowhere vanishing and ape-riodic by Kroneckers theorem on DiophantineFigure 3. A periodic orbit in the case fora = 1, = 2 a = 1, = 2 a = 1, = 2, and = 1 = 1 = 1.approximations. Moreover, it is easy to constructa constant coecient angular one-form for sucha vector eld. Another class of counterexam-ples can be constructed from the Heisenberggroup N = H3(R) of nonsingular upper-triangular3 3 real matrices and its discrete subgroupsk = H3(kZ), where H3(kZ) consists of the uppertriangular Heisenberg matrices with integer en-tries all divisible by k Z with k 1. Explicitly,L. Auslander, Hahn, and L. Markus have shownthat there exist (left invariant) vector elds onN that descend to vector elds on the compact3-manifolds Nk = N/k having Nk as their small-est closed invariant set. In particular, any suchvector eld is aperiodic, and I have constructedexamples which also possess angular one-forms[4]. Actually, these are the only counterexamplesin dimension 3.Theorem5.1 ([4]). Suppose that M is a compact ori-entable 3-manifold without boundary. Every X Vect(M) having an angular variable has a periodicorbit except when M is a nilmanifold, i.e., exceptwhen either(1) M T3, or(2) M N/k.If J is an angular variable for a complete vectoreld on a compact 3-manifold without boundary,then S = J1(0) is a compact surface that is aglobal section for X on M. Since M is orientableand X is transverse to S, S is orientable and canbe shown to be connected [4]. Therefore S is acompact orientable connected surface Sg with gholes, and there is a Poincar map P : Sg Sg.In particular, periodic orbits will exist provided Phas a periodic point, i.e., a xed point of Pkfork Z with k 1.For what follows, I will alsoneedtoassume somefamiliarity with algebraic topology, particularlyhomology or cohomology and the notion of theEuler characteristic of a space. For example, the1088 Notices of the AMS Volume 57, Number 9surfaces Sg have Euler characteristic (Sg) = 2 2g. At about the same time that Birkhopublished[1], S. Lefschetz publisheda remarkable xedpointtheorem that vastly generalized Brouwers xedpoint theorem. As a special case of the generaltheorem, if f : M M is a continuous mapon a smooth compact manifold, with or withoutboundary, Lefschetz introduced an integer (f ),which can be computed in terms of f and thehomology or cohomology vector spaces of M, andfor which (f ) 0 implies that f has a xedpoint. In 1953, one of Lefschetzs students, F. B.Fuller, extended this result to a neat theorem thatimplies that if P : N N is a homeomorphism ona compact manifold N, with or without boundary,then(5.1)(N) 0 (Pk) 0, for some k = 1, 2, . . .and hence P has a periodic orbit. In particular, ifS Sg for g 1, then the Poincar map always hasa periodic point ,and therefore X has a periodicorbit. Incaseg = 1, thesectionSis a2-torus andtheremainder of the proof of Theorem 5.1 consistsof checking when (f ) 0 by hand and what(f ) = 0 means geometrically, following [18]. Theremarkable fact is the role played by nilmanifolds,which can also be expressed algebraically in termsof fundamental groups.The Existence of Nonlinear Oscillations onn-Manifolds With or Without BoundaryIn the decade following Fullers publication of histheoremon periodic points, there were substantialapplications of algebraic topology to the studyof the existence of periodic orbits for dynamicalsystems having an angular variable. If a vector eldX generates a solution backward and forward forall time, then the solutions of the correspondingdierential equation dene a mapping(6.1) : RM M,which is said to be a ow. The case of ows is moretractable than semiows and was developed quitegenerally by S. Schwartzman [17]. An importantspecial case of his results is the following.Theorem 6.1 ([17]). Let M be a compact man-ifold, with or without boundary, and supposeX Vect(M) denes a ow on M. The followingconditions on a closed submanifold S M areequivalent:(1) S is a cross section.(2) The smooth map : R S M denes acovering space with an innite cyclic groupof covering transformations.(3) The map : RS M is a surjective localdieomorphism.(4) There exists a smooth angular variable J :M S1.In particular, if the covering space has a non-vanishing Euler characteristic, then X has a peri-odic orbit. On the other hand, for systems thatdissipate energy, the objects of interest are of-ten asymptotically stable invariant sets, positivelyinvariant submanifolds, and semiows. In this di-rection, Fuller [8] also used the method of angularvariables in the more dicult case of a semiowina general setting that includes the case of a com-pact manifold. In this case, following [8], a smoothconnected hypersurface S is said to be a positivecross section for X Vect+(M) provided S is alocal section for X everywhere in S and, for eachx M, there is a time tx > 0 such that tx(x) S.Among the additional topological challenges inthe fundamental work of Fuller on the existence ofnonlinear oscillations for such dissipative systemsis that, while the Poincar map is an embedding, itis typically not a (surjective) dieomorphism, andhis theorem on periodic points does not apply.Nonetheless, Fuller was able to prove the existenceof periodic orbits in several interesting situations.Fortunately, a renement of the notion of an-gular one-forms provides a general approach tosurmounting this technical diculty.Denition 6.1. When M , is said to bea nonsingular angular one-form provided it is anangular one-form for X and |M is nonsingular.By Sards theoremfor manifolds with boundary[14] and the compactness of M, it follows thatthe period map (3.7) of a nonsingular angularform is a ber bundle, since both J and J|Mare submersions. Moreover, using the existenceof a nonsingular angular one-form, one can showthat P is homotopic to a dieomorphism ofS, and therefore P is an automorphism of theintegral homology ring H(S) of S. This key factenables us to use a corollary of a result of Halpern,generalizing Fullers periodic point theorem.Theorem 6.2. Suppose that M is a compact mani-fold with boundary for which (M) 0. Any con-tinuous map f : M M inducing an automorphismf on H(M) satises (fk) 0, for some k 1. Inparticular, f has a periodic point.We summarize these results concerning theexistence of periodic orbits as follows.Theorem 6.3 ([4]). Suppose that M is a smooth,compact connected orientable manifold, with orwithout boundary, and suppose X Vect+(M) hasa nonsingular angular one-form. There exists asmooth compact, connected and oriented subman-ifold S M having codimension one and boundaryS = S M such that(1) S is a global positive section for X, and(2) P : H(S) H(S) is an automorphism.Consequently, if (S) 0, then X has a periodicorbit.October 2010 Notices of the AMS 1089Remark 6.1. All compact submanifolds S satisfy-ing conditions (1)(2) have canonically isomorphicintegral homology rings, so that (S) is intrinsi-cally dened. There are counterexamples due toFuller for n 4 that show the inequality in (3.3)must be strict.Denoting as before the annulus in two dimen-sions by A, the hollow torus, M = AS1, is also theproduct of the torus T2and an interval and there-fore admits nowhere vanishing aperiodic vectorelds, some of which have a nonsingular angularone-form. This is the only source of counterexam-ples to the existence of periodic orbits for vectorelds on a 3-manifold with boundary having anonsingular angular one-form.Theorem 6.4 ([4]). Suppose M is a three-dimensional manifold with boundary. EveryX Vect+(M) that has an angular one-form has a periodic solution whose homotopy classhas innite order in 1(M), except when M isdieomorphic to a hollow torus AS1.There are several other corollaries of Theorem6.3. For example, [4] contains a general resultfor closed 5-manifolds that implies that periodicorbits exist for vector elds with an angular one-form on any closed 5-manifold with 1(M) Z.A similar result is proven in [4] for vector eldsdened on a compact manifold M with boundarythat is homotopy equivalent to S1.ConclusionsNonlinear oscillations are fascinating and impor-tant but hard to rigorously detect or predict. SincePoincars time, the best known and most accessi-ble methods in applied mathematics and relatedelds rely onsmall parameter analysis andprovidelocal existence criteria for periodic motions havingsuciently small amplitudes. On the other hand,since the period of nonlinear oscillations is gener-ally not knowna fortiori, the existence of nonlinearoscillations is a global phenomenon, and thereforeany comprehensive theory would necessarily beglobal in nature. This article continues in the tra-dition of Poincar, Birkho, and others in studyingcross sections for vector elds, creating a globalapproach to developing criteria for the existenceof periodic orbits using methods drawn from theglobal theory of nonlinear dynamical systems thatdissipate some mathematical form of energy andmethods drawn from algebraic and dierentialtopology, particularly the fruitful combination ofcobordism and homotopy theory.One of the major points of departure for thisapproach is the ability to include motions ofa dynamical system that leave a manifold withboundary positively invariant, rather than invari-ant both forward and backward in time. Thisenables one to discuss and characterize whatmust occur topologically when a locally asymp-totically stable periodic orbit exists, a necessarycondition that itself proves to be sucient for theexistence of periodic orbits. Using the languageand methods of dissipative systems formulatedby Hale, Ladyzhenskaya, and others, this sucientcondition is reformulated into a global sucientcondition that is fairly easy to apply in severalspecic examples. The article concludes with aformulation of stricter sucient conditions forthe existence of periodic orbits for vector eldsdened, however, on general compact manifolds,with or without boundary, that ber over a circle.Myowninterest inthis subject is the existence ofasymptotically stable periodic motions in nonlin-ear feedback systems, both manmade and natural,and possible future directions of research shouldinclude the incorporation of stability criteria, in-cluding classical tools such as Hopf bifurcationsand describing function methods and the intrigu-ing possibility of extending this work to the caseof invariant tori.References[1] G. D. Birkhoff, Dynamical systems, Amer. Math.Soc. Coll. Pub. 9 (revised ed.), Providence, 1966.[2] R. W. Brockett, Pattern generation and the con-trol of nonlinear systems, IEEE Trans. on AutomaticControl 48 (2003), 16991712.[3] C. I. Byrnes, On the topology of Liapunovfunctions for dissipative periodic processes, Emer-gent Problems in Nonlinear Systems and Control,Springer-Verlag, to appear.[4] , On the existence of periodic orbits for vectorelds on compact manifolds, submitted to Trans. ofthe Amer. Math. Soc.[5] C. I. Byrnes and R. W. Brockett, Nonlinear os-cillations and vector elds paired with a closedone-form, submitted to J. of Dierential Equations.[6] M. Farkas, Periodic motions, Springer-Verlag,Berlin, 1994.[7] M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton University Press, Princeton,1990.[8] F. B. Fuller, On the surface of section and periodictrajectories. Amer. J. of Math. 87 (1965), 473480.[9] J. K. Hale, Asymptotic behavior of dissipativesystems, AMS Series: Surv Series 25, 1988.[10] M. Kervaire, Le thorm de Barden-Mazur-Stallings, Comment. Math. Helv. 40 (1965),3142.[11] R. C. Kirby and L. C. Siebenmann, FoundationalEssays on Topological Manifolds, Smoothings, andTriangulations, Annals of Math. Studies, AM-88. (re-vised ed.), Princeton University Press, Princeton,1977.[12] O. Ladyzhenskaya, Attractors for semigroups andevolution equations, Lezioni Lincee, CambridgeUniversity Press, 1991.[13] B. Li, Periodic orbits of autonomous ordinarydierential equations: Theory and applications,Nonlinear Analysis TMA 5 (1981), 931958.1090 Notices of the AMS Volume 57, Number 9[14] J. W. Milnor, Topology from a dierentiable view-point, University Press of Virginia, Charlottesville,Virginia, 1965.[15] J. W. Morgan and G. Tian, Ricci ow andthe Poincar conjecture, Amer. Math. Soc, ClayMathematics Monographs, Vol. 3, 2007.[16] V. A. Pliss, Nonlocal problems in the theory of non-linear oscillations, Academic Press, New York andLondon, 1966.[17] S. Schwartzman, Global cross sections of compactdynamical systems, Proc. N.A.S. 48 (1962), 786791.[18] W. P. Thurston, Three-dimensional geometry andtopology, Vol. 1, Princeton U. Press, 1997.[19] F. W. Wilson Jr., The structure of the level sets ofa Lyapunov function, J. of Di. Eqns. 3 (1967), 323329.AMERI CAN MATHEMATI CAL SOCI ETYGo to www.ams.org/bookstore-email to sign up for email notications of upcoming AMS publication catalogs. PUBL I C ATI ONSOF THE AMERICAN MATHEMATICAL SOCIETYwww.ams.org/bookstore/newpubcatalogsView theSummer 2010 Publications Update Catalog Online!The Art of MathematicsA K PETERS508.651.0887www.akpeters.comSave 25% at www.akpeters.comdiscount code: AMSA whole book full of amazingly attractive new modular pieces. This book is highly recommended to all modular folders and those wanting to dabble in this pastime.DAVID PETTY,British Origami SocietyThe comprehensive nature of this book makes it a wonderful selection for both beginners and experts alike.RACHEL KATZ,OrigamiwithRachelKatz.comOrigami Polyhedra Design is a breakthrough collection of original designs ... In this book, Montroll, an origami master who pioneered the folding of origami animals from a single sheet of paper, presents methods for folding various polyhedra from a single sheet of paper.THOMAS HAGEDORN, MAA ReviewsOrigami InspirationsMeenakshi MukerjiOrigami Polyhedra DesignJohn Montroll$24.95; Paperback$29.95; PaperbackIn this gorgeous book, Eric Gjerde has presented clear, easy-to-follow instructions that introduce the reader to the incredible beauty and diversity of origami tessellations.ROBERT LANG, author ofOrigami Design SecretsOrigami TessellationsAwe-Inspiring Geometric DesignsEric Gjerde$24.95; PaperbackOctober 2010 Notices of the AMS 1091 QUANTITATIVE FINANCE RENAISSANCE TECHNOLOGIES, a quantitatively based financial management firm, has openings for research and programming positions at its Long Island, NY research center. 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Send a copy of your resume to: careers@rentec.com No telephone inquiries. An equal opportunity employer. e gpbh lli Wh Wi i i h be bet h t h malle malle e sm sm at is t at is t Wh Wh WWest est sit ive sit ivehe ee in e in at can at cannt e nt eeger t ger tt ha t haos os t po t poger (ph i wbi ( i ifeger egeras as sum sum t he he t t en a t t en a writ writ wwm o m o wo cu wo cuttub ubve int e ve int ebes es (posi (posiit iv t ivww of t w of t w)ys? diff db ) i b c?n t n t diffe diffe wo d wo dbes) in bes) in ub ub cu cuere ere ways wayst w t ws? s? ww ent ent% 0%% 0%% 0%A. A AAA. A. AA 3 513 3 51333 513 513B. B BBB. B. BB 27 112 27 11227 27 112 112CCCCCCCCCC 66 76 114776 76 76 76 147 147 147 1470%%0%%qgghfllft hWhhihifiiooh oh oe folloe follof t heft heWhWhhichhichowowis an is anofofeqeqg ig iwingwing22221yx

2221xyy1222AAA. AAAAAAAA.A.BBB. BBBBBBBB.B.76 776 6D. D DDD. D. DD29172 2917229 29172172High School Students Compete for $10,000www.ams.org/wwtbamI am sure I can speak for everyone involved when I say that this was the best math competition we have ever been a part of. Many thanks to all of you for that wonderful happening.Come see the national Who Wants to Be a Mathematician at the 2011 Joint Mathematics Meetings in New Orleans. High school students from across the U.S. will compete for a top prize of $10,000.eenat)ys? diffd)it?tttn tntdiffediffewo dwodeeeeeessss) ineeeeeesss) inereerewayswayst w t ws? s?wwent entA. A AAA. A. AA3513 351333513513B. B BBB. B. BB27112 2711227 27112112C. C. C. C. CC CCCCCCCCCCCCCC. C. . CCCC. C. . CCCCCCCCCCCC666666 76147 776 614 14 111111177776 76 76 76 76 76 76 76 76 76 76 76 76 76 7614 14 14 14 14 14 14 14 147777777714 14 14 14 14 14 14 14 14 14777777Tilings, Scaling Functions,and a Markov ProcessRichard F. GundyIntroductionWe discuss a class of Markov processes thatoccur, somewhat unexpectedly, in the constructionof wavelet bases obtained from multiresolutionanalyses (MRA). The processes in question havebeen around for a long time. One of the rstreferences that should be cited is a paper byDoeblin and Fortt [10] (1937), entitled Sur leschanes liaisons compltes. In English, they aresometimes called historical Markov processesand have been used extensively to study the Isingmodel. However, their wavelet connection does notseem to have ltered into the standard texts ontime-scale analysis.The material for this article is drawn from thepublications [9], [12], [13], as well as the priorcontributions by various people who are cited inthe appropriate places. To keep the expositionself-contained and as elementary as possible, wediscuss the special case of the quincunx matrixin which the proofs are somewhat simpler and, insome cases, radically dierent from those foundin the above references.In the rst section, we describe a remarkablecoincidence: two discoveries, the rst concerninga gambling strategy and the second concerning awavelet basis, both leading to the same mathemat-ics. It is remarkable that these discoveries, each offundamental importance, were separated by 330years! The wavelet discovery was a particular classof trigonometric polynomials within a class offunctions called quadrature mirror lters (QMFfunctions, for short). These functions turn out to beprobabilities; hence their connection to gambling.Richard F. Gundy is professor of statistics and mathemat-ics at Rutgers University. His email address is gundy@rci.rutgers.edu.The names associated with the gambling strategyare Pascal and Fermat; the wavelet discovery towhich we refer is due to Ingrid Daubechies.In the subsequent sections, we are concernedwith the class of 1-periodic functions p(), quad-rature mirror lters, that generate an MRA on Ror R2. Some do so, but most do not; our problemis to nd out which is which. The probabilitymethods presented here provide new informationon this topic. We show that there is a one-to-onecorrespondence between two disparate classes ofscaling functions, one dened on R, the otherdened on R2. Second, the probability perspectiveallows us to exhibit a large class of continuousp() that generate MRAs. When the function p()is smooth and generates an MRA, it must satisfyknown necessary conditions. However, these nec-essary conditions may be violated in the extremefor MRAs in which the generator p() is smoothexcept at a few points. Few here means as few asfour.In one dimension, all of the MRAs we considerinvolve the dilation Z 2Z; in two dimensions, aninteresting special case is the quincunx dilation,described below. In the rst case, where R1,the QMF function p() is periodic (with periodone); in the second case, when R1, p() isdoubly periodic on the unit square. From this,one might assume that the natural fundamentaldomains for these functions would be (0, 1), Zor (0, 1) (0, 1), Z2. But the natural assumptionis too nave. In one dimension, the appropriatedomain is sometimes a disconnected set calleda C-tile, described below. In two dimensions, thefundamental fun