Working at the frontiers of knowledge

Working at the University of Groningen (RUG) means working at the frontiers of knowledge. The RUG offers researchers and students the opportunity to expand those frontiers, to develop their talents and together with others realise top quali­ ty achievements. The RUG therefore deliberately opts for an interdisciplinary approach to knowledge. Discoveries are born and innovation realised at the interfaces between the various scientific fields. Researchers and students in Groningen bene­ fit from the rich assortment of disciplines that the RUG has built up since its founding nearly four hundred years ago. Knowledge is universal and transcends boundaries. The RUG has deliberately chosen to make a priority of intensive and worldwide cooperation with other leading universities and organisations. As well as being outward looking, the RUG is also closely involved with its own region where it is one of the largest employers. The universityis 22,500 students and 6,000 staff have a distinct academic identity, firmly rooted in the wider social context. They are a vital element of the vibrant city of Groningen - a great place for students and staff.

Faculty of Mathematics and Natural - teaching experience is recommended Sciences - good organizational skills.

The tenure trackers are expected to develop The closing date for appli cations is Two Tenure-Track their own line of research within a particular May 15, 2006. Positions for Assistant field. The appointment will be on a temporary Please send your written appli cation Professor for Dynamical basis for a maximum of 6 years. On completion with a description of current research of 5 years of employment there will be an interest and a curriculum vitae Systems and Applied assessment of performance based on including a list of publications to: Analysis established criteria. If the outcome of the The University of Groningen VACANCY NUMBER 206006-07 Personnel & Organisation Department assessment is positive, the assistant professor will be promoted to the rank of associate P.O. Box 72, 9700 AB Groningen The vacant positions are in the research professor with tenure. The Netherlands units Dynamical Systems and Applied Analysis, The policy is directed at increasing the Please state the vacancy number on the which participate in the NWO-cluster number of women in academic staff positions. envelope and at the top of your letter. 'Nonlinear Dynamics of Natural Systems'.

Additional information about vacancies The University of Groningen can offer The positions at the RUG is available on the university you a salary dependent on qualifications and - performing research in mathematics, in parti­ web site: work experience from € 4049 gross per month cular dynamical systems or applied analysis up to a maximum of € 4605 (scale 11 /12) for - teaching mathematics, including under­ a full-time job. { www.rug.nl ) graduate and service teaching, and the organization there of Information can be obtained from The RUG provides a special career - participating in supervision of MSc and PhD prof.dr. H.W. Broer, Dynamical Systems, advisory service for partners of new projects. phone +31 50 3633959/ 3939, staff who move to Groningen. e-mail < [email protected] > or Personal profile prof.dr. A.). van der Schaft, Applied Analysis, The RUG is an equal opportunities - PhD in the field of Dynamical Systems or phone +31 50 3633731/3939, employer. Because women are still Applied Analysis or related disciplines e-mail < [email protected] > underrepresented in a number of fields, - an excellent list of publications (at least 5 they are particularly encouraged to publications in international peer-reviewed apply. journals) - at least two years of international experience as a postdoc (industrial experience can partly compensate this requirement) - experience with grant proposals is recommended

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Communications A Tribute to 410 A Photographic Look at the KurtGodel Joint Meetings, San Antonio 2006 April2006 marks the centennial of the birth 456 WHAT IS ... aSyzygy? of Kurt Godel. The Notices marks this occasion Roger Wiegand with a collection of articles about Go del, his 458 Harvey Mudd M work, and its impact on mathematics. athematics Department Garners AMS · Award Allyn jackson 412 A Tribute in Photographs 462 Godel, Inconsistency, 414 The Incompleteness Theorem Provability, and Truth: An Exchange of Letters Martin Davis 464 2006 Steele Prizes 419 How Godel Transformed Set Theory 472 2006 Cole Prize inAlgebr

455 2006 Events Celebrating the Godel Centenary Notices Departments Mathematics People ...... 483 Tomczak-)aegermann Awarded CRM-Fields-PIMS Prize, AWM EDITOR: Andy Magid Essay Contest Winners Announced. ASSOCIATE EDITORS: Susanne C. Brenner, Bill Casselman (Graphics Editor), Mathematics Opportunities ...... 484 Robert]. Daverman, Nathaniel Dean, Rick Durrett, NSF Program in Informal Science Education; Call for Proposals Susan Friedlander, Robion Kirby, Steven G. Krantz, Elliott H. Ueb, Mark Saul, Karen E. Smith, Audrey for NSF Program in Mathematical, Social, and Behavioral Terras, Usa Traynor Sciences; AP Calculus Readers Sought. SENIOR WRITER and DEPUTY EDITOR: Allyn Jackson For Your Information ...... 485 MANAGING EDITOR: Sandra Frost Oberwolfach Photo Collection Now Available Online, Correction: CONTRIBUTING WRITER: Elaine Kehoe NSF Graduate Fellowships. PRODUCTION ASSISTANT: Muriel Toupin Inside the AMS ...... _...... _. 486 PRODUCTION: Kyle Antonevich, Stephen Moye, AMS Presidents: A Timeline, Early Erin Murphy, Lori Nero, Arlene O'Sean, Karen News about Notices Website, Ouellette, Donna Salter, Deborah Smith, Peter Sykes Career Profile Network. ADVERTISING SALES: Anne Newcomb Reference and Book list ...... ___...... 487 SUBSCRIPTION INFORMATION: Subscription prices for Volume 53 (2006) are US$430 list; US$344 institu­ Mathematics Calendar ...... 494 tional member; US$258 individual member. (The sub­ scription price for members is included in the annual New Publications Offered by the AMS ...... 499 dues.) A late charge of 10% of the subscription price will be imposed upon orders received from nonmem­ ...... 508 bers after January 1 of the subscription year. Add for Classified Advertisements ...... postage: Surface delivery outside the United States and India-US$20; in India- US$40; expedited delivery Meetings & Conferences of the AMS ...... 509 to destinations in North Arnerica- US$35; elsewhere­ US$87. Subscriptions and orders for AMS publications Meetings and Conferences Table of Contents ...... 519 should be addressed to the American Mathematical Society, P.O. Box 845904, Boston, MA 02284-5904 USA. All orders must be prepaid. ADVERTISING: Notices publishes situations wanted and classified advertising, and display advertising for publishers and academic or scientific organizations. Advertising material or questions may be faxed to 401-331-3842 (indicate "Notices advertising" on fax cover sheet). SUBMISSIONS: Articles and letters may be sent to the editor by email at noti ces@math. ou . edu, by fax at 405-325-5765, or by postal mail at Department of Mathematics, 601 Elm, PHSC 423, University of Okla­ homa, Norman, OK 73019-0001. Email is preferred. Correspondence with the managing editor may be sent to noti ces@ams. erg. For more information, see the section "Reference and Book List". NOTICES ON THE AMS WEBSITE: Most of this publi­ cation is available electronically through the AMS web­ site, the Society's resource for delivering electronic products and services. Use the URL http: I / www . ams. l org/ noti ces/ to access the Notices on the website. Call for Nominations for the Levi L. Conant,. Ruth Lyttle Satter, Oswald Veblen, and Norber t Wiener Prize's . . : ...... 491 [Notices of the American Mathematical Society (ISSN 0002- 9920) is published monthly except bimonthly in june/July by Call for Nominations for E. H. Moore Research the American Mathematical Society at 201 Charles Street, Prov­ idence, RI 02904-2294 USA, GSTNo. 12189 2046 RT****. Pe­ Article Prize ...... _...... 492 riodicals postage paid at Providence, Rl, and additional mail­ ' ing offices. POSTMASTER: Send address change J:\Otices to Frank and Notices ofth e American Mathematical Society, P.O. Box 6248, Call for Nominations for the 2006 AMS-MAA-SIAM Providence, RI 02940-6248 USA.] Publication here of the So­ Brennie Morgan Prize ...... 1. . ..493 ciety's street address and the other information in brackets above is a technical requirement of the U.S. Postal Service. Tel: 401-455-4000, email: noti ces@ams. org. ©Copyright 2006 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The paper used in this journal is acid-free and falls within the guidelines · established to ensure permanence and durability. Opinions 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. ' Letter from the Editor

selection process which, in the case of the editorial boards, Editor's Log, AMSY will be by Council action upon nomination by chief edi­ tors or chief editors-elect. The Notices Editorial Board plays a crucial role in this publication, and a very differ­ 2005 ent one from those played by the editorial boards of the As I write this in mid-January calendar year (CY) 2006, we Society's research journals. Members of the Notices board are 25% (or 50%) into federal (or most states) fiscal year suggest topics for articles, recruit authors for articles, (FY) 2006 and one-half (semester system) or one-third provide expert advice about received articles in their math­ (quarters) into academic year (AY) 2005-06. For the offi­ ematical specialties, and, perhaps most importantly, pro­ cers and committee members of the American Mathe­ vide nonexpert advice as well, about articles on subjects matical Society, however, it's still AMSY 2005. More pre­ in which they are not specialists. Obviously broad math­ cisely, terms of service for the Society run February 1 ematical learning helps in this, but also required is a sense through January 31, so that old terms end and new ones of what the general Notices reader needs from an article. begin after the session of the AMS Council which takes As with most mathematical editorial advice, recommen­ place during the January Annual Meeting. At its meeting dations from the editorial board to authors is usually this January, in addition to committee appointments and anonymous. But the Notices couldn't function without it. other business, which will be reported on in due course I'm glad the Council has recognized the special character in the "From the AMS Secretary" section, the Council took of the Notices Editorial Board. three actions affecting the Notices. I want to reflect on them. The third action the Council took affecting the Notices The one which will affect readers of the Notices most was to reappoint me as chief editor for a second three-year is the decision to remove the requirement, implemented term. The final year of that term will include the fortieth last spring, that users of online access to the Notices must anniversary of my Ph.D. Someone in a symmetric position log on to the AMS website. The login procedure was partly to me would have received a doctorate in 1929. I would designed to make it more difficult to do scripted automatic have regarded that as ancient history even then, as no doubt downloads of Notices pages. They take place, or took place, current students feel about 1969 now. Nonetheless, I hope in such large numbers that it was impossible to tell from the Notices can improve in providing more articles serv­ Web use statistics how many individuals were really look­ ing graduate students. We have some special features ing at Notices articles online. For example, pre-login, pop­ planned which we hope will address professional needs ular articles like Gerd Faltings's 1995 article about the proof of students, as well as the mathematical needs which we of Fermat's Last Theorem might get 20,000 hits annually; hope are already being addressed. last year it got 3,000, and probably many of those were Notices editors' terms, by the way, follow calendar years, pre-login January and February accesses. Post-login pop­ not AMS years. This is further complicated by the fact that ular articles like Fokas and Sung's November 2005 article Notices cover dates, which are usually two weeks later on generalized Fourier transforms, or Allyn Jackson's than publication date, are ten to twelve weeks after the con­ June/July 2005 interview with Martin Gardner, got about tents go into production. So Notices editor's years actually 1,000 hits in CY 2005. Of course these statistics do not in­ begin a third of a year before AMS years. I would like to clude the number of potential individual users who at­ end these light-hearted calendrical comments with a more tempted to access the articles and were confused or put serious one. Unlike the terms of volunteers, Math Reviews off by the login requirement. (Registration was free, sim­ reviewers points expire on December 31 of the relevant ple, and required only an email address.) The statistics, year. These points, which can be applied to AMS dues or however, were not worth this annoyance factor, although AMS publications purchases, are earned by writing re­ it seems only to have affected a small number of users, views, and reviewers are always needed. Mathematicians and the Council, on the recommendation of the Commit­ interested in reviewing for Math Reviews, both to earn tee on Publications, has eliminated the login requirement points and to serve the community, are invited to inquire to access Notices online. So: to the webcrawlers and down­ at rna th rev@ams . o rg for more information. load scripts reading this online, welcome back. Another purpose of the login was in part to remind No­ -Andy Magid tices readers that the Notices is provided online to the en­ tire mathematical community by the members of the AMS, as is still the case post login. The second action taken by the Council affecting the No­ tices was a revision of the procedures for selecting the chief editor and the editorial board of the Notices (and the Bul­ letin as well). On the recommendation of the Long Range Planning Committee, the Council adopted a revised

APRIL 2006 NOTICES OF THE AMS 405 letters to the Editor Medicine in 1958. His molecular stud­ groups.) Perhaps. But I forsee two ies led to a paper: "Hamilton circuits other categories of math overwhelm­ of convex trivalent polyhedra (up to ing the first two, category 3 "conjec­ Mathematics and Nobel Prizes 18 vertices)", Amer. Math. Monthly 74 tures supported by experimental ev­ Professor Saari mentions the Nobel (1967), 522-527. idence" and category 4 "conditional Prize in Chemistry won by John Pople Finally, the work of Cormack and math", meaning theorems proved (Notices January 2006, page 46) and Hounsfield in developing computer­ under the assumption of a category says that Pople's research centered aided tomography(= CAT scanning), 3 conjecture. An example of category on approximating the solution of the which received the prize in Physiology 3 is the Riemann Hypothesis, and cat­ Navier-Stokes equation. But Pople or Medicine in 1979, was largely the egory 4 is represented by the litera­ worked on approximating the solution numerical inversion of the Radon ture of theorems proven under the of the Schrodinger equation, which is transform. assumption of RH. I expect that as the fundamental PDE in quantum As Saari notes, a great many of the experimental mathematics grows and chemistry, rather than the Navier­ prize winners used substantial math­ prospers, more and more interesting Stokes equation. Professor Saari also ematics in their prize-winning work conjectures will emerge, and mathe­ says that "half of all Economics [Nobel and other readers may suggest further maticians will be impatient to explore Prize] winners and several more from examples. their consequences. Chemistry and Physics ...used a sig­ -David Singmaster These are changes in the practice nificant amount of fairly sophisti­ South Bank University London of mathematics that I view favorably, cated mathematics." But the richest [email protected] no doubt because I am actively in­ source of Nobel Prizes related to (Received January 6, 2006) volved in developing them. But I can mathematics is surely Physics: Ein­ appreciate that some mathematicians stein, Schrodinger, Dirac, Heisenberg, In your January issue Donald Saari may regard them as somewhat alarm­ Glashow, Weinberg, Salam, Feynman, lists some of the mathematicians who ing. Don't they threaten to undermine Schwinger, Gell-Mann, Gross, Politzer, have won a Nobel Prize. Included in the standards of rigor that the math­ Wilczek and many more. Moreover, his list was John Pople whose Ph.D. ematical community has labored for Physics is dripping with mathematics was alleged to have been on PDE. That centuries to enshrine? Perhaps. But I (Lie algebras, PDEs, cohomology, su­ may be true but Pople, whom I knew would rather view it as a clash be­ persymmetry, Calabi-Yau manifolds) at Cambridge in the fifties, was a the­ tween two great impulses: the quest which is more sophisticated and more oretical chemist and the title of his for knowledge, and the quest for cer­ interesting than anything used in Eco­ Ph.D. was remarkably succint, just tainty. As long as these impulses pull nomics. the one word "Water". Another Cam­ together, everyone can be happy. But -Frederick Daum bridge mathematician who went on to when they pull in opposite directions, Raytheon Company win the Nobel Prize in Economics was as has happened before (think of the [email protected] my former student James Mirrlees, development of calculus), I believe we will be happier knowing more, with (Received December 20, 2005) and we should not forget the only mathematician to win the Nobel Prize some uncertainty, rather than know­ ing less, with more certainty. The note by Donald Saari in the Jan­ for Literature-Bertrand Russell. uary 2006 Notices can be expanded a - Michael Atiyah - RobertS. Strichartz bit by including Nobel Laureates who University of Edinburgh Cornell University have also done mathematics. [email protected] [email protected] .edu The most notable of these is Sir (Received January 10, 2006) (Received November 29, 2005) Ronald Ross (1857-1932), Nobel Prize in Physiology or Medicine in1902 for demonstrating that malaria is trans­ The Clash of Knowledge and Status ofthe Classification mitted by mosquitoes. However, he Certainty Proof found less support than he expected Brian Davies forsees the future of We do not challenge the assertion by and spent much of his later life doing mathematics when category 1 math, Brian Davies [Notices, December 2005, mathematics which he had done in­ "theorems proved in the usual sense" "Whither Mathematics?"] that human termittently since his youth. Most of is eclipsed by category 2 math "theo­ beings (even group theorists!) are fal­ his work was unpublished or pub­ rems perhaps proved" ("Whither lible, as are computers. However, the lished either by himself or in a jour­ Mathematics?" December 2005, No­ informal language of his discussion of nal he edited. His principal interest tices). The "perhaps" refers to proofs the classification of the finite simple was in iteration, both numerical iter­ that are too long for any one individ­ groups (CFSG), and his occasional mis­ ation for solving equations and func­ ual to understand, and/or that rely on statements of fact, could mislead tional iteration. He also wrote some lengthy computer programs to check readers into inaccurate conclusions. papers on epidemiology. computations that have no intuitive Here are some examples. Another is Joshua Lederberg, who explanation. (One example he cites is First, Davies's treatment blurs the received the prize in Physiology or the classification of finite simple distinction between two different

406 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Letters to the Editor works: the original proof of CFSG and Response by Davies Classification theorem that at least the Gorenstein-Lyons-Solomon (GLS) Professor Strichartz goes even fur­ one of the authors recognizes may project to provide a new proof of a ther than I have in speculating about be seriously flawed, and the non­ large portion of CFSG. (Cf. p. 737 of the development of mathematics in existence of a proof in the sense that Aschbacher's August 2004 Notices ar­ this century. We have both been in­ many people recognize that term. I ticle.) The original proof was com­ fluenced by our use of computers, fully accept the statement of the au­ pleted in 2004 with the publication of and the new and exciting prospects thors that group theory has made the Aschbacher-Smith proof of the that they open up for the develop­ enormous strides since 1980. The goal Quasithin Theorem (AMS Surveys and ment of the subject. One can discover of my article was to point out that Monographs 111/112). A very small many new ideas and results if one is mathematicians will have to learn to number of constituent theorems exist prepared to accept a degree of inse­ live with such uncertainties, and that only as doctoral dissertations, not curity, and some of them can even be this problem will certainly get steadily published in journals. However the proved to the traditional standards of worse in the future, whatever hap­ assertion that "the proof has never rigour. Numerical experimentation pens in this particular case. By the been written down in its entirety" is can provide not only ideas, but also end of this century many fields will seriously misleading. Reference to supporting evidence for the correct­ be forced to use theorems that have "only about five" (actually, six) vol­ ness of a proof, when one is able to not been proved to the traditional umes being published applies to the give one. In some areas of analysis it standards. GLS project, not to the original proof. can be one of the easiest ways of find­ I would like to correct an error We object even more vigorously to ing a mistake in a proof. brought to my attention by Professor the implication on p. 13 55 that the I next respond to the letter of Pro­ R. L. Griess. Although several of the proof of CFSG does not supply "un­ fessor Aschbacher et al. I welcome sporadic finite simple groups, for ex­ derstanding .. .in full measure". Of their distinction between the original ample, those of Lyons, ]4, O'Nan and course "full measure" is a high stan­ proof of the Classification theorem others, were first constructed using dard for us mortals, but some sense and the GLS project. Both of these computers, Griess proved the exis­ of the enormous advances in the un­ have required brilliant insights and tence of the Monster group itself en­ derstanding of finite groups which heroic commitment. Their statements tirely by hand. have been achieved as by-products of are entirely in line with Professor Ash­ the classification endeavor can be bacher's "The status of the classifi­ - E. B. Davies gained easily by a comparison of the cation of the finite simple groups" King's College London post-CFSG textbooks of Aschbacher Notices Amer. Math. Soc. 51 (2004), [email protected] (Cambridge, 1986) and Kurzweil/Stell­ 736-739, but differ in tone from the macher (Springer, 2004) with the ex­ his more informal paper "Highly com­ cellent pre-CFSG text by Marshall plex proofs and implications", in "The Hall Jr. (Macmillan, 1959). Nature of Mathematical Proof", Phil. Finally, we believe that perusal of Trans. Roy. Soc. A 363 (2005), Serre's interview (Notices, February 2401-2406, which formed the basis 2004) will reveal that the concerns he for this part of my article. On page expressed were in regard to the then 2402 of that article he wrote pre-publication status of the Aschbacher-Smith monographs. If we've made mistakes, so that the [Classification] -Michael Aschbacher theorem is false and there is some H inC - L, then it California Institute of Technology might be possible to re­ [email protected] pair the theorem by adding H to L and making -Richard Lyons minor modifications to the Rutgers University inductive 'proof'. This [email protected] would be true if the struc­ ture of His much like that -Stephen D. Smith of the members of L. But University of Illinois at Chicago if H has a very different [email protected] structure, one could imag­ ine that such a modifica­ -Ronald Solomon tion might not be possi­ The Ohio State University ble. [email protected] I also (deliberately) do not distin­ (Received January 7, 2006) guish between a proof of the

APRIL 2006 NOTICES OF THE AMS 407 A.\lt.IUCA:\' 1\lATHE,\IATICAL SOCIETY

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The Incompleteness Theorem Martin Davis

n September 1930 in Konigsberg, on the third represented by a string of symbols, and a proof, day of a symposium devoted to the founda­ by a finite sequence of such strings. Since these sys­ I tions of mathematics, the young Kurt Godel tems were simple combinatorial objects, it seemed launched his bombshell announcing his in­ quite possible to apply mathematical methods to completeness theorem. At that time, there study their properties. Hilbert's program aimed to were three recognized "schools" on the foundations prove, by utterly unimpeachable methods, that of mathematics: the logicism based on the work of these systems were consistent and complete: that Frege, Russell, and Whitehead that saw mathe­ they were safe from the catastrophic inconsistency, matics as simply part of logic, Brouwer's radical in­ due to Russell's paradox, that had struck Frege's tuitionism, and Hilbert's proof theory (also called ambitious attempt to bridge the gap between the "formalism"). In fact two days earlier, lectures rep­ elements of formal logic and mathematics proper, resenting these schools had been delivered by Car­ and that with respect to some specified class of nap, Heyting, and von Neumann respectively. Von statements, each statement of the class could be Neumann may have been the only person in the either proved or refuted within the system. Godel's room to have grasped the significance of what incompleteness theorem did away with the sec­ Godel had done. He saw that the goals of Hilbert's ond of these goals, and shortly thereafter Godel was proof theory had been shown to be simply unat­ able to show that the first was likewise unachiev­ tainable. Logicism had also been dealt a death blow, able. Go del's theorem had made it clear that no sin­ but Carnap, who had known about Godel's in­ gle formal system could be devised that would en­ completeness theorem for over a week when he able all mathematical truths, even those expressible gave his address, seemed not to realize its signif­ in terms of basic operations on the natural num­ icance. bers, to be provided with a formal proof.

Formalization of Mathematics Godel's Proof It was Gottlob Frege in his Begriffsschrift of 1879 Godel proceeded to define a code by means of who had shown how the logical reasoning used in which each expression of a formal system would mathematical proofs can be reduced to the com­ have its own natural number, what has come to be binatorial manipulation of symbols. By the 1920s called its Gddel number, associated with it. Thus, foundational work had made it clear that the full expressions of the system that represent proposi­ expanse of classical mathematics could be encap­ tions about the natural numbers might be seen by sulated in such formal combinatorial systems. In someone privy to the code as also making asser­ these systems, a proposition of mathematics was tions, incidentally as it were, about the system it­ Martin Davis is professor emeritus of mathematics and self. Working with a particular formal system computer science, New York University, and is a Visiting loosely based on that of Whitehead and Russell and Scholar in mathematics at the University of California, exploiting this idea, Godel showed how to con­ Berkeley. His email address is marti n@ei pye. com. struct a remarkable expression of the system we

414 NOTICES OF THE AMS VOLUME 53, NUMBER 4 may designate as U. To someone who didn't know Computability theory provides a perspective the code, U would be seen as expressing a com­ from which it can be seen that incompleteness is plicated and peculiar statement about the natural a pervasive fundamental property not dependent numbers. But to someone who could decipher it, on a trifling trick. From this point of view the for­ U would be seen as also asserting that some state­ mal systems studied by logicians are simply com­ ment expressible in the system is unprovable. Look­ putable functions that spew out theorems (more ing more closely, it would be found that the state­ precisely, Gbdel numbers of theorems). Such sys­ ment asserted to be unprovable is U itself. Thus we may say: tems are usually given in terms of a set of axioms and rules of inference. One can then imagine anal­ U asserts that it is unprovable. gorithm that begins with the axioms and proceeds Thus, if U were false, it would be provable, and by iteratively applying the rules of inference. hence, presumably, true. This contradiction shows To obtain a form of the incompleteness theorem that U is true, and hence, given what it asserts, un­ let us begin with the set K whose existence is given provable. There are true statements unprovable in by the theorem above, and consider propositions the given system. of the form n rt K where n is a fixed natural num­ Of course, this heuristic outline would have ber. We can suppose that, in a particular formal sys­ hardly been convincing. But Go del carefully worked tem these propositions are each represented by a out the details leaving no doubt about the cor­ rectness of his conclusions.1 Nevertheless, a whiff corresponding string of symbols we may write as of paradox hung over the matter; it seemed hard Pn. We need only assume that there is an algo­ to believe that a trick so close to puzzles usually rithm for obtaining Pn given n.4 Let us use the sym­ offered for amusement could really be used to bol J" for some formal system, and write r-;: Pn to demonstrate something profound about mathe­ mean that Pn is provable in J. We will say that J" matics. is sound if Whenever f-;: Pn for a particular n, Computability Theory Makes a it will also be the case that n rt K. Contribution Since Pn is intended to stand for the proposition We write N = {0, 1, 2, ... } for the set of natural n rt K, soundness simply means that the provable numbers. A function f : N ~ N is called computable statements are true. if there is an algorithm that given an x E N will com­ pute f(x). Here the notion of algorithm is assumed Incompleteness Theorem. Let J" be a sound formal to involve no restriction as to the amount of time system. Then there is a number no such that no rt K, 2 or space required to complete a computation. Fi­ but it is not the case that r-;: Pno· nally a set S ~ N is called computable if its char­ acteristic function Again, we have a true sentence that is not prov­ 1 ifxES able. Note that we only succeed in changing the Cs(x) = { value of the particular number no as we attempt 0 otherwise to create stronger and stronger formal systems that can prove more and more. is computable. Proof of the Incompleteness Theorem. Sup­ The following is fundamental: pose that there is no such no. Then we would have: Theorem. There is a computable function f whose r-;: Pn for a particular n, if and only if n rt K. range K = {f(O), f(1), f(2), ... } Recall that K is the range of the computable func­ tion f. Then the following would be an algorithm is not computable. 3 for computing CK(n) for a given value of n, con­ tradicting the fact that K is not computable: Begin 1 Detailed proofs can be found in a number of textbooks, for example [3]. In addition Code/'s clear and meticulous generating the theorems of J" and at the same time original exposition [8] still repays study. begin computing the successive values 2 Computability theory has provided a number ofprecise f(O), f(1), f(2), . . .. If n E K, then n will eventually characterizations to replace this heuristic explanation, show up in the list of values off so CK(n) = 1. Oth­ and they have all been proved equivalent to one another. erwise, Pn will eventually show up in the theorem 3 See for example [1]. Computability theory is also known list of J" so that CK(n) = 0. D as recursion theory and used to also be called recursive function theory. Computable functions are also called re­ 4 In a traditional formal system, for a given number n, Pn cursive. Sets that are the range of a computable function will be obtained by replacing, in a certain specific formula, as well as the empty set are called recursively enumerable, a symbol for a variable by a "numeral" representing the or more recently, computably enumerable, or listable. number n.

APRIL 2006 NOTICES OF THE AMS 415 A Diophantine Perspective and for the successor, sum, and product functions The following result, known variously as MRDP on the natural numbers. The axioms are the familiar and as Matiyasevich's Theorem, enables it to be seen Peano postulates together with equations serving that the truths unprovable in specified formal sys­ to implicitly define sum and product. The induc­ tems can have a straightforward mathematical tion postulate, whose informal statement is that a form. set of natural numbers containing 0 and closed under successor must consist of all natural num­ Theorem. If S is the range of a computable func­ bers, appears in a restricted form: it is stated only tion, then there is a polynomial p(a, x1, . . . , Xm) with for sets definable in terms of the vocabulary.6 integer coefficients such that the equation PA formalizes the elementary number theory of p(a, x1, . . . , Xm) = 0 has a solution in natural num- the textbooks as well as (via clever coding) sub­ bers Xl, . .. , Xm for a given value of a if and only if stantial parts of elementary analysis. In contrast a E S (see [9, 2]). ZFC formalizes the full scope of modern set­ Applying this result to the case S = K, let us theoretic mathematics including such things as call the corresponding polynomial Po. Now we can general topology and transfinite arithmetic. The vo­ think of the expressions Pn as standing for the cabulary can be extremely parsimonious consist­ proposition that Po(n, x1, ... , Xm) = 0 has no solu­ ing only of the symbol E for set membership. For tions in natural numbers, and say that :F is Dio­ our purposes it will be useful to be slightly less fru­ phantine-sound if 1-J' Pn implies that the equation gal, allowing as well symbols 0 (the empty set), Po(n, x1, ... , Xm) = 0 does indeed fail to have solu­ {.. . } (the set consisting of a single element), and tions. Then, the incompleteness theorem of the u (binary union). The axioms are those of Zermelo­ previous section takes the form: Fraenkel together with the axiom of choice, and the resulting system is powerful enough to encapsu­ Diophantine Incompleteness Theorem. Let :J" be late the full scope of classical mathematics, and in­ Diophantine-sound. Then there is a number no such deed, much more (see for example [4]). that the equation Po(no, Xl, ... , Xm) = 0 has no so­ We write 1-PA and 1-zFc for provability in PA lutions in natural numbers although it is not the case and ZFC, respectively. We will use :F as a subscript that 1-J' Pno· to refer ambiguously to either of these systems. It is worth remarking that the proof of MRDP is Also we write f/- J' to express non-provability in the entirely constructive so the polynomial Po could corresponding systems. be produced quite explicitly. In each of PA and ZFC, a simple notation is avail­ able for representing the natural numbers by se­ Two Formal Systems: PA and ZFC quences of strings we call numerals. We will write What Frege showed is that the ordinary reasoning n for the numeral representing the natural num­ in proofs of mathematical theorems amounts to for­ ber n. In PA this may be defined as follows using mal manipulations of the propositional connec­ the letter s for successor and letting 0 be repre­ tives • - v A together with the quantifiers 'v' 3. sented by its usual symbol: Manipulations of the propositional connectives amounts to carrying out the operations of Boolean 0 = 0; n + 1 = sn algebra. The quantifiers get in the way of this, and Following von Neumann, the numerals in ZFC can careful rules are needed to justify removing and re­ be defined as follows: instating them. Once these rules are specified (which can be done in a number of equivalent o = 0 ; n + 1 = n u {n} ways), the way is open to set up formal systems en­ Now, for :F standing for either PA or ZFC, associ­ capsulating greater or lesser portions of mathe­ ated with the polynomial Po of the previous sec­ matics. This involves supplying a vocabulary of tion, there is a formula TTJ'(xo, x1, ... , Xm) such symbols representing various constants, functions, that for arbitrary natural numbers a, a1, .. . , am: and relations appropriate to the part of mathe­ matics being formalized. Finally a list of axioms if Po(a, a1, ... , am) = 0 then 1-J' rr(a, a1, ... , am) must be given: these are written using this vocab­ if Po(a, a1, ... , am) =f 0 then 1-J' • rr(a , a1, ... , am) ulary together with the symbols listed above cor­ responding to the operations of logical inference. For PA, this is almost a triviality because symbols A symbol for equality should also be available. 5 for addition and multiplication are part of its vo­ For the system PA (for "Peano Arithmetic"), the cabulary, and the axioms justify ordinary calcula­ vocabulary consists of symbols for the number 0, tions. For ZFC, some circumlocution is needed, but

5 Equality can be thought of as the most "advanced" part 6 A fu ller account of PA will be found in Fe ferman's arti­ of the underlying logic, or as the most fundamental math­ cle [5] in this issue of the Notices. Full details will be found ematical relation. in textbooks such as [3].

416 NOTICES OF THE AMS VOLUME 53, NUMBER 4 the ordinary facts about addition and multiplica­ property just exhibited that their undecidability in tion of natural numbers can still be replicated. a reasonable formal system implies their truth. It follows from the MRDP theorem that statements Incompleteness Theorem for PA and ZFC. If :J is asserting that some computable property holds consistent, then there is a natural number no such for all natural numbers are provably equivalent that (for example in PA) to a \!-statement. Many fa­ mous problems are thus seen to belong to this and class, in particular, Fermat's last theorem, the Gold­ I/-:;: •(Vxl) ... (Vxm)•rr(no, x1, ... , Xm). bach conjecture, the four-color theorem, and the Riemann Hypothesis (see [2]). Thus the sentence (Vx1) ... (Vxm)• rr(no, x1, ... , Xm) is undecidable in :J: neither it nor its Beyond ZFC negation is provable. However, what that sentence At the same time that the ZFC axioms provide a asserts, namely that the equation Po(no, x1, ... , Xm) foundation for mathematics, they also can be re­ = 0 has no solutions in natural numbers, is true. garded as defining a class of mathematical struc­ Moreover that truth is a consequence of its unde­ tures. From this point of view they can be seen as cidability. For if Po(no, a1, ... , am)= 0 we would providing "closure" under such operations as form­ have 1-:;: rr(no, a1, ... , am) and using elementary ing the set of all subsets of a given set or the union logic we would obtain of all of its elements. In a normal situation of this 1-:;: (:lx1) ... (:lxm)rr(no, x1, ... , Xm) kind it would be natural to find the least set closed under all of the operations called for by the axioms. from which we readily obtain Remarkably, as natural as such an object appears, 1-:;: • (Vx1) ... (Vxm)•rr(no, X1, ... , Xm) its existence cannot be proved in ZFC. This is be­ cause if such existence could be proved, it would contradicting the claimed undecidability. provide a model for the axioms and hence lead to There has been much confusion about this sit­ a proof in ZFC of its own consistency. And this, uation. How is it that we can see that the proposi­ Godel had proved to be impossible. So systems tion is true although a system as powerful as ZFC like ZFC lead in a natural way to extensions, and cannot? The answer is that ZFC can indeed see in each such extension new V -propositions be­ what we can, namely that ifZFC is consistent then come provable. In fact there will be new values of the proposition is true but undecidable by its no for which the fact that the equation means. In fact, it was precisely by analyzing this situation that Gbdel could conclude that systems Po(no, x1, ... , Xm) = 0 has no solutions in natural like PA and ZFC cannot prove their own consistency, numbers becomes provable. What remains unclear thereby shattering Hilbert's hopes. is whether some really mathematically significant The fact that ZFC is stronger than PA (in actual V -propositions, perhaps like some of those men­ fact very much stronger) is exemplified by the fol­ tioned at the end of the previous section, require lowing result: means beyond ZFC for their proof. We conclude this article with a recent example announced by Har­ Theorem. If PAis consistent, then there is a natural vey Friedman (see [7]) of a V -proposition that is un­ number no such that provable in ZFC, but becomes provable with the aid 1/-rA (Vxl) ... (VXm)•TT(no, X1, ... , Xm), of a so-called "large cardinal" axiom, an assump­ and tion of the existence of an infinite set of a size larger than any whose existence can be proved in ZFC.8 Friedman's example concerns finite directed but graphs (no multiple edges allowed) whose vertices 1-zFc (Vx1) ... (Vxm)•rr(no, x1, ... , Xm). are finite sequences of integers. What is striking is that what looks like a harmless additional conclu­ So the undecidability in PAis decided in ZFC! But sion in a theorem provable in ZFC (and even in much then ZFC has its own undecidability and with the weaker systems) results in a proposition that is un­ very same formula rr. Only the number no changes. The values of no for either system will be enormous provable in ZFC but becomes provable on the ad­ since all the complexity of the algorithms for gen­ dition of a large cardinal axiom, an assumption of erating theorems these systems provide must be the existence of a set too large for that existence 9 contained in those numbers. to be provable in ZFC. Some preliminary defini­ We will refer to statements to the effect that tions are needed. For a natural number n we write some polynomial equation has no solutions in nat­ 8 The article by juliet Floyd and Akihiro Kanamori [6] in ural numbers as V -statements? They all have the this issue of the Notices contains some discussion of large 7 Logicians call these IT~ statements. As this notation sug­ cardinal axioms. gests, they find their place in a hierarchy. 9 Specifically, an axiom of the Mahlo type.

APRIL 2006 NOTICES OF THE AMS 417 fi for the set {1, 2, . .. , n}. So fik is the set of all se­ [3] HERBERT ENDERTON, A Mathematical Introduction to Logic, Academic Press, New York, 1972. quences of these numbers of length k. If x, y E nk, [4] __ , Elements of Set Theory, Academic Press, New we write x y for the element of 2k obtained by * n York, 1977. concatenating x andy. We will work with directed [5] SoLOMON FEFERMAN, The Impact of the Incompleteness graphs G whose vertex set V(G) consists of ele­ Theorems on Mathematics, Notices, April 2006. ments of fik for certain fixed n, k. For x, y E V(G) (6] }ULIET FLOYD and AKIHIRO KANAMORl, How Godel Trans­ we write (x, y) for a possible edge proceeding from formed Set Theory, Notices, April 2006. x to y. G is called an upgraph if for every edge (x, y) [7] HARVEY FRIEDMAN, "II~ Incompleteness: Finite Graph of G, we have max(x) < max(y). We say that Theory 1." http: //www.math.ohio-state.edu/ u, v E fi-e are order equivalent if for all1 ::::; i, j ::::; -8, %7Efriedman/ pdf/ Pi01013006.pdf. [8] KuRT GODEL, "Dber formal unentscheidbare Satze der we have Ui < UJ if and only if vi < VJ . An upgraph Principia Mathematica und verwandte Systeme I" with G is called order invarant if whenever x * y is page-facing English translation, in Solomon Feferman, order equivalent to z * w, we have (x, y) is an edge et al. (eds.), Kurt Gbdel, Collected Works, val. I., Oxford, of G if and only if (z, w) is an edge of G. For New York, 1986. A<;:::: V(G) we write GA = {y I :lx E A say that (x, y) [9] YuRJ MAT!YASEVICH, Hilbert's Tenth Problem, MIT Press, is an edge of G}. A is called independent if no two Cambridge, Massachusetts, 1983. elements of A are connected by an edge of G. Sets B, C <;:::: V(G) are G-isomorphicif there is a bijection h from B to C such that for all x, y E B, (x, y) is an edge in G if and only if (hx, hy) is also an edge in G. Finally, we call x E V( G) two-powered if each Xi is a member of the set {1, 2, 4, 8, ... } of powers of 2. Now, we have:

Theorem. For all n, k, r ~ 1 every order-invariant upgraph G on nk has an independent set A such that if B <;:::: V(G) - A and lEI ::::; r, then B is G­ isomorphic to a set C <;:::: G A such that B and C have the same two-powered elements. This is provable not only in ZFC but also in PA and even in still weaker systems. However consider the following variant:

Proposition. For a ll n, k, r ~ 1 ev e ry order­ invariant upgraph G on fik has a n independent set A such that if B <;:::: V(G) - A and lEI ::::; r, then B is G -isomorphic to a set C <;:::: GA such that B and C have the same two-powered elements, and fur­ thermore, the particular number 2(4kr)2 - 1 doesn't occur in any elem ent of C. Harvey Friedman has announced that this V -statement is not provable in ZFC but becomes provable on the addition of a large cardinal axiom. Acknowledgments: I'm grateful to Solomon Fe­ ferman and to Allyn Jackson for their helpful com­ ments on a previous version of this article.

References [1] NIGEL CUTLAND, Computability: An Introduction to Re­ cursive Function Theory, Cambridge University Press, Cambridge, England, and New York, 1980. [2] MARTIN DAVIS, YURI MATIJASEVIC, and }ULIA ROBINSON, "Hilbert's Tenth Problem: Diophantine Equations: Pos­ itive Aspects of a Negative Solution", Proceedings of Symposia in Pure Mathematics, vol. 28 (1 976), pp. 323- 378; reprinted in Feferman, Solomon, ed. The Collected Works of julia Robinson, Amer. Math. Soc. 1996, pp. 269-378.

418 NOTICES OF THE AMS VOLUME 53, N UMBER 4 How GOdel Transformed Set Theory juliet Floyd and Akihiro Kanamori

urt Godel (1906-1978), with his work on of reals and the like. He stipulated that two sets he constructible universe L, established have the same power if there is a bijection between he relative consistency of the Axiom of them, and, implicitly at first, that one set has a hoice and the Continuum Hypothesis. higher power than another if there is an injection ore broadly, he secured the cumulative of the latter into the first but no bijection. In an hierarchy view of the universe of sets and ensured 1878 publication he showed that R, the planeR x R, the ascendancy of first-order logic as the framework and generally Rn are all of the same power, but for set theory. Godel thereby transformed set the­ there were still only the two infinite powers as set ory and launched it with structured subject mat­ out by his 1873 proof. At the end of the publica­ ter and specific methods of proof as a distinctive tion Cantor asserted a dichotomy: field of mathematics. What follows is a survey of prior developments in set theory and logic in­ Every infinite set of real numbers ei­ tended to set the stage, an account of how Godel ther is countable or has the power of the marshaled the ideas and constructions to formu­ continuum. late L and establish his results, and a description This was the Continuum Hypothesis (CH) in its of subsequent developments in set theory that res­ nascent context, and the continuum problem, tore­ onated with his speculations. The survey trots out solve this hypothesis, would become a major mo­ in quick succession the groundbreaking work at the tivation for Cantor's large-scale investigations of beginning of a young subject. infinite numbers and sets. In his Grundlagen of 1883, Cantor developed the Numbers, Types, and Well-Ordering transfinite numbers and the key concept of well­ Set theory was born on that day in December 18 73 ordering. The progression of transfinite numbers when Georg Cantor (1845- 1918) established that could be depicted, in his later notation, in terms the continuum is not countable: There is no bijec­ of natural extensions of arithmetical operations: tion b e tween the natural numbers N = {0, 1, 2, 3, ... } and the real numbers R, since 0, 1, 2, . . . W, W + 1, W + 2, . . . W + W(= W·2), for any (countable) sequence of reals one can spec­ ... w·3, ... w·w(= w 2), ... w 3 , ... ww, ... ify nested intervals so that any real in the inter­ section will not be in the sequence. Cantor soon in­ A relation -< is a well-ordering of a set if and only vestigated ways to define bijections between sets if it is a strict linear ordering of the set such that every nonempty subset has a -< -least element. Well­ juliet Floyd is professor of philosophy at Boston University. orderings carry the sense of sequential counting, Her email address is j f l oyd@bu. e du. and the transfinite numbers serve as standards Akihiro Kanamori is professor of mathematics at Boston for gauging well-orderings. Cantor called the set of University. His email address is aki @math. bu. edu. natural numbers N the first number class (I) and

APRIL2006 NOTICES OF THE AMS 419 the set of numbers whose predecessors are in bi­ are now to be the cardinal numbers of the succes­ jective correspondence with (I) the second number sive number classes from the Grundlagen and thus class (II). The infinite numbers in the above display to exhaust all the infinite cardinal numbers. Can­ are all in (II). Cantor conceived of (II) as bounded tor pointed out that 2tDavid Hilbert's famous list of problems by his numbers and number classes. With this at the 1900 International Congress of Mathemati­ framework Cantor had transformed CH into the cians; Hilbert drew out Cantor's difficulty by sug­ positive assertion that (II) and R have the same gesting the desirability of "actually giving" a well­ power. However, an emerging problem for Cantor ordering of R. was that he could not even define a well-ordering Bertrand Russell (1872-1970), a main architect of R; the continuum, at the heart of mathematics, of the analytic tradition in philosophy, focused in could not be easily brought into the fold of the 1900 on Cantor's work. Russell was pivoting from transfinite numbers. idealism toward a realism about propositions and Almost two decades after his initial1873 proof, with it logicism, the thesis that mathematics can Cantor in 1891 came to his celebrated diagonal ar­ be founded in logic. Taking a universalist approach gument. In various guises the argument would be­ to logic with all-encompassing categories, Russell come fundamental in mathematical logic. Cantor took the class of all classes to have the largest car­ himself proceeded in terms of functions, ushering dinal number but saw that Cantor's 1891 result collections of arbitrary functions into mathemat­ leading to higher cardinal numbers presented a ics, but we cast his result as is done nowadays in problem. Analyzing that argument, by the spring terms of the power set P(x) = {y I y 5:::; x} of a set of 1901 he came to the famous Russell's Paradox, x. For any set x, P(x) has a higher power than x. a surprisingly simple counterexample to full com­ First, the function associating each a E x with prehension, the assertion that for every property {a} is an injection: x - P(x). Suppose now that F A(x) the collection of objects having that prop­ is any function: x - P(x). Consider the "diagonal" erty, the class {x I A(x)}, is also an object. Consider set d = {a E x I a $ F(a)}. If d itself were a value Russell's {x I x $ x}. If this were an object r, then ofF, say d = F(b), then we would have the contra­ we would have the contradiction r E r if and only diction: b E d if and only if b $ d. Hence, F cannot if r $ r. Gottlob Frege (1848-1925) was the first to be surjective. systematize quantificationallogic in a formalized Cantor had been shifting his notion of set to a language, and he aimed to establish a purely logi­ level of abstraction beyond sets of real numbers cal foundation for arithmetic. Russell famously and the like; the diagonal argument can be drawn communicated his paradox to Frege in 1902, who out of the earlier argument, and the new result gen­ immediately saw that it revealed a contradiction eralized the old since P(N) and R have the same within his mature logical system. power. The new result showed for the first time that Russell's own reaction was to build a complex there is a set of a higher power than R, e.g., P(P(N)). logical structure, one used later to develop math­ Beitrdge Cantor's of 1895 and 1897 presented ematics in Whitehead and Russell's 1910-3 Principia his mature theory of the transfinite. Cantor re­ Mathematica. Russell's ramified theory of types is construed power as cardinal number, now an au­ a scheme of logical definitions based on orders tonomous une fac;on concept beyond de parler and types indexed by the natural numbers. Russell about bijective correspondence. He defined the ad­ proceeded "intensionally"; he conceived this dition, multiplication, and exponentiation of car­ scheme as a classification of propositions based on dinal numbers primordially in terms of set­ the notion of propositional function, a notion not theoretic operations and functions. As befits the reducible to membership (extensionality). Pro­ introduction of new numbers Cantor then intro­ ceeding in modern fashion, we may say that the uni­ duced a new notation, one using the Hebrew letter verse of the Principia consists of objects stratified aleph, N. No is to be the cardinal number of N and into disjoint types Tn, where To consists of the in­ the successive alephs dividuals, Tn+l 5:::; { Y I Y 5:::; Tn}, and the types Tn for n > 0 are further ramified into orders 0~ with

420 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Tn = Ui 0~. An object in 0~ is to be defined either positive use of an arbitrary function operating on in .terms of individuals or of objects in some fixed arbitrary subsets of a set having been made explicit, O/n for some j < i and m .:s; n, the definitions al­ there was open controversy after the appearance lowing for quantification only over O/n. This pre­ of Zermelo's proof. This can be viewed as a turn­ cludes Russell's Paradox and other "vicious cir­ ing point for mathematics, with the subsequent tilt­ cles", as objects consist only of previous objects ing toward the acceptance of AC symptomatic of and are built up through definitions referring only a conceptual shift. to previous stages. However, in this system it is im­ possible to quantify over all objects in a type Tn, Axiomatization and this makes the formulation of numerous math­ In response to his critics Zermelo published a sec­ ematical propositions at best cumbersome and at ond proof of the Well-Ordering Theorem in 1908, worst impossible. Russell was led to introduce his and with axiomatization assuming a general Axiom of Reducibility, which asserts that for each methodological role in mathematics he also pub­ object there is a predicative object consisting of ex­ lished in 1908 the first full-fledged axiomatiza­ actly the same objects, where an object is predica­ tion of set theory. But as with Cantor's work, this tive if its order is the least greater than that of its was no idle structure building, but a response to constituents. This axiom reduced consideration to pressure for a new mathematical context. In this individuals, predicative objects consisting of indi­ case it was not for the formulation and solution of viduals, predicative objects consisting of predica­ a problem but rather to clarify a proof. Zermelo's tive objects consisting of individuals, and so on­ motive in large part for axiomatizing set theory was the simple theory of types. In traumatic reaction to to buttress his Well-Ordering Theorem by making his paradox Russell had built a complex system of explicit its underlying set existence assumptions. orders and types only to collapse it with his Axiom To summarize Zermelo's axioms much as they of Reducibility, a fearful symmetry imposed by an would be presented today, there is an initial axiom artful dodger. asserting that two sets are the same if they contain Ernst Zermelo (1871-1953) made his major ad- · the same members (Extensionality, i.e., membership vances in set theory in the first decade of the new determines equality), and an axiom asserting that century. Zermelo's first substantial result was his there is an initial set 0 having no members (Empty independent discovery of the argument for Russell's Set). Then there are the generative axioms, specific Paradox. He then established in 1904 the Well­ instances of comprehension: For any sets x, y, Ordering Theorem, that every set can be well­ {x,y}={zlz=xorz=y} is a set (Pairs), ordered, assuming what he soon called the Axiom Ux = {z I 3y(y E x and z E y)} is a set (Union), of Choice (AC). Zermelo thereby shifted the notion and 'P(x) = {y I y £; x} is a set (Power Set). There is of set away from Cantor's principle that every well­ an axiom asserting the existence of a particular re­ defined set is well-orderable and replaced that cursively specified infinite set (Infinity). Zermelo principle by an explicit axiom. aptly formulated AC in terms of sets as follows: For In retrospect Zermelo's argument for his Well­ any set x consisting of nonempty, pairwise disjoint Ordering Theorem proved to be pivotal for the de­ sets, there is a set y such that each member ofx in­ velopment of set theory. To summarize, suppose tersects withy in exactly one element. Finally, there that x is a set to be well-ordered, and through Zer­ is the axiom (schema) of Separation: For any set x melo's AC hypothesis assume that the power set and "definite"propertyA(y), {y E x I A(y)} is a set. 'P(x) = {y I y £; x} has a choice function, i.e., a func­ That is, the intersection of a set x and a class tion y such that for every nonempty member y of {y I A(y)} is again a .set. Zermelo saw that Sepa­ 'P(x), y(y) E y . Call a subset y of x a y -set if there ration suffices for a development of set theory is a well-ordering R of y such that for each a E y, that still allows for the "logical" formation of sets y( {z I z R a fails}) = a. That is, each member of y according to property; Russell's Paradox is pre­ is what y "chooses" from what does not R­ cluded since only "logical" subsets are to be al­ precede it. The main observation is that y-sets co­ lowed. But what exactly is a "definite" property? here in the following sense: If y is a y-set with This was a central vagary that would be addressed well-ordering R and z is a y-set with well-ordering in the subsequent formalization of Zermelo's set S, then y £; z and Sis a prolongation of R, or vice theory. versa. With this, let w be the union of all the y-sets. With his axioms Zermelo ushered in a new, ab­ Then w too is a y-set, and by its maximality it stract view of sets as structured solely by member­ must be all of x, and hence x is well-ordered. ship and built up iteratively according to governing Cantor's work had served to exacerbate a grow­ axioms, a view that would soon come to dominate. ing discord among mathematicians with respect to Zermelo's work also pioneered the reduction of two related issues: whether infinite collections can mathematical concepts and arguments to set-theo­ be mathematically investigated at all, and how far retic concepts and arguments from axioms, based the function concept is to be extended. The on sets doing the work of mathematical objects.

APRIL 2006 NOTICES OF THE AMS 421 Unlike the development of classical mathematics recursive definition along well-orderings. The proof from marketplace arithmetic and Greek geometry, had an antecedent in the Zermelo 1904 proof, but sets were neither laden with nor bolstered by well­ Replacement was necessary even for the very for­ worked antecedents. Zermelo axiomatization, un­ mulation, let alone the proof, of the theorem. With like Russell's cumbersome theory of types, provided the ordinals in place von Neumann completed the a simple system for the development of mathe­ incorporation of the Cantorian transfinite by defin­ matics. Set theory would provide an underpinning ing the cardinals as the initial ordinals, those or­ of mathematics, and Zermelo's axioms would res­ dinals not in bijective correspondence with any of onate with mathematical practice. their predecessors. In the 1920s fresh initiatives structured the Replacement has been latterly regarded as some­ loose Zermelian framework with new features and how less necessary or crucial than the other axioms, corresponding developments in axiomatics, the the purported effect of the axiom being only on most consequential moves made by John von Neu­ large-cardinality sets. Initially, Abraham Fraenkel mann (1903-1957) in his dissertation, with antic­ (1891-1965) and Thoralf Skolem (1887-1963) had ipations by Dimitry Mirimanoff (1861-1945). The independently proposed adjoining Replacement transfinite numbers had been central for Cantor but to ensure that E(a) = {a, P(a), P(P(a)), ... } would peripheral to Zermelo, and in Zermelo's system be a set for a, the infinite set given by Zermelo's not even 2N o = N1 could be stated directly. Von Axiom of Infinity, since, as they pointed out, Zer­ Neumann reconstrued the transfinite numbers as melo's axioms cannot establish this. However, even bona fide sets, the ordinals, and established their £(0) cannot be proved to be a set from Zermelo's efficacy by formalizing transfinite recursion. axioms, and if his Axiom of Infinity were refor­ Ordinals manifest the idea, natural once iterative mulated to accommodate £(0), there would still be set formation is assimilated, of taking the relation of many finite sets a such that E(a) cannot be proved precedence in a well-ordering simply to be mem­ to be a set. Replacement serves to rectify the situ­ bership. A set (or class) xis transitive if and only if ation by admitting new infinite sets defined by "re­ whenever a E b for b E x, a E x. A set x is a (von placing" members of the one infinite set given by Neumann) ordinal if and only if x is transitive, and the Axiom of Infinity. In any case, the full exercise themembershiprelationrestricted tox = {y I y E x} of Replacement is part and parcel of transfinite re­ is a well-ordering of x. The first several ordinals are cursion, which is now used everywhere in modern 0 ,{0},{0,{0}},{0 ,{0 },{0,{0}} }, ... , to be set theory, and it was von Neumann's formal in­ taken as the natural numbers 0,1,2,3, .... The union corporation of this method into set theory, as ne­ of these finite ordinals is an ordinal, to be taken as cessitated by his proofs, that brought in Replace­ w; w u {w} is an ordinal, to be taken as w + 1; and ment. so forth. It has become customary to use the Greek Von Neumann (and others) also investigated the letters Oi, {3, ;y, ... to denote ordinals; the class of all salutary effects of restricting the universe of sets ordinals is itself well-ordered by membership, and to the well-founded sets. The well-founded sets are Oi < {3 is written for Oi E {3; and an ordinal without the sets that belong to some "rank" V e<, these de­ an immediate predecessor is a limit ordinal. Von finable through transfinite recursion: Neumann established, as had Mirimanoffbefore him, the key instrumental property of Cantor's ordinal Vo = 0 ; Ve<+l = P(Ve<); and V<5 = U{Ve< I Oi < 8} numbers for ordinals: Every well-ordered set is for limit ordinals 8. order-isomorphic to exactly one ordinal with mem­ bership. The proof was the first to make full use of V w+ 1 contains every set consisting of natural num­ the Axiom of Replacement and thus drew that ax­ bers (finite ordinals), and so already at early levels iom into set theory. there are set counterparts to many objects in math­ For a set x and property A( v, w), the property is ematics. That the universe V of all sets is the cu­ said to be functional on x if for any a E x, there is ex­ mulative hierarchy b A(a, actly one such that b). The Axiom (schema) V = U{V e< I Oi is an ordinal} of Replacement asserts: For any set x and property A(v, w) functional onx, {b I 3a(a Ex and A(a, b))} is thus the assertion that every set is well-founded. is a set. This axiom posits sets that result when mem­ Von Neumann essentially showed that this asser­ bers of a set are "replaced" according to a property; tion is equivalent to a simple assertion about sets, a simple argument shows that Replacement sub­ the Axiom of Foundation: Any non empty set x has sumes Separation. a member y such that x n y is empty. Thus, non­ Von Neumann generally ascribed to the ordinals empty well-founded sets have E -minimal mem­ the role of Cantor's ordinal numbers, and already bers. If a set x satisfies x E x, then {x} is not well­ to incorporate transfinite arithmetic into set the­ founded; similarly, if there are x1 E x2 E x1, then ory he saw the need to establish the Transfinite Re­ {x1, x2} is not well-founded. Ordinals and sets con­ cursion Theorem, the theorem that validates sisting of ordinals are well-founded, and

422 NOTICES OF THE AMS VOLUME 53, NUMBER 4 well-foundedness can be viewed as a generalization Zermelo's 1908 axioms when cast in first-order of the notion of being an ordinal that loosens the logic become a countable collection of sentences, connection with transitivity. The Axiom of Foun­ and so if they have a model at all, they have a dation eliminates pathologies like x Ex and countable model. We thus have the "paradoxical" through the cumulative hierarchy rendition allows existence of countable models for Zermelo's axioms inductive arguments to establish results about the although they entail the existence of uncountable entire universe. sets. Zermelo found this antithetical and repugnant, In a remarkable 1930 publication Zermelo pro­ and proceeded in avowedly second-order terms in vided his final axiomatization of set theory, one that his 1930 work. However, stronger currents were at recast his 1908 axiomatization and incorporated work leading to the ascendancy of first-order logic. both Replacement and Foundation. He herewith completed his transmutation of the notion of set, Constructible Universe his abstract, prescriptive view stabilized by further Enter Godel. Godel virtually completed the math­ axioms that structured the universe of sets. Re­ ematization of logic by submerging "metamathe­ placement provided the means for transfinite re­ matical" methods into mathematics. The Com­ cursion and induction, and Foundation made pos­ pleteness Theorem from his 1930 dissertation sible the application of those means to get results established that logical consequence could be cap­ about all sets. Zermelo proceeded to offer a strik­ tured by formal proof for first-order logic and se­ ing, synthetic view of a procession of natural mod­ cured its key instrumental property of compactness els for his axioms that would have a modern res­ for building models. The main advance was of onance and applied Replacement and Foundation course the direct coding, "the arithmetization of to establish isomorphism and embedding results. Zermelo's 1930 publication was in part a re­ syntax", which together with a refined version of sponse to Skolem's advocacy, already in 1922, of Cantor's diagonal argument led to the celebrated the idea of framing Zermelo's 1908 axioms in first­ 1931 Incompleteness Theorem. This theorem es­ order logic. First-order logic is the logic of formal tablished a fundamental distinction between what languages consisting of formulas built up from is true about the natural numbers and what is prov­ specified function and relation symbols using log­ able and transformed a program advanced by ical connectives and first-order quantifiers V and Hilbert in the 1920s to establish the consistency 3, quantifiers to be interpreted as ranging over of mathematics by finitary means. Gbdel's work the elements of a domain of discourse. (Second­ showed in particular that for a (schematically de­ order logic has quantifiers to be interpreted as finable) collection of axioms A, its consistency, that ranging over arbitrary subsets of a domain.) Skolem from A one cannot prove a contradiction, has a for­ had proposed formalizing Zermelo's axioms in the mal counterpart in an arithmetical formula Con(A) first-order language with E and = as binary rela­ about natural numbers. Gbdel's "second" theorem tion symbols. Zermelo's definite properties would asserts that if A is consistent and subsumes the then be those expressible in this first-order lan­ elementary arithmetic of the natural numbers, then guage in terms of given sets, and Separation would Con(A) cannot be proved from A. become a schema of axioms, one for each first-order Gbdel's advances in set theory can be seen as formula. Analogous remarks apply to the formal­ part of a steady intellectual development from his ization of Replacement in first-order logic. As set fundamental work on incompleteness. His 1931 theory was to develop, the formalization of Zer­ paper had a prescient footnote 48a: melo's 1930 axiomatization in first-order logic would become the standard axiomatization, Zer­ As will be shown in Part II of this paper, melo-Fraenkel with Choice (ZFC). The "Fraenkel" the true reason for the incompleteness acknowledges Fraenkel's early suggestion of in­ inherent in all formal systems of math­ corporating Replacement. Zermelo-Fraenkel (ZF) is ematics is that the formation of ever ZFC without AC. higher types can be continued into the Significantly, before this standardization both transfinite (cf. D. Hilbert, "Uber das Un­ Skolem and Zermelo raised issues about the limi­ endliche", Math. Ann. 95, p. 184), while tations of set theory as cast in first-order logic. in any formal system at most count­ Skolem had established a fundamental result for ably many of them are available. For it first-order logic with the Lbwenheim-Skolem The­ can be shown that the undecidable orem: If a countable collection of first-order sen­ propositions constructed here become tences has a model, then it has a countable model. decidable whenever appropriate higher Having proposed framing set theory in first-order types are added (for example, the type terms, Skolem pointed out as a palliative for tak­ w to the system P [the simple theory of ing set theory as a foundation for mathematics types superposed on the natural num­ what has come to be called the Skolem Paradox; bers as individuals satisfying the Peano

APRIL 2006 NOTICES OF THE AMS 423 axioms]). An analogous situation pre­ M I= formal language and set­ Godel pointed out that L "can be defined and its theoretic assertions about an interpretation of the theory developed in the formal systems of set the­ language and by providing a recursive definition ory themselves." This follows by transfinite re­ of the satisfaction relation, that which obtains when cursion from the formalizability of def(x) in set the­ a formula holds in an interpretation. The eventual ory, the "definability of definability", which was effect of Tarski's mathematical formulation of se­ later reaffirmed by Tarski's systematic definition mantics would be not only to make mathematics of the satisfaction relation in set-theoretic terms. out of the informal notion of satisfiability, but also In modern parlance, an inner model is a transitive to enrich ongoing mathematics with a systematic class containing all the ordinals such that, with method for forming mathematical analogues of membership and quantification restricted to it, the several intuitive semantic notions. For coming pur­ class satisfies each axiom of ZF. Godel in effect ar­ poses, the following specifies notation and con­ gued in ZF to show that L is an inner model and cepts: moreover that L satisfies AC and CH. He thus es­ For a first-order language, suppose that M is an tablished the relative consistency Con(ZF) implies interpretation of the language (i.e., a specification Con(ZFC + CH). of a domain as well as interpretations of the func­ In his 1940 monograph, based on 1938 lectures, tion and relation symbols),

424 NOTICES OF THE AMS VOLUME 53, NUMBER 4 was a veritable Gbdel numbering with ordinals, note. In a comment bringing out the intermixing one that relies on their extent as given beforehand of types and orders, Gbdel pointed out that "there to generate a universe of sets. This approach may are sets of lower type that can only be defined with have obfuscated the satisfaction aspects of the the help of quantifiers for sets of higher type." For construction, but on the other hand it did make example, constructible members of Vw+l in the more evident other aspects: Since there is a direct, cumulative hierarchy will first appear quite high in definable well-ordering of L, choice functions the constructible Lex hierarchy; resonant with abound in L, and AC holds there. Also, L was seen Gbdel's earlier remarks about truth, members of to have the important property of absoluteness V w+ 1, in particular sets of natural numbers, will en­ through the simple operations involved in Gbdel's code truth propositions about higher Lex's. Godel recursion, one consequence of which is that for any had given priority to the ordinals and recursively inner model M, the construction of L in the sense formulated a hierarchy of orders based on defin­ of M again leads to the same class L. Decades later ability, and the hierarchy of types was spread out many inner models based on first-order definabil­ across the orders. The jumble of the Principia Math­ ity would be investigated for which absoluteness ematica had been transfigured into the con­ considerations would be pivotal, and Gbdel had for­ structible universe L. mulated the canonical inner model, rather analo­ Gbdel's argument for CH holding in L rests, as gous to the algebraic numbers for fields of char­ he himself wrote in a brief 1939 summary, on "a acteristic zero. generalization of Skolem's method for construct­ In a 1939lecture about L Gbdel described what ing enumerable models", now embodied in the amounts to the Russell orders for the simple situ­ well-known Skolem Hull argument and Condensa­ ation when the members of a countable collection tion Lemma for L. It is the first significant appli­ of real numbers are taken as the individuals and cation of the Lbwenheim-Skolem Theorem since new real numbers are successively defined via Skolem's own to get his paradox. Ironically, though quantification over previously defined real num­ Skolem sought through his paradox to discredit set bers, and he emphasized that the process can be theory based on first-order logic as a foundation continued into the transfinite. He then observed for mathematics, Gbdel turned paradox into that this procedure can be applied to sets of real method, one promoting first-order logic. Godel numbers and the like, as individuals, and moreover, showed that in L every subset of Lex belongs to that one can "intermix" the procedure for the real some L13 for some {3 of the same power as Oi (so numbers with the procedure for sets of real num­ that in L every real belongs to some L13 for a count­ bers "by using in the definition of a real number able {3, and CH holds). In the 1939 lecture he as­ quantifiers that refer to sets of real numbers, and serted that "this fundamental theorem constitutes similarly in still more complicated ways." Gbdel the corrected core of the so-called Russellian axiom called a constructible Set "the most general [object] of reducibility." Thus, Godel established another that can at all be obtained in this way, where the connection between L and Russell's ramified the­ quantifiers may refer not only to sets of real num­ ory of types. But while Russell had to postulate his bers, but also to sets of sets of real numbers and Axiom of Reducibility for his finite orders, Godel so on, ad transfinitum, and where the indices of it­ was able to derive an analogous form for his trans­ eration .. .can also be arbitrary transfinite ordinal finite hierarchy, one that asserts that the types are numbers." Gbdel considered that although this de­ delimited in the hierarchy of orders. finition of constructible set might seem at first to Gbdel brought into set theory a method of con­ be "unbearably complicated", "the greatest gener­ struction and argument and thereby affirmed sev­ ality yields, as it so often does, at the same time eral features of its axiomatic presentation. First, the greatest simplicity." Gbdel was picturing Rus­ Gbdel showed how first-order definability can be sell's ramified theory of types by first disassociating formalized and used in a transfinite recursive con­ the types from the orders, with the orders here struction to establish striking new mathematical re­ given through definability and the types repre­ sults. This significantly contributed to a lasting sented by real numbers, sets of real numbers, and ascendancy for first-order logic which beyond its so forth. Godel's intermixing then amounted to a sufficiency as a logical framework for mathemat­ recapturing of the complexity of Russell's ramifi­ ics was seen to have considerable operational effi­ cation, the extension of the hierarchy into the cacy. Godel's construction moreover buttressed transfinite allowing for a new simplicity. the incorporation of Replacement and Foundation Gbdel went on to describe the universe of set the­ into set theory. Replacement was immanent in the ory, "the objects of which set theory speaks", as arbitrary extent of the ordinals for the indexing of falling into "a transfinite sequence of Russellian L and in its formal definition via transfinite re­ [simple] types", the cumulative hierarchy of sets. cursion. As for Foundation, underlying the con­ He then formulated the constructible sets as an struction was the well-foundedness of sets. Gbdel analogous hierarchy, the hierarchy of his 1939 in a footnote to his 1939 note wrote: "In order to

APRIL 2006 NOTICES OF THE AMS 425 give A [the axiom V = L, that the universe is L] an described a new inner model, the class of ordinal intuitive meaning, one has to understand by 'sets' definable sets. all objects obtained by building up the simplified In his 194 7 article on the continuum problem hierarchy of types on an empty set of individuals Godel pointed out the desirability of establishing (including types of arbitrary transfinite orders)." the independence of CH, i.e., in addition to his rel­ Some have been baffled about how the cumulative ative consistency result, that also Con(ZF) implies hierarchy picture came to dominate in set-theoretic Con(ZFC + the negation of CH). However, Gbdel practice; although there was Mirimanoff, von Neu­ stressed that this would not solve the problem. mann, and especially Zermelo, the picture came in The axioms of set theory do not "form a system with Godel's method, the reasons being both the­ closed in itself", and so the "very concept of set on matic and historical: Godel's work with L with its which they are based" suggests their extension by incisive analysis of first-order definability was read­ new axioms, axioms that may decide issues like CH. ily recognized as a signal advance, while Zermelo's New axioms could even be entertained on the ex­ 1930 paper with its second-order vagaries remained trinsic basis of the "fruitfulness of their conse­ somewhat obscure. As the construction of L was quences". Go del concluded by advancing the re­ gradually digested, the sense that it promoted of markable opinion that CH "will turn out to be a cumulative hierarchy reverberated to become the wrong" since it has as paradoxical consequences basic picture of the universe of sets. the existence of thin, in various senses he de­ scribed, sets of reals of the power of the contin­ New Axioms uum. How Gbdel transformed set theory can be broadly Later touted as his "program", Godel's advo­ cast as follows: On the larger stage, from the time cacy of the search for new axioms mainly had to of Cantor, sets began making their way into topol­ do with large cardinal axioms. These postulate ogy, algebra, and analysis so that by the time of structure in the higher reaches of the cumulative Go del, they were fairly entrenched in the structure hierarchy, often by positing cardinals whose prop­ and language of mathematics. But how were sets erties entail their inaccessibility from below in viewed among set theorists, those investigating strong senses. Speculations about large cardinal sets as such? Before Gbdel, the main concerns were possibilities had occurred as far back as the time what sets are and how sets and their axioms can of Zermelo's first axiomatization of set theory. serve as a reductive basis for mathematics. Even Godel advocated their investigation, and they can today, those preoccupied with ontology, questions be viewed as a further manifestation of his foot­ of mathematical existence, focus mostly upon the note 48a idea of capturing more truth, this time by set theory of the early period. After Godel, the positing strong closure points for the cumulative main concerns became what sets do and how set hierarchy. In the early 1960s large cardinals were theory is to advance as an autonomous field of vitalized by the infusion of model-theoretic meth­ mathematics. The cumulative hierarchy picture ods, which established their central involvement in was in place as subject matter, and the meta­ embeddings of models of set theory. The subject mathematical methods of first-order logic mediated was then to become a mainstream of set theory the subject. There was a decided shift toward epis­ after the dramatic introduction of a new way of get­ temological questions, e.g., what can be proved ting extensions of models of set theory. about sets and on what basis. Paul Cohen (1934-) in 1963 established the in­ As a pivotal figure, what was Gbdel's own stance? dependence of AC from ZF and the independence What he said would align him more with his pre­ of CH from ZFC. That is, Cohen established that decessors, but what he did would lead to the de­ Con(ZF) implies Con(ZF + the negation of AC) and velopment of methods and models. In a 1944 ar­ that Con(ZF) implies Con(ZFC + the negation of ticle on Russell's mathematical logic, in a 1947 CH). These results delimited ZF and ZFC in terms article on Cantor's continuum problem (and in a of the two fundamental issues raised at the be­ 1964 revision), and in subsequent lectures and cor­ ginning of set theory. But beyond that, Cohen's respondence, GOdel articulated his philosophy of proofs were soon to flow into method, becoming "conceptual realism" about mathematics. He es­ the inaugural examples of forcing, a remarkably poused a staunchly objective "concept of set" ac­ general and flexible method for extending models cording to which the axioms of set theory are true of set theory by adding "generic" sets. Forcing has and are descriptive of an objective reality schema­ strong intuitive underpinnings and reinforces the tized by the cumulative hierarchy. Be that as it notion of set as given by the first-order ZF axioms may, his actual mathematical work laid the ground­ with conspicuous uses of Replacement and Foun­ work for the development of a range of models and dation. With L analogous to the field of algebraic axioms for set theory. Already in the early 1940s numbers, forcing is analogous to making tran­ Godel worked out for himself a possible model for scendental field extensions. If Gbdel's construction the negation of AC, and in a 1946 address he of L had launched set theory as a distinctive field

426 NOTICES OF THE AMS VOLUME 53, NUMBER 4 of mathematics, then Cohen's method of forcing recently provided a variety of propositions of finite began its transformation into a modern, sophisti­ combinatorics that are equi-consistent with the ex­ cated one. Set theorists rushed in and were soon istence of large cardinals; this incisive work serves establishing a cornucopia of relative consistency re­ to affirm the "necessary use" of large cardinal ax­ sults, truths in a wider sense, some illuminating ioms even in finite mathematics. In set theory it­ problems of classical mathematics. In this sea self, Hugh Woodin has developed a scheme based change the extent and breadth of the expansion of on a new logic in an environment of large cardinals set theory dwarfed what came before, both in terms that argues against CH itself, and with an additional of the numbers of people involved and the results axiom, that 2No = Nz. These results serve as re­ established. markable vindications for Godel's original hopes Already in the 1960s and into the 1970s large for large cardinals. cardinal postulations were charted out and elabo­ rated, investigated because of the "fruitfulness of References their consequences" since they provided quick KURT GODEL, [1986], Collected Works, Volume I: Publications proofs of various strong propositions and because 1929-1936, (Solomon Feferman, editor-in-chief), Ox­ they provided the consistency strength to establish ford University Press, New York. __ , [1990], Collected Works, Volume II: Publications new relative consistency results. A subtle connec­ 1938-1974, (Solomon Feferman, editor-in-chief), Ox­ tion quickly emerged between large cardinals and ford University Press, New York. combinatorial propositions low in the cumulative __ , [1995] Collected Works, Volume III: Unpublished Es­ hierarchy: Forcing showed just how relative the says and Lectures, (Solomon Feferman, editor-in-chief), Cantorian notion of cardinality is, since bijections Oxford University Press, New York. could be adjoined easily, often with little distur­ [2002] THOMAS ]ECH, Set Theory, third millennium edition, bance to the universe. In particular, large cardinals, revised and expanded, Springer, Berlin. highly inaccessible from below, were found to sat­ [2003] AKIHIRO KANAMORI, The Higher Infinite. Large Car­ isfy substantial propositions even after they were dinals in Set Theory from their Beginnings, second edition, Springer-Verlag, Berlin. "collapsed" by forcing to N1 or Nz, i.e., bijections [1982] GREGORY H. MOORE, Zermelo's Axiom of Choice. Its were adjoined to make the cardinal the first or Origins, Development, and Influence. Springer-Verlag, second uncountable cardinal. Conversely, such New York. propositions were found to entail large cardinal hy­ potheses in the clarity of an L-like inner model, sometimes the very same initial large cardinal hy­ pothesis. In a subtle synthesis, hypotheses of length concerning the extent of the transfinite were cor­ related with hypotheses of width concerning the fullness of power sets low in the cumulative hier­ archy, sometimes the arguments providing equi­ consistencies. Thus, large cardinals found not only extrinsic but intrinsic justifications. Although their emergence was historically contingent, large car­ dinals were seen to form a linear hierarchy, and there was the growing conviction that this hierar­ chy provides the hierarchy of exhaustive principles against which all possible consistency strengths can be gauged, a kind of hierarchical completion of ZFC. In the 1970s and 1980s possibilities for new complementarity were explored with the develop­ ment of inner model theory for large cardinals, the investigation of minimalL-like inner models hav­ ing large cardinals, models that exhibited the kind of fine structure that Godel had first explored for L. Also, determinacy hypotheses about sets of reals were explored because of their fruitful consequences in descriptive set theory, the definability theory of the continuum. Then in a grand synthesis, certain large cardinals were found to provide just the con­ sistency strength to establish the consistency of ADL(R), the Axiom of Determinacy holding in the minimal inner model L(R) containing all the reals. In a different direction, Harvey Friedman has

APRlL2006 NOTICES OF THE AMS 427 Pictures at an Exhibition Karl Sigmund

n 1965 Kurt Godel wrote to his mother Mari­ place other than the Institute for Advanced Study anne: "I am happy not to have to take part in (lAS) could have provided him with more peace and I the Viennese festivities, as I hate these things." ease, or better intellectual companionship. Yet So we must concede: the Viennese festivities within a dozen years his productivity trickled off, planned for April 2006, on the occasion of his although he certainly did not relax in either ambi­ centenary, would probably have made him wince. tion or hard work. In April 2006 a large scientific congress will be The city of Vienna cannot boast about the way held at the University of Vienna, organized by the it treated Godel, but his centenary is a good occa­ International Kurt Godel Society and generously sion for making amends. The same applies to sponsored by the Templeton Foundation (see Mozart. And as luck will have it, Mozart's 2 50th an­ http://www.logic.at/goedel2006/). It will be niversary also takes place in 2006. And Freud's attended by the few scientists who had close per­ 150th. Some competition! sonal exchanges with Go del, such as Georg Kreisel, Exhibitions are costly. Aspiring to just one or two Dana Scott, and Gaisi Takeuti. There will also be an percent of the sum lavished by Berlin, for example, exhibition on the life and times of Godel, and this on its splendid Albert Einstein Exhibition 2005 is provides me, who rashly volunteered for the job, not a trivial matter. Fortunately, among officials in with a n ew set of experiences. some ministries and magistrates I found several The University of Vienna is well-advised, of dedicated enthusiasts, true crypto-Godelians who course, to celebrate Kurt Godel as much as it can. were ready to help. I also encountered some who The game theorist Oskar Morgenstern, one of had never heard of him: but as soon as I men­ Godel's few friends in later years, was surely right tioned that Time magazine had listed Go del among when he wrote: "Among all those who taught at the the hundred most important persons of the twen­ University of Vienna, there is probably nobody tieth century, conversation flowed more easily. whose name outshines that of Godel." Godel did Whoever drew up that list deserves an accolade! most of his best work in Vienna. The university did Exhibitions also need space. Eventually, the list not exactly pamper him, however. He remained a of possible locations boiled down to three candi­ lowly Privatdozent (meaning he had the right to lec­ dates. One was the main building of the University ture, but no salary worth speaking of), and he even­ of Vienna. During the Congress, in the week of tually had to escape, under hair-raising circum­ Gbdel's birthday (April 28), hundreds of experts stances, to the safety offered by Princeton. No from set theory and mathematical logic will stroll Karl Sigmund is professor of mathematics at the Univer­ its neo-Renaissance arcades, in addition to the sity of Vienna and a research scholar at the Institute for daily stream of students; but many other Viennese Applied Systems Analysis in Laxenburg, Austria. His email will be reticent to cross the university's entrance address is karl. si gmund@uni vie. ac. at. stairs. The second location is Palais Pallfy, on the

428 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Josefsplatz, the most beautiful piazza in town, Next to the univer­ just opposite the Hofburg of the Habsburgs. Since sity, the richest source this will be the venue of many high-level confer­ on Go del is in City Hall: ences in the first half of 2006 (when Austria pre­ the municipality sides over the council of the European Union), one bought hundreds of let­ hopes that some visitors will take a look at Gbdel. ters which he wrote to But people around the Josefsplatz are usually busy. his mother on every Finally, there is the beautiful MuseumsQuarter, other Sunday evening, whose baroque halls are available during the sum­ for the first twenty mer months, when flocks of tourists throng be­ years after the war. tween the Museum for the History of Art, the They were bought, Leopold Museum with its collections of Klimt and thanks to Werner Schi­ Schiele, and a lively scene of bars and restaurants. manovich and Peter No drawback here, except that Gbdel's birthday Weibel (producers of a does not fall in summer. It finally was decided that documentary on the exhibition will travel and visit all three spots Gbdel), from the heirs in different incarnations, from the end of April to of Gbdel's brother Rudolf. Unfortunately, mid-August 2006. there is no trace of the What can one show in an exhibition on Kurt letters Go del must have Gbdel? His work was abstract and his life with­ sent from Princeton drawn. There is nothing equivalent to, say, the during his visits there STAMP OF APPROVAL "I replied very couch of Sigmund Freud, or the fiddle of Albert Ein­ in the 1930s, and after gently to the magistrate, saying that I stein. Gbdel's famous spectacles have not been Godel's death his wife owed my education to the University," preserved, so it seems, but his optometric pre­ Adele destroyed all let­ wrote Godel to his mother. He had scription from 192 5 has survived. For Go del, who ters by her mother-in­ experienced superb teachers but grew up as an avid stamp collector, was not one to law. obnoxious administration. During his throw things lightly away. He kept the bill for his The gist of Godel's time as Privatdozent, he mostly was wedding meal, for instance, as well as the testy re­ major discoveries can on leave, either in Princeton or in minder, signed by Helmut Hasse, that he had not be explained fairly eas­ sanatoria, and lectured only for three paid his membership dues for the German Mathe­ ily to a large public: (a) semesters altogether. matical Society. Godel also kept the receipt for the incompleteness, (b) the purchase of Principia Mathematica, acquired dur­ consistency of the continuum hypothesis, and (c) ing his student days. The Godel Nachlass, which be­ time travel in rotating universes. But I do not be­ longs to the lAS and is kept at Princeton Univer­ lieve that an exhibition can explain the finer points sity's Firestone Library, is a gold mine of tidbits like of "Gbdel's proof" at the level of detail which sev­ these, but also of more serious information, and eral trade books have achieved. Visitors stroll from Gbdel's biographer John Dawson, who knows that one spot to another. Exhibitions are meant for me­ Nachlass like no one else, has kindly agreed to co­ andering around. This implies some superficiality. curate the exhibition. He writes, in this issue of the Anyone wanting greater depth should sit down to Notices, about his experiences cataloguing the boxes read or listen to a lecture. In contrast, the format of Nachlass material (see also Dawson 199 7). of an exhibition seems well suited to giving an idea The only other place with a sizeable amount of of Gbdel's intellectual surroundings. That is the information on Kurt Godel is Vienna (Kohler at al, right topic for an easy stroll. 2002). There are the archives of the university, So let us take a stroll through Gbdel's Vienna, containing much on his brilliant Ph.D. thesis, on his even if that means walking in a Circle. Godel was epochal Habilitation, and on the sinister corre­ an unusually quiet and withdrawn person, but by spondence between academic officials and high­ no means a hermit in his Viennese years. He was level Nazis after 1938. Godel managed twice, after a member of the Vienna Circle. This shaped him Hitler annexed Austria, to go on leave to visit the profoundly, but in an indirect, almost contrary U.S., first at the height of the Munich crisis and again way. In a questionnaire that Godel filled out much in the tense months of the "phony war". More than later (but which he never sent off- Dawson exca­ a year after Go del had settled down for good in the vated it from the Nachlass), he states that the most U.S., the German ministry sent him (c/o University decisive influences on his intellectual development of Vienna) a lavishly emblazoned diploma with his were the lectures by Heinrich Gomperz in philos­ promotion to "Dozent Neuer Ordnung" and the ophy and Philipp Furtwangler in mathematics. One pompous guarantee of the Fuhrer's special pro­ would have expected the names of Moritz Schlick tection. The document was never collected, though, and Hans Hahn instead, the professors of philos­ and the receipt still waits to be signed. ophy and mathematics who had founded the Vienna

APRIL 2006 NoTicEs oF THE AMS 429 Circle (and were certainly no including Georg Nobeling, Franz· Alt, and Abraham mean lecturers). But no: Gom­ Wald. It was to this group that Gbdel lectured first perz and Furtwangler, who on incompleteness. "That's very interesting," said held the introductory lec­ a voice in the awestruck silence ending Godel's tures for Godel's cohort, im­ lecture. "You should publish that." (Alt, 1998) "I am printed him for life. There is consumed with unjustifiable pride," wrote Natkin little reason to doubt that from Paris. "So you have proved that Hilbert's sys­ Godel was a Platonist by the tem of axioms contains unsolvable problems­ age of nineteen and never wa­ why, this is not a trifling matter." vered in this conviction (Fe­ The young prodigy was soon pointed out by ferman 1984). Karl Menger and John von Neumann to Oswald Ve­ Karl Menger, the profes­ blen, who toured Europe as talent scout for the lAS. sor of geometry who was just The newly founded institute invited Godel to be four years older than Kurt among the first group of visiting scholars. Feigl, who Godel and who became his had been the first member of the Vienna Circle to mentor for many years, de­ emigrate to the U.S ., wrote from Iowa: "So you too, scribed how Go del usually re- my son, like Einstein and all other celebrities, could RIPE IDEALS. Philipp frained from speaking out, Furtwangler, a cousin of the but when he disagreed with not help it and had to cross the great water. Well famous conductor, was a first- something he heard, showed then, probably a permanent position will come out rate number theorist. He was his disagreement by a slight of it in the end, and the Germans and Austrians will partially paralyzed from the movement of his head again have lost a scientist (racially pure, this time)." neck down and had to be carried (Menger 1994). The sessions The words of Feigl were prophetic. The Circle of into the lecture hall in his chair. of the Vienna Circle gave Vienna disbanded rapidly. Menger was one of His greatest achievement came Godel many opportunities to those who left for the U.S. In 1937 he wrote to Alt, in 1929 (the year of Godel's do so, during discussions on who was still in Vienna: "I believe you should get completeness theorem), when Wittgenstein or Russell. Most he proved another conjecture of of the younger members of Hilbert, the Hauptidealsatz the Circle seem to have dis­ (principal ideal theorem) for played a healthy but discreet class fields, at the respectable skepticism towards their age of sixty. more outspoken seniors. Godel's correspondence in­ dicates that his closest friends in those days were Marcel Natkin and Herbert Feigl, two students of philosophy and mathematics who both did their Ph.D. with Schlick. Both venerated their professor but were not above poking gentle fun at him. "For consolation, I'll send you Schlick's essay, an ex­ ample that one can talk sensibly only about non­ sense. Did you hear from Feigl," wrote Marcel Natkin to Kurt Godel in the summer break of 1928, "how Wittgenstein and Schlick enjoyed speaking for hours about the unspeakable?" Hahn became Godel's thesis adviser but did not have to do much. A careful analysis of the Ph.D. the­ sis (as found in the university archives, reprinted in the Collected Works) and of the version pub­ I lished in the Monatshefte, suggests that Godel -· slightly adapted the latter to better fit the "party line" of Hahn (Feferman 1984). It was only twelve years later, after having solved two and a half of Hilbert's problems, that Godel started to express CLASSICAL TASTE. Heinrich Gomperz, his Platonism publicly. He then could argue that his philosopher, in a drawing by Egon Schiele. An success was due to the firmness of his conviction intriguing literary vignette of Gomperz can be on the reality of abstract concepts. found in the autobiography of Elias Canetti, In unison with Menger, Godel drifted away from who studied chemistry in the same building the Vienna Circle and became a member of an­ and at the same time as Godel and who later other circle, this time of younger mathematicians, won a Nobel Prize in literature.

430 N OTICES OF THE AMS VOLUME 53, N UMBER 4 together from time to time, and especially see that Godel takes part in the Kolloquium. It would not only be of greatest benefit for all other partici­ pants, but also for himself, though he might not realize it. Heaven knows into what he could entangle himself if he does not talk to you and the other friends in Vienna from time to time. If necessary, be pushy, on my say-so." But by the time that let­ ter reached him, Alt had to work urgently on his own escape. Menger would later ruminate that Gbdel needed to move within a sympathetic group, which would stimulate him to lecture frequently and gently re­ mind him to write things down, and even push him a little to do so, if necessary (Menger 1994). This is what Vienna had been able to provide, for a few blessed years. Hahn and Schlick had managed to create a critical mass (as one says nowadays) of young people studying both philosophy and math­ SKEPTICAL ATTITUDE. Marcel Natkin, a young philosopher ematics. But in those Viennese days the two top­ and student friend of Godel, eventually became one of the ics were in the air. The writers Hermann Broch and most eminent photographers in Paris. When Natkin, Feigl, and Robert Musil, and the philosophers Ludwig Wittgen­ Godel met in New York thirty years later, Godel wrote to his stein and Karl Popper had the same twin interests. mother: "The two have hardly changed. I do not know whether Their opinions were widely divergent (as were opin­ t he same t hing can be said of me." ions within the Circle); but that heterogeneity was could have found splendid teachers in Gbttingen also probably an advantage for Kurt Gbdel. He or Cambridge, but probably nowhere else a simi­ lar variety of views. For someone who was per­ fectly aware that his opinions were unfashionable to the extreme, and who wished to explore a com­ pletely new approach, this must have been en­ couraging. The main contribution of the Vienna Circle, then, rrtight have been to give everyone some clear state­ ments to clearly disagree with. Karl Popper is a case in point. For the last sixty years of his life he kept repeating that he did not mind at all never having been invited to a meeting of the Vienna Circle. But his huge first book, Die Logik der Forschung, ap­ peared in the series edited by Frank and Schlick, and Hahn had gently said "as kind words as I could have wished". (Popper 1995) Nevertheless, Popper took an almost oedipal pleasure in claiming, later, that the Vienna Circle was dead. "Who killed it? .. .I am afraid I did it." (Stadler 1997) Popper was as vehemently anti-Plato as Gbdel was Platonist. The two met but never were close. "I recently met a Mr. Popper (philosopher)," wrote Godel to Menger in 1934. "He has just finished a huge book in which, so he claims, all phil. problems are solved. -Do you think he is any good?" (Gbdel, Collected Works, 1986-2002) PHILOSOPHICAL CONFIDENCE. Herbert Feigl The novelist Hermann Broch also turned away (with bare feet) and Moritz Schlick (without) on from logical empiricism, but for entirely different the shores of Lake Millstaedter. Feigl, a student reasons. In the 1920s, Broch was a minor celebrity frie111d of Godel, later became professor of in the Viennese coffee-house scene, the well-off heir philosophy at the universities of Iowa and to a textile firm and a woma:nizer of renown. He Minnesota, and president ofthe American decided, at the age of forty, to study mathematics Philosophical Association. and philosophy and sat through many of the same

APRIL 2006 NOTICES OF THE AMS 431 Go del had no wish to return to Vienna after the war. "I am so happy to have escaped from beauti­ ful Europe," he would write to his mother. But if you happen, between May and August, to stop­ over in beautiful Europe, do come and visit the Gbdel exhibition. It is free! Acknowledgments The author wishes to acknowledge gratefully the use of material from the Go del Nachlass kept at the Firestone Library for Rare Manuscripts at Prince­ ton University, the archives of the lAS and the Uni­ versity of Vienna, the Handschriftensammlung of the Wiener Stadt- und Landesbibliothek and the Vi­ enna Circle Foundation in Amsterdam, in particu­ lar Monika Cliburn (-Schlick) and G. M. H. van de Vel de (-Schlick). Friedrich Stadler and John Dawson helped generously. Vas Orrf.,r des Morda"s~Da A-

432 NOTICES OF THE AMS VOLUME 53, NUMBER 4 About the Cover Bndgmg Support tor Physical/Computational Scientists Entenng Biology Mathematical theory of the Enigma machine 2007 Career Awards at the Scientific Interface This month's cover shows the first few pages of Alan Turing's treatise on the Enigma machine, from Deadline: May 1, 2006 the master copy typed by Turing himself in 1940, and $500,000 award over five years for postdoctoral fellows also containing Turing's sketches and annotations. I BWF IS ACCEPTING ELECTRONIC APP LI CATIONS ONLY I • These portable awards support up to two years of advanced postdoc­ toral training and the first three years of a faculty appoinrmenr • Candidates must hold a Ph.D. in mathematics, physics, biophysics, chemistry (physical, theoretical, or computational), computer science, statistics, or engineering and must not have accepted, either verbally or in writing, a faculty appoinrmenr at the time of application • Candidates should propose innovative approaches to answer important biological questions • BWF encourages proposals that include experimental validation of theoretical models • Degree-granting institutions in the U.S. and Canada may nominate It is now held as document HW2 513 in the up to two candidates National Archives of the U. K. (http: I lwww. • Complete program information, eligibility guidelines, and application nationalarchives.gov.uk). A copy made from instructions are available on BWF's website at www.bwfund.org microfilm, originating in American archives, can also be found in the Turing Digital Archive at t 919.991.5100 BURROUGHS j 919.991.5160 http:llwww.turingarchive.orglbrowse. WELLCOME www.bwfund.org phpiCI30. Post Office Box 13901 This year's theme for Mathematics Awareness 21 T. W. Alexander Drive Month is "Mathematics and Internet Security". (The FUND~ Rese.arch Triangle Park, NC 27709-3901 The Burroughs Wei/come Fund is an independent private foundation official website is http: I lwww. mathaware. dedicated to advancing the biomedical sciences by supporting research orgli ndex. html .) Mathematics pervades security and other scientific and educational activities. measures on the Internet, as Susan Landau showed in her article on hash functions in the March issue of the Notices, but it's a tough topic to illustrate. The theme is closely connected to the more general one of mathematics in cryptography, however, and in this regard little can compare in dramatic interest to the British work on reading German codes and ciphers at Bletchley Park during World War II. Polish math­ ematicians began the process early in the 1930s and British mathemati­ cans, most promi­ nently Alan Turing, ommunications in Partial were major contribu­ C Differential Equations publishes tors during the war. high quality papers concerning Turing also made ex­ any theoretical aspect of partial differential equations, as tremely important well as its applications to other areas of mathematics. contributions to the theory of com­ The journal presents the most significant advances in partial putability, not far re­ differential equations to a wide readership which includes moved from the in­ researchers and graduate students in mathematics and the terests of Kurt Go del, more mathematical aspects of physics and engineering. so it seems particu­ larly appropriate that PUBLICATION DETAILS his work appear in Volume 31, 2006 • Print ISSN: 0360-5302 • Online: 1532-4133 this issue, which con­ tains several articles To view more information on this journal, visit on Godel. http:/ /www.tandf.co.uk/journals/titles/03605302.asp. -Bill Casselman, Graphics Editor To subscribe, contact Customer Service at customerservice@ ([email protected]) taylorandfrancis.com or call Toll Free at: 800-354-1420, x216

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APRIL 2006 NOTICES OF THE AMS 433 The Impact of the Incompleteness Theorems on Mathematics Solomon Feferman

n addition to this being the centenary of Kurt ting that aside, my view of Godel's incompleteness Godel's birth, January marked 75 years since theorems is that their relevance to mathematical I the publication (1931) of his stunning in­ logic (and its offspring in the theory of computa­ completeness theorems. Though widely tion) is paramount; further, their philosophical rel­ known in one form or another by practicing evance is significant, but in just what way is far from mathematicians, and generally thought to say some­ settled; and finally, their mathematical relevance thing fundamental about the limits and potential­ outside of logic is very much unsubstantiated but ities of mathematical knowledge, the actual im­ is the object of ongoing, tantalizing efforts. My portance of these results for mathematics is little main purpose here is to elaborate this last assess­ understood. Nor is this an isolated example among ment. famous results. For example, not long ago, Philip Davis wrote me about what he calls The Paradox Informal and Formal Axiom Systems of Irrelevance: "There are many math problems One big reason for the expressed disconnect is that have achieved the cachet of tremendous sig­ that Godel's theorems are about formal axiom sys­ nificance, e.g., Fermat, four-color, Kepler's packing, tems of a kind that play no role in daily mathe­ Go del, etc. Of Fermat, I have read: 'the most famous matical work. Informal axiom systems for various math problem of all time'. Of Godel, I have read: kinds of structures are of course ubiquitous in 'the most mathematically significant achievement practice, viz. axioms for groups, rings, fields, vec­ of the 20th century' .... Yet, these problems have tor spaces, topological spaces, Hilbert spaces, etc., engaged the attention of relatively few research etc.; these axioms and their basic consequences are mathematicians-even in pure math." What ac­ so familiar it is rarely necessary to appeal to them counts for this disconnect between fame and rel­ explicitly, but they serve to define one's subject mat­ evance? Before going into the question for Godel's ter. They are to be contrasted with foundational theorems, it should be distinguished in one re­ axiom systems for the "mother" structures-the spect from the other examples mentioned, which natural numbers (Peano) and the real numbers in any case form quite a mixed bag. Namely, each (Dedekind)-on the one hand, and for the general of the Fermat, four-color, and Kepler's packing concepts of set and function (Zermelo-Fraenkel) problems posed a stand-out challenge following ex­ used throughout mathematics, on the other. Math­ tended efforts to settle them; meeting the challenge ematicians may make explicit appeal to the prin­ in each case required new ideas or approaches and ciple of induction for the natural numbers or the intense work, obviously of different degrees. By con­ least upper bound principle for the real numbers trast, Godel's theorems were simply unexpected, or the axiom of choice for sets, but reference to and their proofs, though requiring novel tech­ foundational axiom systems in practice hardly goes niques, were not difficult on the scale of things. Set- beyond that. One informal statement of the basic Peano ax­ Solomon Feferman is professor of mathematics and phi­ ioms for the natural numbers is that they concern losophy, emeritus, at Stanford University. His email address a structure (N, 0, s) where 0 is inN, the successor is sf@csl i. stanford. edu. function s is a unary one-one map from N into N

434 NOTICES OF THE AMS VOLUME 53, NUMBER 4 which does not have 0 in its range, and the Induc­ Consistency, Completeness, and tion Principle is satisfied in the following form: Incompleteness (IP) for any property P(x), if P(O) holds All such formal details are irrelevant to the work­ and if for all x inN, P(x) implies P(s(x)) ing mathematician's use of arguments by induction then for all x inN, P(x) holds. on the natural numbers, but for the logician, the way a formal system S is specified can make the But this is too indefinite to become the subject difference between night and day. This is the case, of precise logical studies, and for that purpose in particular, concerning the questions whether S one needs to say exactly which properties Pare ad­ is consistent, i.e., no contradiction is provable from missible in (IP), and to do that one needs to spec­ S, and whether S is complete, i.e., every sentence ify a formal language L within which we can sin­ A is decided by S in the sense that either S proves gle out a class of well-formed formulas (wffs) A A or S proves •A. If neither A nor •A is provable which express the admitted properties. And to do in S then A is said to be undecidable by S, and S that we have to prescribe a list of basic symbols is said to be incomplete. and we have to say which finite sequences of basic As an example of how matters of consistency and symbols constitute well-formed terms and which completeness can change dramatically depending constitute wffs. Finally, we have to specify which on the formalization taken, consider the subsys­ wffs are axioms (both logical and non-logical), and tem of PA obtained by restricting throughout to which relations between wffs are instances of rules terms and formulas that do not contain the mul­ of inference. The wffs without free variables are tiplication symbol x . That system, sometimes called those that constitute definite statements and are Presburger Arithmetic, was shown to be complete called the closed formulas or sentences of L. All by Moses Presburger in 1928, and his proof of of this is what goes into specifying a formal axiom completeness also gives a finite combinatorial systemS. proof ofits consistency.l Go del's discovery in 19 31 In the case of a formal version of the Peano ax­ was that we have a radical change when we move ioms, once its basic symbols are specified, and the to the full axiom system PA. What Gbdel showed logical symbols are taken to be • ("not"), 1\ ("and"), was that PAis not complete and that, unlike Pres­ v ("or"), = ("implies"), V ("for all"), and 3 ("there burger Arithmetic, its consistency cannot be es­ exists"), one puts in place of the Induction Princi­ tablished by finite combinatorial means, at least not ple an Induction Axiom Scheme: those that can be formalized in PA. Before going (IA) A(O) 1\ Vx(A(x) = A(s(x))) = VxA(x), into the mathematical significance of these results, where A is an arbitrary wff of the let us take a closer look at how Gi:idel formulated language L and A(t) indicates the result of and established them not only for PA, but also for 2 substituting the term t for all free a very wide class of its extensions 5. To do this occurrences of the variable x in A. he showed that the language of PAis much more expressively complete than appears on the sur­ N.B. (IA) is not a single axiom but an infinite col­ face. A primitive recursive (p.r.) function on N (in lection of axioms, each instance of which is given any number of arguments) is a function generated by some wff A of our language. from zero and successor both by explicit definition But what about the basic vocabulary of L? Be­ and definition by recursion along N. A p.r. relation sides zero and successor, nothing of number­ (which may be unary, i.e., a set) is a relation whose theoretical interest can be derived without ex­ characteristic function is p.r. Gi:idel showed that panding it to include at least addition and multi­ every p.r. function is definable in the language of plication. As shown by Dedekind, the existence of PA, and its defining equations can be proved there. those operations as given by their recursive defin­ For example, the operations of exponentiation xY, ing conditions can be established using (IP) ap­ the factorial x!, and the sequence of prime plied to predicates P involving quantification over functions. But for a formal axiom system PA ("Peano 1 Presburger's work was carried out as an "exercise" in a Arithmetic") for elementary number theory, in seminar at the University of Warsaw run by Alfred Tarski. which one quantifies only over numbers, one needs His proof applies the method of elimination of quantifiers to posit those operations at the outset. The basic to show that every formula is equivalent to a proposi­ vocabulary of PAis thus taken to consist of the con­ tional combination of congruences. At its core it makes use of the Chinese Remainder Theorem giving conditions for stant symbol 0 and the operation symbols s, +, the existence of solutions of simultaneous congruences. and x together with the relation symbol=. Then 2 the axioms indicated above for zero and successor Gdde/'s initial statement of his results was for extensions of a variant P of the system of Principia Mathematica, but are supplemented by axioms giving the recursive a year later he announced his results more generally for characterizations of addition and multiplication, a system like P A in place of P; no new methods of proof namely: x + 0 = x, x + s(y) = s(x + y), x x 0 = 0, and were required. Nowadays it is known that much weaker X X s(y) =(X X y) +X. systems than P A suffice for his results.

APRIL 2006 NOTICES OF THE AMS 435 numbers Px, each of which is p.r., can all be rep­ Automatically, every 1-consistent system is con­ resented in this way in PA, facts that are not at all sistent, but the converse is not true: by (1), if S is obvious.3 Each instance of a p.r. relation is decid­ consistent it remains consistent when we add •Ds able by PA; for example if R is a binary p.r. rela­ to it as a new axiom, and the resulting system is tion then for each n, mEN, either PA proves not 1-consistent. The following is Gbdel's first in­ R(n, m) or it proves •R(n, m). completeness theorem:5 Godel's Incompleteness Theorems (2) ifS is 1-consistent then Ds is To apply these notions to the language and de­ undecidable by S; hence Sis not complete. ductive structure of PA, Godel assigned natural It turns out that only the first part, (1), is needed numbers to the basic symbols. Then any finite se­ for his second incompleteness theorem. Let C be quence (]" of symbols gets coded by a number#(]", the sentence •(0 = 0), so S is consistent if and only say, using prime power representation;#(]" is nowa­ if Cis not provable inS; this is expressed by the days called the Go del number (g.n.) of(]". A relation '\/-sentence •Provs(#C), which is denoted Cons. By R between syntactic objects (terms, formulas, etc.) formalizing the proof of (1) it may be shown that is said to be p.r. if the corresponding relation be­ the formal implication Cons = Provs(#Ds) is tween g.n.'s is p.r. For example, with a basic finite provable in PA. But since •Provs(#Ds) = Ds in vocabulary, the sets of terms and wffs are both p.r. PA by the diagonal construction, we have Finally a formal systemS for such a language is said Cons Ds too. Hence: to be p.r. if its set of axioms and its rules of infer­ = ence are both p.r. The formal system PA and its sub­ (3) if Sis consistent then S does not prove Cons. systems defined above are all p.r. That is Gbdel's second incompleteness theo­ Throughout the following, S is assumed to be rem. Its impact on Hilbert's consistency program any p.r. formal system that extends PA. Denote by has been much discussed by logicians and histo­ Proofs(x, y) the relation which holds just in casey rians and philosophers of mathematics and will not is the g.n. of a proof inS x. of a wff with g.n. Then be gone into here, except to say that it is generally Proofs(x, y) is p.r. Using its definition in PA, the for­ agreed that the program as originally conceived can­ mula :Jy Proofs(x, y) expresses that x is the g.n. of not be carried out for PA or any of its extensions. a provable formula; this is denoted Provs (x). Finally, However, various modified forms of the program for each wff A, Provs(#A) expresses in the language have been and continue to be vigorously pursued of PA that A is provable. By a diagonal argument, within the area of logic called proof theory, inau­ Godel constructed a closed wff Ds which is prov­ gurated by Hilbert as the tool to carry out his pro­ ably equivalent in PA to •Provs(#Ds ); more loosely, gram. I recommend Zach (2003) (readily accessible "Ds says of itself that it is not provable inS." And, online) for an excellent overview and introduction indeed, to the literature on Hilbert's program. (1) if Sis consistent, Ds is not provable inS. Godel's Theorems and Unsettled Hence, in ordinary informal terms, Ds is true, Mathematical Problems so S cannot establish all arithmetical truths. This With this background in place we can now return is one way that Godel's first incompleteness theo­ to the question of the impact of the incompleteness rem is often stated, but actually (1) is only the first theorems on mathematics; in that respect it is part of the way that he stated it. For that we need mainly the first incompleteness theorem that is of a few more slightly technical notions. A sentence concern, and indeed only the first part of it, namely A of the language of PAis said to be in :3-form if, (1). A common complaint about this result is that up to equivalence, it is of the form :JyR(y) where it just uses the diagonal method to "cook up" an R defines a p.r. set, and A is said to be in '\/-form example of an undecidable statement. What one if, up to equivalence, •A is in :3-form, or what would really like to show undecidable by PA or comes to the same, if A can be expressed in the some other formal system is a natural number­ form '\/yR (y) with R1 p.r.4 Thus Ds is in'\/ -form 1 theoretical or combinatorial statement of prior in­ and its negation is in :3-form. S is said to be terest. The situation is analogous to Cantor's use 1-consistentif every :3-sentence provable inS is true. of the diagonal method to infer the existence of tran­ scendental numbers from the denumerability of 3 The way that is done might interest number theorists; see Franzen (2004), Ch. 4, for an exposition. the set of algebraic numbers; however, that did not provide any natural example. The existence of 4 Standard logical terminology for :3 -form and '

436 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Neither argument helped to show that e and rr, p(x1, ... , Xn) = 0, where pis a suitable polynomial among other reals, are transcendental, but they with integer coefficients. So each '\1 -sentence A is did at least show that questions of transcendence equivalent to the non-solvability of a suitable dio­ are non-vacuous. Similarly, Godel's first incom­ phantine equation, in particular, sentences known pleteness theorem shows that the question of to be undecidable in particular systems such as the undecidability of sentences by PA or any one of its Godel sentences Ds. The trouble with this result consistent extensions is non-vacuous. That sug­ compared to open questions in the literature about gests looking for natural arithmetical statements which have resisted attack so far to try to see the solvability of specific diophantine equations in whether that is because they are not decided by sys­ two or three variables or of low degree is that the tems that formalize a significant part of mathe­ best value known for the above representation in matical practice, and in particular to look for such terms of number of variables is n = 9, and in terms statements in '\1 -form. An obvious candidate for a of degree d with a much larger number of variables long time was the Fermat conjecture; now that we is d = 4 (cf. Jones 1982). know it is true, it would be interesting to see just what principles are needed to establish it from a log­ Combinatorial Independence Results ical point of view. Some logicians have speculated Things look more promising if we consider '\13- that it has an elementary proof that can be for­ sentences, i.e., those that can be brought to the form malized in PA, but we don't have any evidence so "ix3yR(x, y) with R p.r.6 The statement that there far to settle this one way or the other. Another ob­ are infinitely many y's satisfying a p.r. condition vious candidate is the Goldbach conjecture; in­ P(y) is an example of a '\1 3-sentence, since it is ex­ deed, Godel often referred to his independent state­ pressed by "ix3y(y > x A P(y)). ments as being "of Goldbach type", by which he In particular, the simply meant that they are both expressible in '\1- twin prime conjecture has this form. Again, no form. A far less obvious candidate is the Riemann problems of prior mathematical interest that are Hypothesis; Georg Kreisel showed that this is equiv­ in '\1 3-form have been shown to be undecidable in alent to a '\1 -statement (see Davis, Matijasevic, and PA or one ofits extensions. However, in 1977, Jeff Robinson (1976), p. 335, for an explicit presenta­ Paris and Leo Harrington published a proof of the tion of such a sentence). No example like these has independence from PA of a modified form of the been shown to be independent of PA or any of its Finite Ramsey Theorem. The latter says that for presumably consistent extensions. each n, r and k there is an m such that for every r-colored partition rr of the n-element subsets of Unsolvable Diophantine Problems M = {l, m} there is a subset H of M of cardinality Consider any systemS containing P Athat is known at least k such that His homogeneous for rr, i.e., or assumed to be consistent and suppose that A all n-element subsets of H is a '\1 -sentence conjectured to be undecidable by are assigned the same S. It turns out that proving its undecidability would color by rr. The Paris-Harrington modification automatically show A to be true, since •A is equiv­ consists in adding the condition that card(H) 2: alent to a 3-sentence 3yR(y); thus if •A were true min(H). It may be seen that this statement, call it there would be an n such that R(n) is true, hence PH, is in '\13-form. Their result is that PH is not provable in PA and thence in S. So, finally, •A provable in PA. But they also showed that PH is would be provable in S, contradicting the sup­ true, since it is a consequence of the Infinite Ram­ posed undecidability of A by S. The odd thing sey Theorem. The way that PH was shown to be in­ about this is that if we want to show a '\1 -sentence dependent ofPA was to prove that it implies ConrA; undecidable by a givenS, we better expect it to be in fact, it implies the formal statement of the true. And if we can show it to be true by one means 1-consistency of PA. That work gave rise to anum­ or another, who cares (other than the logician who ber of similar independence results for stronger sys­ is interested in exactly what depends on what) tems S, in each case yielding a '\1 3 -sentence As whether it can or can't be proved inS? Still, the first incompleteness theorem is tantalizing for its which is a variant of a combinatorial result already prospects in this direction. The closest that one has in the literature such that As is true but unprovable come is due to the work of Martin Davis, Hilary from S on the assumption that S is 1-consistent. The Putnam, Julia Robinson, and Yuri Matiyasevich re­ proof consists in showing that As implies (and is sulting, finally, in the latter's negative resolution in some cases equivalent to) the formal statement of Hilbert's lOth Problem on the algorithmic solv­ of its 1-consistency. However, no example is known ability of diophantine equations (cf. Matiyasevich of an unprovable '\13-sentence whose truth has 1993). It follows from their work that every 3- been a matter of prior conjecture. sentence is equivalent in P A to the· existence of natural numbers X1, ... ,xn such that 6 Standard logical terminology for these is rrg sentences.

APRIL 2006 NOTICES OF THE AMS 437 Set Theory and Incompleteness latter axiom, roughly speaking, means Godel signaled a move into more speculative ter­ nothing else but that the totality of sets ritory in footnote 48a (evidently an afterthought) obtainable by exclusive use of the of his 1931 paper: processes of formation of sets ex­ pressed in the other axioms forms again As will be shown in Part II of this paper, a set (and, therefore, a new basis for a the true reason for the incompleteness further application of these processes). inherent in all formal systems of math­ Other axioms of infinity have been for­ ematics is that the formation of ever mulated by P. Mahlo .... These axioms higher types can be continued into the show clearly, not only that the axiomatic transfinite ... For it can be shown that the system of set theory as known today is undecidable propositions constructed incomplete, but also that it can be sup­ here become decidable whenever ap­ plemented without arbitrariness by new propriate higher types are added ... An axioms which are only the natural con­ analogous situation prevails for the ax­ tinuation of those set up so far. (Godel iom system of set theory_? 1947, as reprinted in 1990, p. 182). The reason for this, roughly speaking, is that for Whether or not one agreed with Godel about the each systemS the notion of truth for the language ontological underpinnings of set theory and in par­ of S can be developed in an axiomatic system S' ticular about the truth or falsity of CH, in the fol­ for the subsets of the domain of interpretation of lowing years it was widely believed to be indepen­ S; then inS' one can prove by induction that the dent of ZFC; that was finally demonstrated in 1963 statements provable inS are all true, and hence that by Paul Cohen using a new method of building S is consistent. Implicit in the quote is that S' models of set theory. And, contrary to Godel's ex­ ought to be accepted if S is accepted. Later, in his pectations, it has subsequently been shown by an famous article on Cantor's Continuum Problem expansion of Cohen's method that CHis undecid­ (1947), Godel pointed to the need for new set­ able in every plausible extension of ZFC that has theoretic axioms to settle specifically set-theoretic been considered so far, at least along the lines problems, such as the Continuum Hypothesis (CH). of Godel's proposal (cf. Martin 1976 and Kanamori At that point, it was only known as a result of 2003). For the most recent work on CH, see the con­ his earlier work that AC (the Axiom of Choice) and clusion of Floyd and Kanamori (this issue). CH are consistent relative to Zermelo-Fraenkel ax­ But what about arithmetical iomatic set theory ZF. 8 In Godel's 1947 article he problems? For a argued thatCH is a definite mathematical problem, number of years, Harvey Friedman has been work­ and, in fact, he conjectured that it is false while all ing to produce mathematically perspicuous finite combinatorial 'v'3-statements A whose proof re­ the axioms of ZFC (= ZF + AC) are true. Hence CH must be independent of ZFC; he thus concluded quires the use of many Mahlo cardinals and even that one will need new axioms to determine the car­ stronger axioms of infinity (such as the so-called dinal number of the continuum. In particular, he subtle cardinals) and has come up with a variety of suggested for that purpose the possible use of ax­ candidates; for a fairly recent report, including work ioms of infinity not provable in ZFC: in progress, see Friedman (2000).9 The strategy for establishing that such a statement A needs a sys­ The simplest of these ... assert the ex­ temS incorporating the strong axioms in question istence of inaccessible numbers .. . The is like that above: one shows that A is equivalent to (or in certain cases is slightly stronger than) the 7 Part II of Code/ (1931) never appeared. Also promised !-consistency of S. In my discussion of this in Fe­ for it was a full proof of the second incompleteness theo­ ferman (2000), p. 407, I wrote: "In my rem, the idea for which was only indicated in Part I. He view, it is beg­ later explained that since the second incompleteness the­ ging the question to claim this shows we need ax­ orem had been readily accepted there was no need to ioms of large cardinals in order to demonstrate the publish a complete proof Actually, the impact of Code/'s truth of such A, since this only shows that we work was not so rapid as this suggests; the only one who 'need' their !-consistency. However plausible we immediately grasped the first incompleteness theorem might find the latter for one reason or another, it was John von Neumann, who then went on to see for him­ doesn't follow that we should accept those axioms self that the second incompleteness theorem must hold. Others were much slower to absorb the significance of 9 More recently, Friedman has announced the need for such Code/'s results (cf Dawson 1997, pp. 72-75.) The first de­ large cardinal axioms in order to prove a certain combi­ tailed proof of the second incompleteness theorem for a natorial statement A that can be expressed in V -form; see system Z equivalent to PA appeared in Hilbert and Bernays the final section ofDavis (this issue). Here, A implies Cons (1939). and is itself provable in S' for S and S' embodying suit­ 8 See Floyd and Kanamori in this issue of the Notices. able large cardinal axioms.

438 NOTICES OF THE AMS VOLUME 53, NUMBER 4 themselves as first-class mathematical principles." Mathematics 28 (1976), Amer. Math. Soc., Providence, (Cf. also op. cit. p. 412). RI. [2] M. DAVIS, The incompleteness theorem (this issue of Prospects and Practice the No tices). As things stand today, these explorations of the set­ [3] M. DAVIS, Y. MATIJASEVIC, and]. ROBINSON, Hilbert's tenth theoretical stratosphere are clearly irrelevant to problem. Diophantine equations: positive aspects of the concerns of most working mathematicians. A a negative solution, in Browder (1976), 323-378. reason for this was also given by Godel near the [4]]. W. DAWSON ]R., Logical Dilemmas. The Life and Work outset of his 1951 Gibbs lecture (posthumously of Kurt Code/, A. K. Peters, Ltd., Wellesley, MA, 1997. published in 1995), where he said that the "phe­ [5] S. FEFERMAN, In the Light of Logic, Oxford University nomenon of the inexhaustibility of mathematics" Press, New York, 1998. follows from the fact that "the very formulation of [6] __ , Why the programs for new axioms need to be the axioms [of set theory over the natural numbers] questioned, Bull. Symbolic Logic 6 (2000), 401-413. up to a certain stage gives rise to the next axiom. [7]]. FLOYD and A. KANAMORJ, How Godel transformed set theory (this issue of the Notices). It is true that in the mathematics of today the [8] H. FRIEDMAN, Normal mathematics will need new axioms, higher levels of this hierarchy are practically never Bull. Symbolic Logic 6 (2000), 434-446. used. It is safe to say that 99.9% of present-day [9] T. FRANZEN, Inexhaustibility, Lecture Notes in Logic 28 mathematics is contained in the first three levels (2004), Assoc. for Symbolic Logic, A.K. Peters, Ltd., of this hierarchy." ln fact, modern logical studies Wellesley (distribs.). have shown that the system corresponding to the [10] K. GODEL, Ober formal unentscheidbare Satze der second level of this hierarchy, called second-order Principia Mathematica und verwandter Systeme I, arithmetic or analysis and dealing with the theory Monatshefte fiir Mathematik und Physik 38 (1931), of sets of natural numbers, already accounts for the 173-198. Reprinted with an English translation in bulk of present-day mathematics. Indeed, much Godel (1986), 144-195. weaker systems suffice, as is demonstrated in [11] __ ,What is Cantor's continuum problem?, A mer. Simpson (1999). Even more, I have conjectured and Math. Monthly 54 (1947), 515-525, errata, 55, 151. verified to a considerable extent that all of current Reprinted in Godel (1990), 176-187. scientifically applicable mathematics can be for­ [12] __ , Some basic theorems on the foundations of malized in a system that is proof-theoretically no mathematics and their implications, in Godel (1995), stronger than PA (cf. Feferman 1998, Ch. 14). pp. 304-323. [The 1951 Gibbs lecture.] Whether or not the kind of inexhaustibility of [13] __ , Collected Works, Vol. I. Publications 1929- 1936, mathematics discovered by Godel is relevant to (S. Feferman, et al., eds.), Oxford University Press, New present-day pure and applied mathematics, there York, 1986. is a different kind of inexhaustibility which is [14] __ , Collected Works, Vol. II. Publications 1938- clearly significant for practice: no matter which 19 7 4, (S. Feferman, et al., eds.), Oxford University axiomatic systemS is taken to underlie one's work Press, New York, 1990. at any given stage in the development of our sub­ [15] __ , Collected Works, Vol. III. Unpublished Essays ject, there is a potential infinity of propositions that and Lectures, (S. Feferman, et al., eds.), Oxford Uni­ can be demonstrated in S, and at any moment, versity Press, New York, 1995. only a finite number of them have been estab­ [16] D. HILBERT and P. BERNAYS, Grund/agen der Mathe­ matik, Vol. II, Springer-Verlag, lished. Experience shows that significant progress Berlin, 1939. Second, re­ vised edition, 1968. at each such point depends to an enormous extent [17] ]. P. ]ONES, Universal Diophantine equation, ]. Sym­ on creative ingenuity in the exploitation of ac­ bolic Logic47 (1982), 549-571. cepted principles rather than essentially new prin­ [18] A. KANAMORJ, The Higher Infinite, Springer-Verlag, ciples. But Godel's theorems will always call us to Berlin, 2nd edition, 2003. try to find out what lies beyond them. [19] D. A. MARTIN, Hilbert's first problem: The continuum Note to the reader: Godel's incompleteness pa­ hypothesis, in Browder (1976), 81-92. per (1931) is a classic of its kind; elegantly organized [20] Y. MATIYASEVICH, Hilbert's Tenth Problem, MIT Press, and clearly presented, it progresses steadily and ef­ Cambridge, 1993. ficiently from start to finish, with no wasted energy. [21] ]. PA.rus and L. HARRINGTON, A mathematical incom­ The reader can find it in the German original along pleteness in Peano Arithmetic, in Handbook of Math­ with a convenient facing English translation in ematical Logic, (]. Barwise, ed.), 1133-1142, North­ Vol. I of his Collected Works (1986). I recommend Holland, Amsterdam, 1977. it highly to all who are interested in this landmark [22] S. G. SIMPSON, Subsystems of Second Order Arithmetic, in the history of our subject. Springer-Verlag, Berlin, 1999. [23] R. ZAcH, Hilbert's program, Stanford Encyclopedia References of Philosophy, (E. N. Zalta, ed.), http: I /plato. [1] F. E. BROWDER (ed.), Mathematical Developments Aris­ stanford.edu/archives/fall2003/ entries/ ing from Hilbert Problems, Proc. Symposia in Pure hilbert-program/(2003).

APRIL 2006 NOTICES OF THE AMS 439 The Popular Impact of GO del's Incompleteness Theorem Torkel Franzen

mong Godel's celebrated results in logic, strong axiomatic theory is either incomplete or in­ there are two that can be formulated in consistent." Many nonmathematicians at once find terms that are intelligible in a general this fascinating and are ready to apply what they A way even to those unfamiliar with the take to be the incompleteness theorem in many dif­ technicalities involved. The first is his ferent contexts. The task of the expositor becomes, completeness theorem for first order logic. This the­ rather, to dampen their spirits by explaining that orem, which is not widely known outside the world the theorem doesn't really apply in these contexts. of logic, can be formulated as saying that every But as experience shows, even the most deter­ statement that follows logically from a set of ax­ mined wet blanket cannot prevent people from ap­ ioms in a formalized language, such as that used pealing to the incompleteness theorem in contexts in Zermelo-Fraenkel set theory with the axiom of where its relevance is at best a matter of analogy choice (ZFC) or first order Peano Arithmetic (PA), or metaphor. This is true not only of the first in­ can be proved using those axioms and the rules of completeness theorem (as formulated above), but logic. A general audience of nonmathematicians also of the second incompleteness theorem, about would probably find this statement of the com­ the unprovability in a consistent axiomatic theory pleteness theorem intelligible but unexciting. After T of a statement formalizing "Tis consistent." all, isn't that what "follows logically" means? It is Supposed applications of the first incomplet ~­ no easy task to explain the distinction between ness theorem in nonmathematical contexts usually the semantic concept of logical consequence and disregard the fact that the theorem is a statement the purely formal notion of derivability so as to about formal systems and is stated in terms of bring out the importance of this result for an au­ mathematically defined concepts of c on~istency dience unacquainted with logic or mathematics. By and completeness. This mathematically ~'ssent

440 NOTICES OF THE AMS VOLUME 53, N UMBER 4 the Bible is inconsistent, they are arguing that it con­ only to formal mathematical theories. This is a line tains apparently irreconcilable statements, and of thought that tends to appeal to postmodernists those who regard the Bible as a complete guide to and theologians. From this point of view, the in­ life presumably mean that they can find answers completeness theorem shows that even in mathe­ in the Bible to all questions that confront them matics, that supreme bastion of reason, truth is ei­ about how to live their lives. Einstein, in regarding ther beyond us or a matter of more or less arbitrary quantum mechanics as incomplete (although con­ consensus rather than objective fact. Given our sistent), believed that it is possible to find a more most powerful mathematical theory, we know that, fundamental physical theory. The system of laws assuming its consistency, we can produce an arith­ of any country is incomplete or inconsistent or metical statement such that we can add either that both in the sense that there are always situations statement or its negation to the theory, obtaining in which conflicting legal arguments can be brought incompatible theories that are still consistent. So to bear, or in which no statute seems applicable. either reason is powerless in this context (as in the But the incompleteness theorem adds nothing to wider context of the universe as a whole, with truth such claims or observations, for two reasons. The ultimately resting only in God), or there is no other first is that these "systems" are not at all formal truth than that which we more or less arbitrarily systems in the logical sense. There is no formally agree upon (just as in the physical sciences, ac­ specified language, and there are no formal rules cording to this line of thought). Either way, after of inference in the logical sense associated with Godel's theorem, mathematics flounders in a sea quantum mechanics, the Bible, a system of laws, of undecidability. and so on. What follows or does not follow from a When we look at mathematical practice, however, religious or philosophical text, a scientific theory, we find that mathematicians, although generally or a system of laws is not determined by any for­ aware of the phenomenon of incompleteness, and mal rules of inference, such as might be imple­ therewith of the theoretical possibility that a prob­ mented on a computer, but is very much a matter lem they are working on may be unsolvable in the of interpretation, argument, and opinion, where the current axiomatic framework of mathematics, are relevant reasoning is limited only by the vast bound­ by no means floundering in a sea of undecidabil­ aries of human thought in scientific, religious, po­ ity. litical, or philosophical contexts. In the year 1900 David Hilbert made a famous The second reason for the irrelevance of Godel's affirmation in his presentation of twenty-three theorem in such discussions is that the incom­ problems facing mathematics in the twentieth cen­ pleteness of any sufficiently strong consistent ax­ tury [Browder 1976). At first glance, it might be iomatic theory established by that theorem con­ thought that the incompleteness theorem scuttles cerns only what may be called the arithmetical the confidence expressed in this affirmation: component of the theory. A formal system has such Take any definite unsolved problem, a component if it is possible to interpret some of such as the question as to the irra­ its statements as statements about the natural tionality of the Euler-Mascheroni con­ numbers, in such a way that the system proves stant C, or the existence of an infinite some basic principles of arithmetic. Given this, we number of prime numbers of the form can produce (using Rosser's strengthening of zn + 1. However unapproachable these Godel's theorem in conjunction with the proof of problems may seem to us, and however the Matiyasevich-Davis-Robinson-Putnam theorem helpless we stand before them, we have, about the representability of recursively enumer­ nevertheless, the firm conviction that able sets by Diophantine equations) a particular their solution must follow by a finite statement of the form "The Diophantine equation number of purely logical processes. p(x1, ... , Xn) = 0 has no solution" which is unde­ .. .This conviction of the solvability of cidable in the theory, provided it is consistent. every mathematical problem is a pow­ i While it is mathematically a very striking fact that I erful incentive to the worker. We hear any sufficiently strong consistent formal system is within us the perpetual call: There is the incomplete with respect to this class of statements, problem. Seek its solution. You can find it is unlikely to be thought interesting in a non­ it by pure reason, for in mathematics mathematical context where completeness or con­ there is no ignorabimus. sistency (in some informal sense) is at issue. No­ body expects the Bible, the laws of the land, or the Although Hilbert did not specify just what he philosophy of Ayn Rand to settle every question in meant by a "definite" problem, it is no doubt sig­ arithmetic. nificant that his two examples are statements that There is also a different kind of appeal to the can be formulated in arithmetical terms. Today, first incompleteness theorem outside mathemat­ mathematicians have accepted 'that some appar­ ics, one that recognizes that the theorem applies ently straightforward questions in set theory, such

APRIL 2006 NOTICES OF THE AMS 441 as the very first problem on Hilbert's list, that of point of view, it would be immensely interesting if the truth or falsity of Cantor's continuum hy­ some famous conjecture in arithmetic turned out pothesis, cannot in fact be settled by a mathemat­ to be undecidable in ZFC, but it seems too much ical proof as proof is ordinarily understood in to hope for. In short, while Hilbert's affirmation mathematics today. Instead, we must either rest does not have any theoretical support from logic, content with proving hypothetical statements such logic does not refute that affirmation, as naturally as "Assuming CH, there is a group G with the prop­ understood from the point of view of the working erties ..." or extend set theory by new axioms. Also, mathematician. mathematicians who work in set theory or areas It is thus understandable that the first incom­ closely connected with set theory have learned to pleteness theorem has not had much of a "popu­ recognize the kind of problem or conjecture that lar impact" among mathematicians, who are un­ may well be affected by the incompleteness of set likely to seek to apply a mathematical theorem to theory. (It should be emphasized that this category the Bible and so on, and who are, for the reasons of incompleteness, although established on the indicated, not overly concerned about the possi­ basis of the pioneering work in set theory by Godel, bility of an arithmetical problem that they are and later Paul Cohen, is not a consequence of the working on being unsolvable in current mathe­ incompleteness theorem.) matics. Mathematics may be "floundering" as far The situation is different with Hilbert's exam­ as solving a particular problem is concerned, but ples of "definite unsolved problems." It would be this neither leads to any inclination to regard prob­ startling indeed if it turned out that ZFC does not lems such as those mentioned by Hilbert as in any settle whether or not there are infinitely many Fer­ way solvable by fiat or consensus, nor instills any mat primes. In such a case, very few mathemati­ sense that the problem may be unsolvable. This nat­ cians would be content to note that we can con­ ural attitude is, furthermore, supported both by ex­ sistently take the number of Fermat primes to be perience and by logical and philosophical argu­ either finite or infinite, and leave it at that. Rather, ment, as briefly touched on above. mathematical instinct, if nothing else, tells us that The second incompleteness theorem, although whether or not there are infinitely many Fermat not as often referred to in nonmathematical con­ primes is not a question that can be meaningfully texts, has also prompted theologians and post- · settled by stipulation, but if it can be settled at all modernists to reflect that since mathematics can­ calls for an argument that we perceive as mathe­ not prove its own consistency, reason is powerless matically compelling. Given such an incompleteness to justify itself, so that either there is no justifica­ result, the search for new axioms in mathematics tion to be had, or reason can be supported only by would take on a new urgency. faith. Without going into detail about such ideas, However, no famous arithmetical conjecture has it is a relevant observation that a somewhat simi­ been shown to be undecidable in ZFC. We do know lar line of thought seems to have had considerable that certain natural statements formalizable in "popular impact" even among mathematicians, in arithmetic are undecidable in ZFC (given the con­ their more philosophical moments. What I have in sistency, or more accurately what is called in logic mind here is the following. Mathematicians often the 1-consistency, of ZFC), typically consistency tend to regard proofs of consistency, not only of statements, such as a statement formalizing "ZFC ZFC but of such very much weaker theories as PA, is consistent." Here again mathematical instinct as somehow more unattainable or problematic tells us that whether or not ZFC is consistent can­ than proofs of ordinary arithmetical statements. In­ not be meaningfully settled by stipulation, but deed, it is not uncommon for mathematicians ~ to statements of this kind are not at all what mathe­ say that arithmetic cannot be proved consistent. maticians normally seek to prove. Mathematicians Thus Ian Stewart, in summing up the second in­ tend to be content with accepting that the consis­ completeness theorem for popular con&umption, tency of the most powerful formal theory to which remarks that "Goedel showed that.. .if an)fone finds they ordinarily refer in foundational contexts can­ a proof that arithmetic is consistent, then it isn't!" not be proved in ordinary mathematics, without ([Stewart 1996], p. 262) thereby concluding that their own mathematical ef­ What is odd about such remarks is that we can forts are likely to run up against the barrier of un­ easily, indeed trivially, prove PA consistent using decidability. For, while we have no basis for a gen­ reasoning of a kind that mathematicians other­ eral claim that every arithmetical problem that wise use without qualms in proving theorems of arises naturally in mathematics is decidable in ZFC, arithmetic. Basically, this easy consistency proof ob­ we don't have a single example of an arithmetical serves that all theorems of PA are derived by valid problem-about primes or Diophantine equations logical reasoning from basic principles true of the or other such things-that has occurred to math­ natural numbers, so no contradiction is derivable ematicians in a natural mathematical context being in PA. It appears that many mathematicians have shown to be unsolvable in ZFC. From a logician's come to absorb the view that a consistency proof

442 NOTICES OF THE AMS VOLUME 53, NUMBER 4 for PAis not really a consistency proof unless it existence of the objects studied in mathematics or convinces somebody who does not accept the ax­ the philosophical justifiability of mathematics, and ioms of PA as expressing valid principles of math­ as far as we know, mathematics as it stands today ematics that PA is nevertheless consistent. The is consistent. But such a view is at odds with how second incompleteness theorem and general ex­ we actually think about arithmetical problems in perience do indeed indicate that no such proof is mathematics. For example, there is no logical basis to be expected. If we were to make similar de­ for claiming that there are infinitely many twin mands on proofs of arithmetical statements in primes if all we know is that PAis consistent and general, we would be forced to the conclusion that proves the twin prime conjecture. Consistent the­ it is equally impossible to prove the prime num­ ories of arithmetic may prove false theorems (when ber theorem, Dirichlet's theorem, and so on. The we are not talking about theorems having the log­ insight underlying the idea that it is impossible to ical form of Goldbach's conjecture), and if we con­ prove, in this sense, the consistency of arithmetic clude that there are infinitely many twin primes on is a perfectly valid one, but it has nothing to do with the basis of a proof in some particular mathemat­ Gbdel's theorem. Instead it is the insight, familiar ical framework, the mere consistency of that frame­ since antiquity, that we cannot prove everything. work is insufficient to justify our conclusion. We need to start from some basic principles in our There is of course much more that could be said mathematical reasoning, principles that we can about the impact of the incompleteness theorem justify only in informal terms. The principles for­ outside the field of logic proper. For one thing, malized in PA are the infinity of the natural num­ there is the whole subject of Lucas-Penrose argu­ ber series, the basic properties of addition and ments in the philosophy of mind, which seek to es­ multiplication, and the principle of mathematical tablish that the human mind does not work on me­ induction. As long as we accept these principles as chanical principles in mathematics by appealing to mathematically valid-as a large majority of math­ the incompleteness theorem. A more extensive dis­ ematicians do in practice-there is no reason why cussion can be found in [Franzen 2005]. we should not accept a proof of the kind described as proving the consistency of PA, just as we accept References other mathematical proofs that depend on the va­ [1] F. BROWDER (ed.), Mathematical Developments Arising lidity of these principles. Those who do have gen­ from Hilbert's Problems, American Mathematical Soci­ uine doubts about the consistency of PA will of ety, 1976. (Hilbert's address was translated into Eng­ lish by Mary Winston Newson for the Bulletin of the course not accept this proof of consistency, but American Mathematical Society, 1902, and a reprint is then there is no reason why they should accept stan­ included in this AMS volume.) dard proofs of the prime number theorem, Dirich­ [2] T. FRANZEn, Gddel's Theorem: An Incomplete Guide to let's theorem, and so on, either. its Use and Abuse, A K Peters, 2005. It should be noted that in the logical literature, [3]lAN STEWART, From Here to Infinity, Oxford University there are various nontrivial consistency proofs for Press, 1996. PA, but the question of their interest and content is a subtle one, and I think it can be safely said that they will not convince anybody who has genuine doubts about the consistency of PA. Of course, the above comments do not apply to every question of consistency. For example, the consistency of PA extended with the axiom "PAis inconsistent" is established only through the proof of the second incompleteness theorem itself, and proving the consistency of PA extended with Gold­ bach's conjecture as a new axiom is equivalent to proving Goldbach's conjecture. In these cases, the theory whose consistency is at issue is not one that formalizes basic principles of mathematical reasoning. A point that deserves to be made in this con­ nection is that the significance of consistency proofs as a means of justifying our mathematical reason­ ing is easily overstated. For a mathematician, it may at times seem convenient to refer to consistency in response to philosophical prodding about the truth or validity of mathematical axioms and methods of reasoning: only consistency matters, not the

APRIL 2006 NOTICES OF THE AMS 443 In Quest of Kurt GOdel: Reflections of a Biographer john W. Dawson]r.

ince the days of E. T. Bell the writing of in part, perhaps, because the private nature of mathematical biographies has gradually mathematical research is less amenable to the so­ Smatured. Nevertheless, chronicling the life ciological kind of analysis that is presently so fash­ of a mathematician remains a difficult un­ ionable in historical studies of the natural sciences. dertaking. The question of audience is Because of the highly technical nature of mod­ paramount, and the response to it determines both ern mathematics one might presume that only a the style of presentation and the level of detail. Bal­ mathematician can adequately understand and ex­ ancing the demands of lucidity and mathematical plain the work of another. But a biography is not accuracy against the constraints of popular un­ a textbook. It is a portrayal of a life. And recount­ derstanding presents a particularly vexing chal­ ing the life of a mathematician requires a sensitivity lenge. to human values as well as an understanding of the The observations that follow are based upon details of theorems and proofs. my own experiences in writing a biography of Kurt All too often, biographical memoirs written by GOdel [Dawson 1997]. I hope that others who con­ mathematicians are anecdotal in nature and focus sider becoming involved in biographical endeavors on mathematical results rather than on the per­ may find them of value. sonality or habits of life and work of the individ­ ual in question. The authors are frequently former Who Should Write Mathematical colleagues, now past the most productive years of Biographies? their own careers (understandably enough, since Interest in the lives of mathematicians and in the few who are actively involved in mathematical re­ history of mathematics in general has increased search or concerned about professional advance­ markedly in recent years. More mathematical bi­ ment can afford the time to engage in serious bi­ ographies are now being written, they are receiv­ ographical scholarship). In many cases they are or ing greater attention from reviewers and readers, were close friends of the subject and so may be un­ and the standards and sophistication of the genre able to evaluate the person's character and con­ have improved substantially. Writing a mathemat­ tributions objectively. They may also be too close ical biography has accordingly become both a more to the mathematical subject matter, so that, despite rewarding and a more demanding task-the more having a thorough understanding of the technical so the broader the audience to whom the biogra­ details, they may lack historical perspective con­ phy is addressed. cerning the development of the underlying math­ Most mathematical biographies, unsurprisingly, ematical ideas. In particular, many mathematicians are written by and for mathematicians. Some, aimed cleave to the Whiggish view that the development at a more general readership, have been written by of mathematics has been a story of continuing and journalists, among whom Constance Reid is the pre­ inevitable progress. eminent example; and there are a few, such as To Whom Should Mathematical Biographies [Ltitzen 1990], whose authors are trained historians be Addressed? of mathematics. The latter, however, are a rare breed. In general, historians of science have displayed an For the mathematical biographer there is a strong astonishing lack of interest in mathematics, temptation to preach to the converted. To be sure, writing for an audience of mathematicians de­ John W Dawson ]r. is professor of mathematics at the York mands great precision in the description of math­ campus ofPennsylvania State University. His email address ematical details. But much less effort need be de­ is jwd7@psu. edu. voted to explaining the concepts involved, or to

444 NOTICES OF THE AMS VOLUME 53, NUMBER 4 developing interest in what the person in question colleagues with whom I could discuss research did. And outside the mathematical community questions in logic, and the joys of teaching calcu­ sales of any mathematical biography are likely to lus term after term had begun to pall. I felt that I be few. That, however, is disturbing and suggests was losing contact with my discipline and was no that there are larger purposes that mathematical longer able to contribute productively to mathe­ biography might serve. matical research. In that regard the biographical writings of E. T. To avoid intellectual stagnation I resolved to go Bell are instructive. They have been much dis­ back and study the works of the masters. The ob­ praised by professional historians, both because of vious place for a logician to start was Godel's writ­ Bell's tendency to romanticize his subjects and be­ ings; but to my surprise, I found that no list of them cause of factual errors in some of his accounts. In had ever been compiled. Preparing an annotated his Mathematics, Queen and Servant of Science, for bibliography of his published works thus appeared example, Bell asserts that Godel received a degree to be a worthwhile endeavor-one I felt I was ca­ in engineering from the University of Brno-a state­ pable of undertaking and that was a necessary first ment that has no basis in fact. Such sloppy schol­ step toward the larger goal of compiling a com­ arship cannot be defended, of course. Yet the pos­ prehensive edition of those works. itive impact that Bell's writings have had cannot be I made a firm decision to take on the biblio­ denied either. Indeed, several mathematicians of graphic task when, following a tip from my friend stature have attributed the awakening of their in­ Fred Rickey, I discovered three short papers on terest in the discipline to their reading of Men of geometry that Godel had published in the 1930s, Mathematics. Julia Robinson, for one, recalled that which had not been cited by any previous com­ she hardly knew what mathematics was until she mentators. That so aroused my curiosity that I read Bell's book. "I cannot," she declared, "overem­ began a detailed search of the literature. The result phasize the importance of such books .. .in the in­ was [Dawson 1983], whose appearance brought tellectuallife of a student like myself [who was] my efforts to the attention of others in the logical completely out of contact with research mathe­ community and led straightaway to my complete maticians." ([Reid 1996], p. 25) immersion in Godel studies. On the strength of that Not all readers would share Robinson's attrac­ compilation I was invited, at almost the same time, tion to mathematics as a career. But surely our both to become one of the editors of Godel's Col­ field would benefit from a wider appreciation of lected Works and to undertake the cataloguing of what it is that mathematicians do. We are not ac­ his Nachlass (literary remains) at the Institute for countants, as so many seem to think, and to com­ Advanced Study (lAS). bat that widespread misimpression it is important While compiling my list of Godel's publications to communicate to laymen how genuinely exciting I began to wonder about his unpublished manu­ the exploration of mathematical ideas can be. scripts as well. I had no idea how extensive they In the end, it is up to each individual author to might be or what condition they might be in. I did decide what audience to address. In doing so, he know that other scholars had tried without success or she must consider the particulars of the life to to gain access to them, and I had heard that many be chronicled, whether the story has been told be­ of the papers were written in some sort of short­ fore, and how accessible the mathematical ideas are hand; so the prospect of my making much head­ that must be discussed. In addition, the would-be way with them seemed remote. author must decide whether or not he or she is in­ How, within a few months, I was offered the op­ tellectually suited to the task at hand. I, for exam­ portunity to catalog those papers is a story too long ple, am too much a child of my own time to con­ to relate in detail here. Suffice it to say that I was sider writing about a figure from an earlier century. persistent in making inquiries to the lAS, and that, unknown to me, the mathematicians there were Becoming a Biographer: My Own faced with the problem of deciding what to do Experience with donated materials for which the lAS had no When I began my studies of Godel, a few years proper storage facilities. I was due for a sabbati­ after his death, very little about him had appeared cal the coming year, so I happened to make my in­ in print. I had no idea what sources might be avail­ quiries at just the right time. able, no reason to expect that details of his life could It took me two full years to complete the cata­ be reconstructed to any considerable extent, and loging. The problem of reading Godel's Gabels­ no intention of becoming his biographer. berger shorthand was overcome with the help of Shortly before, I had achieved tenure at a two­ my wife, who volunteered to learn that now­ year branch campus of Penn State. My department obsolete system. Through our combined efforts I was supportive, I had good access to library re­ became more familiar than anyone else with the sources, and my teaching load was not unduly contents of Godel's Nach/ass, and my work on the heavy. But I found myself out of touch with Collected Works edition brought me into contact

APRIL 2006 NOTICES OF THE AMS 445 with such scholars as Solomon Feferman and Jean The Task of Writing van Heijenoort, whose experience and expertise By the time I returned home from Princeton I had immeasurably deepened my own understanding of done most of the necessary data gathering. I Godel's accomplishments.1 brought back with me many folders of photocopied I soon realized that there was no dearth of doc­ material, and as a result of my cataloguing expe­ umentary sources to work with. Quite the con­ rience I had formed an overall view both of the trary! Godel's Nachlass contained a great deal of sci­ structure of Godel's life and of the sources avail­ entific interest, including correspondence, able to draw upon. I knew that a great deal of read­ manuscripts, and research notebooks. Godel saved ing, note-taking, and reflection lay ahead before I much that others would have discarded, some of could commence writing. The problem was to find which, such as the library slips for books he checked time for all that in the midst of my teaching re­ out over the years, constituted important bio­ sponsibilities. graphical resources. But there was a great deal of Those and other commitments forced me to heed Pais's advice. I spent the next seven years­ chaff as well (letters from cranks, luggage tags, the time until my next sabbatical-studying the var­ miscellaneous memoranda slips, etc.), and there ious documentary materials, fitting pieces of the were also some notable gaps. There were, for ex­ puzzle together, and, above all, developing a view ample, no financial records after his emigration to of what sort of person Godel had been. America, and no letters to or from his mother or Establishing such a viewpoint is crucial to the brother. Nor did Godel ever keep a diary. He did, success of any biographical endeavor. For every bi­ however, keep a daily record of his body temper­ ography, rightly so called, must portray a life from ature and milk of magnesia consumption! a definite perspective. The point of view chosen will In the midst of my cataloging efforts a major vary from one biographer to another-that is why international logic conference took place in there can be more than one biography of the same Salzburg, and that event gave me the opportunity person- and it will necessarily reflect the author's both to organize a commemorative session for own background and biases. Accordingly, no bi­ Godel in his former homeland and to travel to ography can ever be the final word on its subject. Vienna and Brno (Godel's birthplace), where I was But without some perspective around which to or­ able to meet his brother, to photograph sites ganize the narrative, the account will be an un­ associated with Godel's childhood, and to become directed chronicle of events. As with all history, bi­ acquainted with Austrian scholars who were aware ographical writing demands interpretation of the data. of sources unknown to me. In particular, through By the fall of 1991, when my second sabbatical the efforts and generosity of Werner DePauli­ commenced, I was ready to begin writing. By then Schimanovich and Eckehart Kohler I was able to I had extracted a great many details about Godel's obtain photocopies of Godel's surviving letters to life from the sources I had studied, and I felt that his mother (now preserved in the Wiener Stadt- und I must begin to assemble them into a narrative Landesbibliothek), of documents from the archives while my memory of them was still fresh. Although of the University of Vienna, of some early pho­ there were still some loose ends, I had formed a tographs of Godel, and of a memoir about him by definite idea of what made Godel tick; I could en­ Karl Menger (eventually published posthumously, vision the chapter structure of the book quite in a revised English version, as a chapter in [Menger clearly; and I knew from Godel's own example that 1994]). if I waited too long my book might never be writ­ In Princeton and elsewhere I also tried to inter­ ten at all. The experience of cataloguing his Nach­ view as many as I could of those who had known lass had also taught me an important lesson about Godel personally,2 knowing full well that time was tackling big projects: One dare not look too far of the essence if I were to do so before my infor­ ahead, lest the work remaining to be done appear mants' health and memories failed. Among those too daunting. I had never written a book before, but with whom I spoke was Abraham Pais, whose bi­ I knew that in doing so I would have to keep my head down and concentrate on one chapter at a ography of Einstein [Pais 1982] had been published time. not long before. When I asked him if he had any During my sabbatical year I wrote seven of the advice for a would-be biographer, he replied sim­ fourteen chapters, at the rate of about one a month. ply, "Patience". I was pleased with the progress I was able to make, but only too well aware that once my teaching du­ 1 For a retrospective account of the Godel Collected Works ties resumed my pace would slow abruptly. project see [Feferman 2005]. In fact, it took four more years to complete the 2 I myself never met Godel, nor did I ever have any cor· manuscript. Yet, as things turned out, had I begun respondence with him. earlier I would have had much rewriting to do; for,

446 NOTicES OF THE AMS VOLUME 53, NUMBER 4 quite by chance, in March of 1992 I discovered a In all those regards, research in the history of new source of information that significantly en­ mathematics should be congenial to those trained riched my knowledge of Godel's later years: the di­ to do traditional mathematical research. What is dif­ aries of the economist Oskar Morgenstern. ferent is the lack of finality that historical conclu­ About six months earlier I had seen an an­ sions possess: the most carefully supported inter­ nouncement in the newsletter of the History of pretations of historical events may be upset by Science Society that Morgenstern's papers had been the discovery of new artifacts or data, whereas the donated to the archives at Duke University. I knew truth of a mathematical statement, once proven, is that Morgenstern had had a long-standing friend­ seldom called into question. Nevertheless, stan­ ship with Godel, so I was mildly interested. But since dards of mathematical proof do slowly change, his widow Dorothy was one of those I had inter­ and the recognition of that fact is perhaps what viewed during my stay in Princeton, and since she most distinguishes the viewpoint of the mathe­ had been quite willing to share her recollections of matical historian from that of most mathematical Godel with me and to talk about her husband's re­ practitioners. lationship with him, I didn't expect to find much It is comforting to think, after proving a theo­ in Morgenstern's papers that I didn't already know. rem, that one has settled a question once and for I was not aware that he had kept any diaries, but all. That feeling of security is one of the attractions when I later learned of their existence, I thought it unlikely that they would contain very much of rel­ of doing mathematical research, and it can be hard evance to my interests. It did not occur to me that to turn away from that and accept the vulnerabil­ his widow might not read German and so might her­ ity inherent in doing historical work. Those reluc­ self have been unaware of their contents. tant to do so should probably leave historical stud­ As luck would have it, that spring the Associa­ ies to others. But a degree of risk can also be tion for Symbolic Logic met at Duke. I took that op­ exciting, and to the extent that mathematical re­ portunity to look at the Morgenstern papers, and search is "safe" it is also abstractly removed from quickly found a slim folder labeled "Godel". There the affairs of the world. were only a few pages of interest in it, but the li­ Speaking personally, I believe that biographical brarian suggested that references to Godel might or historical work can be a valuable adjunct to tra­ also be found in Morgenstern's diary entries. Ire­ ditional mathematical pursuits, especially for those quested to see them, not realizing until she said who are remote from centers of mathematical re­ "Which volumes?" that his diaries covered a period search and do not have regular contact with ad­ of nearly sixty years. A cursory glance at a couple vanced students or colleagues in their discipline. of the volumes was enough to reveal what a wealth I recommend it to readers who may be dissatisfied of information about Godel they contained. I later with their present situation and are seeking an al­ spent a full week examining the diaries in detail. ternative way to remain productive and intellec­ What I found filled many gaps in the record of tually alive. Go del's life and gave me an insider's view of his final years unobtainable elsewhere. References [1]]. W. DAWSON, 1983, The published work of Kurt GOdel: Similarities and Contrasts Between an annotated bibliography, Notre Dame journal of Historical and Mathematical Research Formal Logic 24, 255-284. Addenda and corrigenda, Proving theorems is an analytical endeavor, whereas 25 (1984), 283-287. historical research is synthetic in character. Nev­ [2] __ , 1997, Logical Dilemmas: The Life and Work of ertheless, the standards of logical rigor to which Kurt Godel, A K Peters, Ltd., Wellesley, MA. mathematicians are accustomed have much in com­ [3] S. FEFERMAN, 2005, The GOdel editorial project: a syn­ mon with the standards of historical evidence. Nor opsis, Bulletin of Symbolic Logic 11, 132-149. is there any less satisfaction or excitement in set­ [4]]. LoTZEN, 1990, joseph Liouville 1809-1882: Master of tling historical questions or discovering new his­ Pure and Applied Mathematics, Studies in the History torical facts than there is in discovering new math­ of Mathematics and Physical Sciences, val. 15, (G. ]. Toomer, ed.), Springer-Verlag, New York et al. ematical results. Historical research is constrained [5] K. MENGER, 1994, Reminiscences of the Vienna Circle and by the available data and guided by historical acu­ the Mathematical Colloquium, (Brian McGuinness, ity just as mathematical research is constrained by Louise Golland, and Abe Sklar, eds.), Kluwer Academic axioms or the behavior of the real world and guided Publishers, Dordrecht-Boston-London. by mathematical insight. Both activities can be [6] A. PAIS, 1982, Subtle Is the Lord. The Science and the equally frustrating or intellectually rewarding, and Life of Albert Einstein, Oxford University Press, New if done properly, both demand equal standards of York, NY. scholarship and should be accorded equal schol­ [7] C. REm, 1996, julia, A Life in Mathematics, Mathemat­ arly respect. ical Association of America, Washington, D.C.

APRIL 2006 NOTICES OF THE AMS 447 Book Review Incompleteness: The Proof and Paradox of Kurt Gbdel Reviewed by juliette Kennedy

Incompleteness: The Proof and Paradox of Kurt Incompleteness Theorems. This means that some Godel of these areas are covered more comprehensively Rebecca Goldstein than others. W. W. Norton & Company In places the author succeeds creditably; for ex­ February 2005 ample, her portrayals of behind-the-scenes acade­ $22.95, 296 pages, ISBN 0393051692 mic life will likely be of interest to readers who enjoy such material-indeed, such portrayals seem Popular books on mathematics play an impor­ to be her forte. tant role in the lay public's education. But as is As is often the case with books about mathe­ known to anyone who has given a popular mathe­ matics lecture or written about a famous theorem matics written by nonmathematicians though, for an audience of nonmathematicians, doing jus­ shortfalls of precision occurring here and there will tice to the mathematics in question is almost im­ leave mathematicians unsatisfied; and the mis­ possible in those circumstances. Rebecca Gold­ statement of the Fixed Point Theorem on page 180, stein, the MacArthur Foundation fellow and author the heart of the matter technically, makes it, of The Mind-Body Problem (a novel which seems to unfortunately, quite impossible for anyone to be quite popular among mathematicians) attempts reconstruct the proof of the First Incompleteness an even more difficult task in her short new book Theorem from Goldstein's account. Incompleteness: The Proof and Paradox of Kurt In this review I will comment on the three main Godel, namely, to place a significant piece of math­ aspects of Goldstein's book, apart from her tech­ ematics-Godel's Incompleteness Theorems-in nical account of the theorems: first, her portrayal the context of the wider intellectual currents of the of the life and personality of Godel and the social twentieth century, both within the mathematical surroundings in which he worked; second, her logic and the philosophy of mathematics commu­ main claim, which is that Godel's work has been nities, as well as within the intellectual culture at misunderstood and misused by postmodernists large. and other intellectuals; and third, her rather sub­ The theorems are presented in the context of a stantial discussion of foundational issues, which somewhat detailed personal and intellectual biog­ raphy of Godel as well as in that of the various unfortunately is the weakest of these three aspects schools in foundations of mathematics in exis­ of the book. tence at the time. A vast amount of material is The author's novelistic skills are at their most covered: everything from the history of the Vienna conspicuous in the section of the book devoted to Circle, to Godel's philosophical differences with her colorful portrait of Godel. From p. 59: Wittgenstein, to the Hilbert Program, to Godel's I think it is fair to say ... that like so views on appointments at the Institute for Ad­ many of us Godel fell in love while an vanced Study, to many aspects of his personal bi­ ography, in addition to the account of the two undergraduate. He underwent love's ec­ static transfiguration, its radical re­ juliette Kennedy is university lecturer in the Department ordering of priorities, giving life new ofMathematics and Statistics at the University ofHelsinki. focus and meaning. One is not quite Her email address is juliette. kennedy@he lsi nki . fi . the same person as before.

448 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Kurt Godel fell in love with Platonism, Those familiar and he was not quite the same person with Godel's life may as he was before. find all this some­ what reductive; al­ Another example of this theme occurs on p. 110, though it is clear where Godel is described as "a man whose soul had from the author's been blasted by the Platonic vision of truth." portrayal of him that Goldstein includes all of the standard anecdotes she very much sym­ about Godel, as reported by the occasional, usually pathizes with Godel. confounded, eyewitness. Some of these are quite Some occasional amusing, for example, the story of Godel's citi­ wrong notes include zenship hearings (pp. 232-233), in which a logical the discussion on pp. inconsistency Godel discovered in the American 226-227, in which Constitution threatened to upset the proceedings. some key back­ The psychological analysis the author sprinkles ground facts are here and there into the biographical material is omitted, as well as nothing if not enterprising. For example, Goldstein the discussion on p. 223, in which the author ven­ has a theory about the source of Godel's psycho­ tures to describe Godel's marriage to his wife Adele, logical difficulties (pp. 48-49): a lively and witty woman who seems to have been as I hope will become ever clearer in the somewhat out of place in Princeton, as "weird", "ac­ chapters to come, the internal para­ cording to just about everyone". One wonders doxes in Godel's personality were at about this characterization of their marriage, when least partially provoked by the world's many of the Godels' friends seem to give a differ­ ent impression in their reports of it. In fact, on the paradoxical responses to his famous whole the portrait of Adele is a bit ungenerous, with work. the author making very heavy weather, for exam­ See also p. 57: ple, about such things as Mrs. Godel's appalling­ to Goldstein- taste in home decoration. . . .the precocious Godel grasped the lim­ Does the real person come through in this ac­ its of parental omniscience at about the count? Go del was an extremely private person who age of five. It would be comforting, in at the same time suffered, as many creative peo­ the presence of such a shattering con­ ple do, from disabling episodes of anxiety and de­ clusion, especially when it's reinforced pression. The episodes became worse with age. It by serious illness a few years later, to is not a very pretty story; but his productivity, derive the following additional conclu­ given the circumstances, makes it a very moving sion ... the grownups around me may be one-albeit one that in the end, at least in this a sorry lot, but luckily I don't need to book, may remain to be told. depend on them. I can figure out every­ That said, interviews the author has conducted thing for myself. The world is thor­ with the principals, for example Armand Borel, oughly logical and so is my mind-a have yielded valuable new information about perfect fit. Godel's relationship with his colleagues at the In­ Quite possibly the young Godel had stitute for Advanced Study, answering the question some such thoughts to quell the terror why Godel was so isolated from his colleagues dur­ of discovering at too young an age that ing his later years there. Goldstein also draws upon he was far more intelligent than his par­ her experience as a graduate student in philosophy at Princeton, which has put her in the position of ents. It would explain much about the being able to speak firsthand about the Princeton man he would become.1 academic culture of the time-even if her per­ 1 Some of the important memoirs about G6del include spective is very much that of an awestruck student. those by his brother Rudolf G6del, his classmate Olga The book is centered around the claim that, in Taussky-Todd [18], the obituary of G6del for the Royal So­ an ironic twist of events, the "intellectual commu­ ciety by Georg Kreisel [11], as well as other writings of nity", as Goldstein refers to it, used Godel's own Kreisel on G6del, Hao Wang's three books based on his con­ incompleteness theorems to discredit his philo­ versations with G6del [14], [15}, [16}, as well as Stephen sophical Platonism; that the Incompleteness The­ Kleene's memoir [10], to name just a few. The reader is orems became "grist for the postmodern mill", if also referred to the biography of G6del by john Dawson not the main weapon in the contemporary "revolt entitled Logical Dilemmas [4], as well as to Palle Yourgrau's against objectivity"; that consequently Godel, to portrayal of G6del in A World without Time [17], which focuses mostly on G6del's friendship with Einstein and the whom the notion of mathematical truth was an ab­ scientific work which grew out of it. solute and objective one, had to battle to the end

APRIL 2006 NOTICES OF THE AMS 449 of his days against postmodem misconceptions and view of Gbdel's time. For example, the Hilbert Pro­ misconstruals of his theorems, which were misin­ gram aimed to show that all that was required to terpreted to show that there is no such thing as formalize mathematics were finitary axioms stated truth. in a precise syntax together with finitary rules of Platonism in the context of foundations of math­ proof (proving consistency was the second desider­ ematics is essentially the view that mathematics is atum of it). In showing that mathematics can be con­ a descriptive science, although, unlike the empir­ strued as a "formal game of symbols", a slogan ical sciences, the domain described is thought to which came into use at the time, mathematics consist of abstract objects. Another tenet of Pla­ would be put on a firm foundation by eliminating tonism is that the concept of mathematical truth reference to infinite objects, as well as the use of is a meaningful one. Godel held this view from unstated assumptions or proof procedures that about 1925 onwards (though he wavered a bit be­ might lead to paradoxes. The part of mathematics tween then and about 1940). Gbdel also thought in­ which remains, and to which it is possible to reduce finitistic methods were fully acceptable, something the infinitary part, was called the inhaltlich, or con­ which, in his case, had to do with his Platonism. tentual, part. Formalism was Brouwer's term for the The author's description of Godel's mature Pla­ school associated with the Hilbert Program. tonism is essentially correct. But her interpretation Indeed the influence of the Hilbert Program on of the "intellectual community's" reaction to the In­ Gbdel can be seen from the fact that Gbdel stated completeness Theorems, although making for a and proved the Incompleteness Theorems "syn­ dramatic story, and probably correct in some par­ tactically", i.e., so as to avoid any reference to the ticulars, is an oversimplification of the facts. One notion of truth in the standard interpretation-that problem is that there seem to be at least two com­ is, truth in the domain of the natural numbers. He munities described as misconstruing Gbdel's the­ himself viewed the informal argument, which in­ orems: his colleagues in the logic and foundations volves the concept of truth in the standard inter­ of mathematics communities, and secondly the pretation, as sufficient. culture at large together with the postmodem phi­ As for whether the Incompleteness Theorems losophy community-quite a different audience served as a further stimulus for the anti­ for those theorems. Although it is the second of semantical or anti-truth view within the mathe­ these that is referred to more often, the claim is a matical logic and foundations global one. Unfortunately this leads to some con­ community, to some extent the influence flowed fusion, especially since it is often unclear to which in the opposite direc­ community the author is referring in particular in­ tion, as the author notes, upsetting the entrenched stances; and the abundance of vague references to, view. This is because, with respect to carrying out for example, "eminent thinkers", (p. 198) or "in­ the envisaged formal reconstruction of mathe­ tellectual gurus", (p. 40), or, simply, "they", (pp. matics, eliminating semantical notions, such as 135-6) is no help: meaning and truth, depends on showing that for­ mal provability is all that is needed. The irony of course is that while his The First Incompleteness Theorem refutes this theorems were accepted as of para­ by showing that the concept of mathematical truth mount importance, others did not al­ (if one accepts the concept at all) properly con­ ways hear what he was attempting to tains, in a sense, the concept of formal provabil­ say in them. They heard-and continue ity.2 Therefore the latter cannot bear the burden to hear-the voice of the Vienna Circle of the formalization. or of existentialism or postmodernism Gbdel demonstrated, then, if not the indis­ or of any other of the various fashion­ pensability of semantic notions altogether, at least able outlooks of the twentieth century. some grounds for their necessity. Many accepted They heard everything except what the view that semantics were ineliminable, at least Gbdel was trying to say. those, unlike strict finitists, who were not predis­ The text is full of such assertions, without their ever posed against the idea. And on the other hand, being pursued. The reader may find this vexing: many did not: the fact that the Incompleteness Who are "they"? Theorems can be formulated and proved com­ At least within the foundational community, a pletely syntactically led some to conclude that the critical attitude toward semantic notions, such as theorems say nothing at all about the validity-or the concept of mathematical truth, together with lack thereof-of semantic methods. As for vindi­ a bias against infinitistic methods, had been en­ cating the use of infinitistic methods, Go del had al­ trenched for decades before the Incompleteness ready achieved this, to some extent, with his Theorems. It was the driving force behind most of the foundational schools of the first half of the 2 Gddel's way of putting it was to say that the activity of twentieth century and was simply the dominant the mathematician cannot be mechanized. Seep. 164, [8].

450 NOTICES OF THE AMS VOLUME 53, NUMBER 4 completeness theorem, the main result of his doc­ hypothesis, namely the question whether there are toral thesis (submitted October 1929). any infinite cardinals strictly between No, the car­ In sum, differing views as to what constitutes dinality of the natural numbers, and 2No, that of the contentual mathematics led to the establishment reals, is misstated-the author confuses ordinals of various foundational schools long before the In­ with cardinals. The First Incompleteness Theorem completeness Theorems were published. But the itself is misstated on p. 191 (although the author complicated story of how those foundational gets it right elsewhere). schools absorbed the impact of those theorems is Also on the continuum hypothesis, Goldstein touched upon so briefly here, and is blended in so claims that Godel's landmark paper "What is Can­ indistinguishably with references to the second tor's Continuum Hypothesis?", " ... explains how community's absorption of those theorems, that it Cantor's continuum hypothesis has been shown to is difficult to support the central claim about the be independent of the axioms of set theory .... " But misuse and misinterpretation of Godel's works. as the paper was written in 1947, well before the The response of experts to the theorem aside, independence had been shown, it cannot possibly the author's claim that postmodern philosophers explain how the independence of the continuum or thinkers have used Godel's results to show, for hypothesis has been shown. 7 example, that truth is relative, or nonexistent, The author's lengthy discussion of the Hilbert seems plausible.3 How Godel's theorems were ul­ Program, which she characterizes as being dedi­ timately stirred into the mix, though, seems to be cated to "eliminating intuitions", is not exactly er­ a very large topic, deserving, possibly, a book of its roneous, but does violence to the spirit of that own.4 program, in the opinion of this reviewer, and will On the side of scholarship, it is to be regretted justifiably perplex mathematicians. that a book which has and will continue to gather From p. 129 and p. 133: such a wide and enthusiastic readership (at least judging from the satisfied customer reviews on The drive for limiting our intuitions Amazon) should contain at the same time such a went even further. The aim became to truly alarming preponderance of factual and con­ eliminate intuitions altogether. ceptual errors regarding other matters. 5 Names of If it could be shown that logically con­ major figures, such as Georg Kreisel's, are mis­ sistent formal systems are adequate for spelt. Many crucial dates are incorrect, including, proving all the truths of mathematics, twice in the book, the date of Godel's death (off by then we would have successfully ban­ two years). Hilbert's famous list of problems num­ ished intuitions from mathematics. bered 23 and not 10, as Goldstein has it, and Tarski's original surname was Tajtelbaum (usually She also remarks (p. 131) that written Teitelbaum), not Tannenbaum.6 The dia­ gram on page 12 5 is incorrect. The existence of non­ We don't have to appeal to our intu­ standard models of arithmetic follows already from itions about numbers or sets or space Godel's Completeness Theorem-the First Incom­ in laying down the givens of a formal pleteness Theorem is not needed. The continuum system. Giving a finitary formal reconstruction of math­ ematics of the kind the Hilbert school envisaged 3 The author seems to use "postmodernism" to refer to the in no way eliminates intuitions from mathematics. views associated with a much wider group of philoso­ Formal systems are to be set up exactly on the phers, writers, sociologists, and so on, than those associ­ ated with the French school around, for example, Lacan basis of our intuitions. Of course one can arbi­ and Derrida. This may confuse European readers. trarily set up a great variety of formal systems; but 4 There is a growing literature in the area of postmodern only those which are conceived on the basis of our commentaries of Code/'s theorems. For example, Regis Debray has used G6de/'s theorems to demonstrate the 7 The revised 1964 version of G6del's paper contains a very logical inconsistency of self-government. For a critical brief appendix commenting on P.]. Cohen's proof of this view of this and related developments, see Bricmont and result, published while G6de/'s revised paper was in proofs; Sakal's Fashionable Nonsense [13]. For a more positive view perhaps this explains Goldstein's error. What is striking see Michael Harris's review of the latter, "I know what you about Code/'s paper is that in it he predicts that the inde­ mean!" [9}. See also the recently published [6] by Torkel pendence of the continuum hypothesis will be shown. That Franzen as well as Franzen's "The popular impact of is, he predicts the 1963 result due toP.]. Cohen, that the G6del's incompleteness theorem", this issue of the Notices. negation of the continuum hypothesis is consistent with the Zermelo-Fraenkel axioms for set theory. This in spite of the 5 Apart from the error which has been noted earlier, in the fact that G6del himself proved in 193 7 that the continuum treatment of the First Incompleteness Theorem itself hypothesis is itself consistent with the axioms of set the­ 6 Though Stanley Tennenbaum was an important logi­ ory, a result that is in some sense opposite to what Cohen cian and Princeton figure. proved.

APRIL2006 NOTICES OF THE AMS 451 intuitions about number, set, and so on, can pos­ in some sense of the term. There are different ways sibly be of interest. Again, what the Hilbert school to interpret this, but for the sake of the present dis­ wanted was to capture mathematical content in the cussion let us take the assertion that mathematics kind of formal system in which reference to infi­ is complete to mean that any mathematical state­ nite entities does not occur. It is only in that pre­ ment, suitably precisely stated, is going to be de­ cise sense one can say those entities have been cided one way or the other by mathematical means "eliminated". For this to be possible one needs to -as following from some suitable set of axioms for define what it is that is to be eliminated, for a example (at least theoretically, that is, leaving aside start, and infinite entities such as N1 are suffi­ questions such as those involving resources). In fact ciently clearly defined- whereas the concept of in­ this very question, namely whether mathematics is tuition is not. complete in a more general sense than the techni­ On the philosophical side of things, connections cal one with respect to which he answered the between the notion of pure intuition and the Hilbert question negatively, occupied Godel himself to a Program are deep and important. Getting the con­ very great degree-and long after he proved the In­ cept of intuition right, whether it be the Kantian completeness Theorems. For example, in the late notion or otherwise, was a project of considerable 1930s he pondered the existence of absolutely un­ importance to the adherents of the Hilbert Pro­ decidable sentences, by which he meant precisely gram, throughout the 1920s.8 Simplyput, the Kant­ stated mathematical assertions "undecidable, not ian notion of pure intuition was seen by them to just within some particular axiomatic system, but be the very basis of what Hilbert called "the finite by any mathematical proof the human mind can mode of thought". As Bernays put it in 1928: conceive." ([8], p. 310) Interestingly, after a short The "finitistic attitude" required by period during which he entertained the possibility that there might Hilbert as a methodological basis must be absolutely undecidable math­ be characterized epistemologically as ematical sentences, Gbdel came to the conclusion a form of pure intuition. ([12], p. 170) around the early 1940s-and held to it for the re­ mainder of his life-that this would turn out not And indeed Gbdel's so-called Dialectica paper of to be the case. There would be no absolutely un­ 1958 (see [8]) contains an extensive discussion of decidable sentences in mathematics. What he called the topic. "Hilbert's original rationalistic conception" -that For pellucid discussions of these matters by the "for any precisely formulated mathematical ques­ principals, mathematicians are referred to Philos­ tion a unique answer can be found" ([8], p. 163)­ ophy of Mathematics [1], a collection of landmark was, in his view, the correct one. papers by Poincare, von Neumann, Godel, and How is this possible? Didn't Gbdel prove that others involved in these developments. For the In­ mathematics is incomplete (if consistent)? completeness Theorems themselves, mathemati­ The answer, of course, is no. What Go del proved cians are referred to Gbdel's original1931 paper, is that certain formal systems, including canonical "Dber formal unentscheidbare Satze der Principia ones like Peano Arithmetic or Zermelo-Fraenkel Mathematic a und verwandter Systeme, I, "9 a set theory, are incomplete. He did not prove that powerful piece of mathematics written in majes­ mathematics is incomplete in the sense we have de­ tic prose.10 fined completeness above; and therefore in some As an aside, on the topic of different commu­ sense the question is still an open one. nities' response to the Incompleteness Theorems, An issue which bears on the question being the readers of this review may see themselves as open is the fact that, as it turns out, there are dif­ comprising a third community-that of working ferent classes of undecidable sentences. The so­ mathematicians. What was the impact of the In­ called Go del sentences published by Go del in 19 31, completeness Theorems on this community? This do not seem to have much to do with mathemati­ is an interesting question. cal practice as such. The "I am unprovable" sen­ One might think of the average mathematician tences arising from the First Incompleteness the­ as taking the view that mathematics is complete- orem are somewhat ad hoc. (Indeed Gbdel himself referred to the theorems occasionally, in conver­ s Consequently they wrote about this extensively. See for sations with Kreisel, as the result of a "parlor trick", example Hilbert's "Foundations of Elementary Number see Theory", reprinted in [12]. [11].) In any case they, as well as the other Gbdel sentences, namely those involving 9 consis­ Reprinted with a facing English translation in [7}. The tency, can be decided simply by passing in a nat­ excellent introductory note to it by Kleene is also strongly 11 recommended. ural way to systems of so-called "higher type" . lO There are many modern accounts of the theorem. See, for example "The Incompleteness Theorem", by Martin 11 See [3}, as well as ''The impact of the incompleteness the­ Davis, this issue of the Notices. orem on mathematics", by Solomon Feferman, this issue.

452 NOTICES OF THE AMS VOLUME 53, NUMBER 4 In Conversation with Rebecca Goldstein Rebecca Goldstein's 1983 novel The Mind-Body Problem has been widely admired among mathematicians for its authentic depiction of academic life, as well as for its exploration of how philosophical issues impinge on every­ day life. Her new book, Incompleteness: The Proof and Paradox of Kurt Godel, is a volume in the "Great Discov­ eries" series published by W. W. Norton. The series aims to take a fresh look at great stories in science and math­ ematics. Another mathematical book in this series is Infinity and More by novelist David Foster Wallace (reviewed in the June/July 2004 issue of the Notices). In March 2005 the Mathematical Sciences Research Institute (MSRI) in Berkeley held a public event in which its special projects director, Robert Osserman, talked with Goldstein about her work The conversation, which took place before an audience of about fifty people at the Commonwealth Club in San Francisco, was taped and later broadcast on radio. A member of the audience posed a question that has been on the minds of many of Goldstein's readers: Is The Mind-Body Problem based on her own life? She did indeed study philosophy at Princeton, finishing her Ph.D. in 1976 with a thesis titled "Reduction, Realism, and the Mind". She said that while there are correlations be­ tween her life and the novel, the book is not autobiographical. Many have speculated that the model for one of the main characters, Noam Himmel, is Princeton philosopher Saul Kripke. Not so, said Goldstein: All of the char­ acters are fictional composites, except for the protagonist's father, who is based on Goldstein's own father. One of the attractions of writing about Godel's work for the general public, she said, is that his main theo­ rems seem to say something about the nature of mathematics itself and even to reach beyond boundaries of the field. "To have a result that has the rigor of mathematics and the reach of philosophy is beautiful," Goldstein re­ marked. She also talked about the relationship between GOdel and his colleague at the Institute for Advanced Study, Albert Einstein. The two were very different: As Goldstein put it, "Einstein was a real mensch, and Go del was very neurotic." Nevertheless, a friendship sprang up between the two. It was based in part, Goldstein speculated, on their both being exiles-exiles from Europe and intellectual exiles. Godel's work was sometimes taken to mean that even mathematical truth is uncertain, she noted, while Einstein's theories of relativity were seen as imply­ ing the sweeping view that "everything is relative." These misinterpretations irked both men, said Goldstein. "Ein­ stein and Godel were realists and did not like it when their work was put to the opposite purpose." -Allyn jackson

In that spirit some (see [5]) refer to this type of in­ interesting-and much more threatening to the completeness as "residual incompleteness", a Platonist viewpoint. phrase meant to capture the idea that this kind of As was noted above, the continuum hypothesis incompleteness arises only as an artifact of for­ is independent of the (highly canonical) Zermelo­ malization-after all, such sentences simply do Fraenkel axioms. But the continuum hypothesis is not arise in ordinary unformalized mathematics­ an elementary statement from the multiplication and is an exception to completeness of an entirely table of cardinal numbers, as Godel put it in his inessential nature, more or less on a par with in­ 1947 "What is Cantor's Continuum Problem?" cluding zero as a counterexample to the axiom (reprinted in [1]); it has a clear and unambiguous that every element of a field has a multiplicative meaning. Therefore it ought to have a definite truth inverse. The basic question whether mathematics value. As Godel wrote in that paper: is complete in the sense we defined it is therefore Only someone who ... denies that the not affected by GOdel's examples. As Godel put it concepts and axioms of classical set in referring to the undecidable sentences of 19 31: theory have any meaning (or any well­ "As to problems with the answer Yes or No, the con­ defined meaning) could be satisfied with viction that they are always decidable remains un­ such a solution, 13 not someone who touched by these results."12 believes them to describe some well­ The situation regarding statements emerging determined reality. For in this reality from set theory, however, such as the continuum Cantor's conjecture must be either true hypothesis, is much more complicated, much more or false, and its undecidability from the axioms as known today can only mean 12 See [8], pp. 174-5. There is a point of view that em­ phasizes more the importance of what we have called "residual incompleteness". But this would not have been 13 That there is no way to settle the continuum problem Godel's view. definitively. (Footnote the author's.)

APRIL 2006 NOTICES OF THE AMS 453 that these axioms do not contain a com­ is easy enough to find set theorists who disagree plete description of this reality .. . with Godel; in fact many, perhaps even a majority, of set theorists see themselves as standing in-or But there is a problem with deciding whether the near-the formalist camp.l4 continuum hypothesis is true or false in this sense­ Returning to the book under review, of its three that is, with deciding the continuum hypothesis by facets-Godel's incompleteness theorems, Godel's finding a natural extension of the Zermelo-Fraenkel theorems set against the background of the intel­ axioms that decides it (not to mention the problem lectual currents of his time, and finally Godel the with deciding other set-theoretical statements such man as well as the behind-the-scenes look at the as that asserting the existence of measurable car­ academic life of his contemporaries: as noted ear­ dinals). The reason is that there seem to be anum­ lier, the author's account of the incompleteness the­ ber of natural extensions of the Zermelo-Fraenkel orems is not sufficient to reconstruct them, while axioms that decide the continuum hypothesis in some defects in the treatment of the second aspect different ways-natural, at least, at first glance. of the book have also been indicated. As for the But then, just as in the case of the parallel axiom third aspect of the book, as mentioned, a rather col­ in geometry, the question of the truth value of the orful portrait of Godel is to be found in it. continuum hypothesis takes on a different mean­ Acknowledgments: For helpful discussions and ing. Is the parallel axiom true or not? To most correspondence during the preparation of this re­ mathematicians this is a meaningless question. view I would like to express my gratitude to Mark The answer depends upon which geometry one van Atten, John Burgess, John Crossley, Sol Fefer­ is referring to. Similarly with the continuum man, Allyn Jackson, Roman Kossak, Georg Kreisel, hypothesis, some argue that its truth value de­ Jouko Vaananen, Palle Yourgrau, and Norma Yunez­ pends upon which model of set theory one is in, Naude. so to speak, among a spectrum of natural models to choose from. This is a kind of formalism to References which, for example, Cohen has subscribed (see (1] PAUL BENACERRAF and HILARY PUTNAM, eds., Philosophy of his Set Theory and the Continuum Hypothesis Mathematics: Selected readings, Prentice-Hall, Inc., En­ [2]). glewood Cliffs, N.J., 1964. Of course many argue against the notion that the [2] PAUL J. COHEN, Set Theory and the Continuum Hypoth­ continuum hypothesis is analogous to the parallel esis, W. A. Benjamin, Inc., New York-Amsterdam 1966. axiom, and indeed Godel was among the first (if [3] WILUAM CRAIG, Satisfaction for n-th order languages de­ fined in n-th order languages, journal of Symbolic not the very first) to argue against the analogy in Logic 30, 1965, pp.l3-25. his 194 7 paper mentioned above. Among other ar­ [4] ]OHNW. DAWSON ]R., Logical Dilemmas: The life and work guments given there, Godel observes that, "as of Kurt G6del, A K Peters, Ltd., Wellesley, MA, 1997. against the numerous plausible propositions which [5] PATRICK DEHORNOY, Progres recents sur !'hypothese du imply the negation of the continuum hypothesis, continu (d'apres Woodin), Asterisque, 294, 2004, viii, not one plausible proposition is known which pp. 147-172. would imply the continuum hypothesis." (See [8], [6] ToRKEL FRANZEN, G6del's theorem: An incomplete guide p. 264.) to its use and abuse, A K Peters, 2005. This counts in favor of the continuum hypoth­ [7] KURT GODEL, Collected Works. I: Publications 1929-1936, (eds. S. Feferman et al.), Oxford University Press, 1986. esis being false. [8] __ , Collected Works. III: Unpublished essays and The arguments he gave in that paper have grown lectures, (eds. S. Feferman et al.), Oxford University into the so-called "large cardinal program", which Press, 1995. is the program of finding a natural extension of the [9] MicHAEL HARRis, "I know what you mean!", http: I /www. Zermelo-Fraenkel axioms that decides the mathe­ math.jussieu.fr/-harris/Iknow.pd~ matical statements one is interested in, meaning [10] STEPHEN C. KLEENE, The Work of Kurt Gtidel,]. Sym­ the "natural" ones, or more generally, those not aris­ bolic Logic41 (1976), no. 4, pp. 761-778. ing from residual incompleteness. As of today, the technical developments have not l4 There is of course much more to be said about which settled this issue in a definitive way. On the one independent statements need to be taken seriously, at hand the large cardinal program of Godel is alive least by Platonists. The present discussion has omitted the and well. Accordingly, for an important faction of natural statements independent ofPeano arithmetic of the set theorists, the continuum hypothesis is simply Paris-Harrington type, as well as those arising from the a problem to be solved-granted a very difficult one work of Harvey Friedman. For discussion of these and re­ lated matters see the above cited ''The impact of the in­ -just like any other mathematical problem (see [5]). completeness theorem on mathematics", by Solomon Fe­ This means, interestingly enough, that it is still ferman, this issue of the Notices. For a discussion of G6de/'s possible for the discoverer of incompleteness to be results in set theory together with their impact, see "How vindicated in his view that mathematics is, for all G6del transformed set theory", by juliet Floyd and Aki practical purposes, complete. On the other hand it Kanamori, also this issue.

454 NOTICES OF THE AMS VOLUME 53, NUMBER 4 (11] GEORG KREISEL, Kurt G6del: 1906-1978, Biographical Memoirs ofFellows of the Royal Society, vol. 26, (1980), pp. 148-224. 2006 Events Celebrating the Godel Centenary [12] PAOLO MANcosu, From Brouwer to Hilbert. The debate on the foundations of mathematics in the 1920s, Ox­ Several events are taking place around the world to mark ford University Press, 1998. the one hundredth anniversary of the birth of Kurt Godel. [13] ALAN SaKAL and JEAN BRJCMONT, Fashionable Nonsense, Below is a partial list. Picador, 1999. [14] HAo WANG, From Mathematics to Philosophy, Hu­ March 25-26, University ofEdinburgh: Conference on manities Press, New York, 1974. "Truth and Proof: Kurt Godel and the Foundations of [15] __ ,Reflections on Kurt Code/, MIT Press, 1987. Mathematics", organized by Jeffrey Ketland. [16] __ , A Logical Journey, Cambridge, MIT Press, April25-29, Technical University ofBrno: Centennial Cambridge, 1996. conference on Godel's life and work, organized by Masaryk A World Without Time. The Forgot­ [17] PALLE YOURGRAU, the Institute of Philosophy, Czech ten Legacy of Code/ and Einstein, Basic Books, New University, Brno, and York, 2005. Academy of Sciences, Prague. (18] PAUL WEINGARTNER and LEOPOLD SCHMETTERER, Godel Re­ April 2 7-29, University of Vienna: "Horizons of Truth: membered. Papers from the 1983 Godel Symposium Logics, Foundations of Mathematics, and the Quest for held in Salzburg, July 10-12, 1983, History of Logic, Understanding the Nature of Knowledge". This confer­ N, Bibliopolis, Naples, 1987. ence celebrating the Godel centenary is organized by the Kurt Godel Society with the support of the John Templeton Foundation. Website: http: I /www. logic. at/ goede 12006/ . May 17-21, Montreal: The 2006 annual meeting of the Association for Symbolic Logic will include several sessions devoted to Godel, organized by a subcommittee chaired by Charles Parsons. May 18-21, University of Lille: Workshop on "Kurt Godel: The writings", organized by Pierre Cassou-Nogues and Mark van Atten. June 30-]uly 5, UniversityofWales, Swansea:The con­ ference "Computability in Europe 2006: Logical approaches to computational barriers", organized by Arnold Beckman and John Tucker, will include a special session "Godel Cen­ tenary: His Legacy for Computability", organized by John Dawson and Matthias Baaz, and lectures on Godel by Mar­ tin Davis andJohnDawson. Website: http: I j www. cs. swan. ac.uk/cie06/ give-page.php?l. August 12-14, Seattle: The IEEE Symposium on Logic in Computer Science, part of the 2006 Federated Logic Con­ ference, will include a special session on Godel, organized by Moshe Vardi. Website: http: I /www. i nformati k. hu - berlin.de/lics/lics06/. In addition, a major indoor and outdoor exhibition on Gbdel will open in Vienna in April (curated by Karl Sig­ mund and John Dawson; see the article by Karl Sigmund in this issue of the Notices). A panel session on "Godel's Contributions to the Foun­ dations of Mathematics", has also been proposed (by Linda Bercerra and Ron Barnes of the University of Hous­ ton Downtown) for MathFest 2006, August 10-12, in Knoxville.

APRIL2006 NOTICES OF THE AMS 455 W H A T I S a Syzygy?

Roger Wiegand

Linear algebra over rings is lots more fun than resolution of the following form (the rather com­ over fields. The main reason is that most modules plicated matrices defining the maps between free over a ring do not have bases-that is, their gen­ modules need not concern us): erators usually satisfy some nontrivial relations or "syzygies". Given a finitely generated R -module 0 ~ R 3 ~ R 6 ~ R 4 ~ R ~ M ~ 0 M (where R is a commutative ring) and a set This time it takes three steps to resolve the mod­ z1, ... , Zn of generators, a syzygy of M is an element ule: The first module of syzygies of M is ker(E), the (al, ... , an) ERn for which a1z1 + · · · + anZn = 0. second (i.e., the module of syzygies of ker(E)) is The set of all syzygies (relative to the given gen­ ker(01), and the third module of syzygies is ker(o2). erating set) is a submodule of Rn, called the mod­ The 0 at the left end of the exact sequence then tells ule of syzygies. Thus the module of syzygies of M us that 03 maps R 3 isomorphically onto ker(02), is the kernel of the map Rn -~ M that takes the that is, the third syzygy of M is free. standard basis elements of Rn to the given set of Hilbert's famous "Syzygy Theorem", published generators. in 1890, states that every finitely generated graded Suppose, for example, that R = C[x, y] and module M = ®j: Mi over the polynomial ring M = R / m, where m is the maximal ideal consist­ 0 ing of polynomials with no constant term. The IC [x1, ... , Xn] has a free resolution of length at most that is, its nth syzygy is free. (The grading re­ module of syzygies is m, which is generated by x n, andy. These elements are not independent, since spects the action of the variables, in the sense that x JMi <;; Mi+ 1 for all i and all j :o; n. The length is one they satisfy the non-trivial relation (-y)x + xy = 0. So we look at the moduleS of syzygies of m, which less than the number of free modules in the reso­ is the rank-one free module generated by the ele­ lution.) Hilbert used this result to prove that the ment (-y, x) E R 2. Thus after taking syzygies twice function i ~ dilllc Mi is, for large i, a polynomial we have "resolved" M and obtained a free module. function of i. The idea here is that many proper­ We say that S is the second module of syzygies of ties of modules are easy to compute for free mod­ R/m. We can encode this information in the fol­ ules and behave well along exact sequences. Thus, lowing exact sequence (where we have written el­ if a module has a finite free resolution, one can eas­ ements of R2 as columns, so that matrices act on ily read off certain kinds of numerical data. the left): It turns out that every finitely generated mod­ ule M (not necessarily graded) over a polynomial 0 R [ -:] R2 R R /m 0 ~ ~ ~ ~ ring R = k[x1, ... , xnl, where k is a field, has a free For a slightly more complicated example, let resolution of length at most n. By introducing an­ R = C[x, y, z] and resolve the module other variable and "homogenizing", one can obtain M :=R /(x 3y+z4 ,xy2 ,x3 +y3 ,z5). Using the soft­ a free resolution of length at most n + 1. Using ho­ ware package MACAULAY 2, one obtains a free mological ideas available in the 1950s, one can then show that the nth syzygy S of M is stably free, Roger Wiegand is professor of mathematics at the that is, S ® RP ~ Rq for suitable integers p, q. Now University of Nebraska. His email address is one uses "Serre's Conjecture", proved indepen­ [email protected] .edu . dently by Quillen and Suslin in 1976: projective

456 NOTICES OF THE AMS VOLUME 53, NUMBER 4 modules over polynomial rings (such as the R ­ ematicians should know about syzygies, cf. David module S above) are free. Cox's article in the November 2005 Notices. Let's switch gears, from graded rings to local I will close by mentioning a few open problems. rings. Suppose R is a (commutative, Noetherian) First, as might be suggested by our first example, local ring with maximal ideal m, and let M be a fi­ for the simple module RIm over a regular local ring nitely generated R -module. By choosing a minimal R of dimension n, the Betti numbers are binomial generating set forM, and then a minimal generat­ coefficients: bi = (7). The "Horrocks conjecture" as­ ing set for the first syzygy, and so on, one obtains serts that these are the smallest possible Betti num­ a free resolution bers for a module of projective dimension n. Are­ lated problem is the "syzygy conjecture", which ... ~ Rb" ~ . .. ~ Rh ~ Rbo ~ M ~ 0. asserts that if M has projective dimension n then bi- h+l + h+2- · · · ± bn ~ i for each i < The It is not hard to see that the syzygies are uniquely n. conjecture is true for rings containing a field [3), determined up to isomorphism (independent of the but it is still open in the general case. Another choice of generators at each stage). We let syzi(M) open question concerns modules of infinite pro­ denote the ith syzygy, that is, the image of the map jective dimension: Given a finitely generated mod­ Rb; ~ Rb;- 1 • The integers bi = bi(M) are called the ule M over an arbitrary local ring R, are the Betti Betti numbers of M. If, for some h, we have numbers bi(M) eventually nondecreasing? bh(M) =f 0 but bi(M) = 0 for i > h, we say M has projective dimension h. For the ring of formal power Further Reading series C ([x1, ... , Xnll, it follows from Hilbert's [1] L. AVRAMov, Homological asymptotics of modules over theorem that every finitely generated module M local rings, Commutative Algebra (M. Hochster, has projective dimension at most n. On the other C. Huneke, ]. D. Sally, eds.), Mathematical Sciences hand, for the cuspidal ring R := C[[t 2, t3]] = Research Institute Publications, val. 15, 1987. C[[x, y ]] 1(y 2 - x 3), the minimal resolution of the [2] D. EISENBUD, The Geometry of Syzygies, Graduate Texts simple module R / (x, y) is easily seen to be infinite, in Mathematics, val. 229, Springer, 2005. and periodic after one step: [3] E. G. EVANS and P. GRIFFITH, Syzygies, London Mathe­ matics Society Lecture Notes Series, val. 106, 1985.

On the other hand, the Betti numbers of the module C[[t3, t4, t 5]] / (t3, t4 , t 5) over the ring C [[ t3, t4 , tS]J grow exponentially. These examples are symptoms of general be­ havior promised by deep results in the represen­ tation theory of local rings. First and foremost is the theorem of Auslander, Buchsbaum, and Serre, which says that a local ring R is regular (meaning, in the geometric context, that it corresponds to a smooth point on a variety) if and only if every fi­ nitely generated R -module has finite projective di­ mension. Next, if R is any ring of the form C[[x1, ... , XnlJ / (f), where fis a nonzero element in the square of the maximal ideal, then the minimal resolution of every finitely generated R -module is eventually periodic, of period at most two. To place the last example in context, see Avramov's survey [1) of the asymptotic behavior of Betti numbers over local rings. The "WHAT IS ... ?" column carries short (one- or Syzygies and the structure of free resolutions are two-page) nontechnical articles aimed at graduate stu­ dents. Each article focuses on a single mathematical currently the subject of intense research, though object rather than a whole theory. The Notices I have mentioned only a few highlights. For an ac­ welcomes feedback and suggestions for topics. Mes­ count of the use of syzygies in algebraic geometry, sages maybe sent to noti ces- whati s@am s . org. cf. Eisenbud's book [2). To see why applied math-

APRIL 2006 NOTICES OF THE AMS 457 Harvey Mudd Mathematics Department Garners AMSAward

Ten years ago the mathematics department at Har­ Borrelli was deeply involved with the clinic from its vey Mudd College was a very good department inception in the 1970s, and other faculty-Stavros known for excellent teaching and for innovations Busenberg, Courtney Coleman, and Henry Krieger­ like its "clinic" in which student teams work on served as directors of the clinic up to the 1990s. The mathematics problems from industry. The de­ clinic teams up senior mathematics majors to work partment was doing well, and likely there would on mathematical problems supplied by industry. have been no complaints had it remained as it was. These are real-life problems, messy and ill posed, Instead, the department has over the past decade and a crucial factor in making the program work brought itself up to a new level to become one of was the ingenuity of the Harvey Mudd faculty in the best places in the nation to be an undergrad­ figuring out how to distill the problems down to uate mathematics major. The quantitative mea­ workable projects for the student teams. During the sures of the department's success went through the 1970s and 1980s the faculty introduced other in­ roof, with the number of majors tripling and more novations as well; for example, Borrelli and Coleman than half of them going on to graduate school. But encouraged their students to experiment with soft­ what really changed is how the department unified ware for solving partial differential equations at a itself around its core mission of promoting teach­ time when not much of this kind of software was ing and scholarship in ways that inspire faculty and around. This experimentation evolved into the ODE students alike to do their best work. For its out­ Toolkit, now available on the Web. Borrelli and standing performance, the Harvey Mudd mathe­ Coleman also founded an interdisciplinary student matics department this year received the first-ever research journal called Interface. AMS Award for an Exemplary Program or Achieve­ In the mid-1990s the department, facing a big ment in a Mathematics Department. wave of retirements, seized the opportunity to How did the department get to where it is today? reinvent itself. A key step was hiring Michael Moody, One important factor is that it built upon its a mathematical biologist from Washington State strengths, one of which is the Harvey Mudd University, to take over as department chair. "He Mathematics Clinic. Professor emeritus Robert was a wonderful leader, very charismatic, with lots of ideas," noted Lesley Ward, who came to Harvey Mudd in 1997. "He would see what should be don~ in the future and then lay out the steps to get there." Moody set for the department what he called an "animating goal": To be recognized as one the very best undergraduate programs in the country. I 1 During Moody's six years as chair, from 1996 until 2002, the department hired eight new pro­ fessors; the total number of faculty is twelve. He and Arthur Benjamin, who had come to Harvey Mudd in 1989, worked as a team to do much of the hiring. Moody recalled that he and Benjamin resonated on what they wanted: "We wanted peo­ Harvey Mudd mathematics department, fall 2005. Left to ple who would mesmerize and inspire students right: j. Milton (visitor), H. Krieger, F. Su, A. Benjamin, in the classroom and have a passion for their math­ A. Castro, D. 'Vong, L. de Pill is, L. Ward, S. Martonosi, S. Frantz, ematical work." In addition, the department was A. Bern off, W. Gu, J. jacobsen, M. Orrison, and D. Goroff. Not lucky in the timing of its hiring. Current chair pictured are M. Raugh, B. Schade, and C. Connelly. Alfonso Castro, who came to Harvey Mudd three

458 NOTICES OF THE AMS VOLUME 53, NUMBER 4 years ago, noted that because the 1990s were tough (Raugh also cre­ years for job seekers in mathematics, the depart­ ated and serves ment was able to hire junior people who under as the director different circumstances would likely have gone to for a version of top research universities. "These young faculty not the clinic at the only bought into the mission of quality teaching at Institute for the undergraduate level, they also established a Pure and Ap­ first-rate presence in the research community," he plied Mathe­ said. Some of the candidates sensed the depart­ matics at the ment's newfound dynamism and were attracted University of by the prospect of contributing to the building of California, Los a new department. "That was exciting to me," noted Angeles.) The Francis Su, who came to Harvey Mudd in 1996. range of topics In the HMC Mathematics Clinic, teams of Young, newly hired faculty were given big re­ covered in the students work with faculty and company sponsibilities early on. Soon after Ward came to Harvey Mudd liaisons to solve open-ended real-world Harvey Mudd, she and Su, together with senior Mathematics problems. Here Lisette de Pill is (standing, faculty member Henry Krieger, took on the task of Clinic is enor- left) supervises a network optimization restructuring the department's core curriculum. mous; in recent project for ESRI Corporation. This two-year collection of courses is taken by all projects, stu- Harvey Mudd students. After consulting with other dents have worked on such topics as gene expres­ departments about what their majors needed to get sion data, global positioning system algorithms, out of the core mathematics curriculum, Krieger, soliton-like water waves, and cryptography. Su, and Ward reconfigured the four, semester-long The department set about revitalizing its senior courses into eight courses that run half a semes­ thesis by raising standards and by introducing ter each. They also added to the core a new course additional structure, such as having students in statistics and probability and introduced a participate in a weekly seminar in which they strand in discrete dynamical systems. "This bought give talks to each other about their work as it pro­ us huge credibility inside the institution," Moody gresses. Small changes like preparing hardbound noted, because the other departments felt that copies of theses and storing them on shelves where their concerns were heard and taken into account. students can browse through them help to convey He took flak from some senior faculty who con­ the message that the department believes that writ­ sidered the changes too radical, but he was not ing a senior thesis is a serious and substantial un­ perturbed. "I grew up in a system where assistant dertaking. At the end of the year, all graduating se­ professors were seen and not heard to a large ex­ niors must participate in "Presentation Days", a tent," he noted. "But they have creativity and energy, three-day, collegewide miniconference in which and if you leaven that with an experienced person students from all departments present talks about who has a light hand on the tiller, they'll do great their work. "The fact that we as a college decided things. Plus they'll be invested in it." it's more important to have three days of presen­ Also around this time the department revamped tations than three more days of lectures says a the mathematics major. Over the years the major lot about how the college values undergraduate had morphed into a system with several different research," commented Ward. tracks, and there would sometimes be just one or But is it really possible to involve undergradu­ two students per track, leaving them feeling some­ ates in mathematics research? Absolutely, said Su. what isolated. The department now has a single, uni­ "I believe undergraduates at any level can do re­ fied major centered on a core set of six advanced search," he said. "There are many kinds of research courses. Students take electives to customize their experiences. In terms of serving a student, what is own programs in consultation with faculty advisors. valuable for the student is learning to inhabit the In addition, the department launched two new research process." The goal is to give the students majors: the joint major in computer science and an experience in mathematical discovery that allows mathematics that started in 1998 and a major in them to understand what research is like. Su noted mathematical biology that started in 2001. that even in a deep field where there are not many Every senior major in the department must take easy problems lying around-say, differential geom­ part in a "senior research experience", which can etry-there are nevertheless problems that under­ mean either participating in the mathematics clinic graduates can sink their teeth into. For example, his or writing a senior thesis. The strong tradition colleague Weiqing Gu has come up with problems built up over thirty years of running the clinic in M-theory and string theory that boil down to served the department well when directorship of specific systems of partial differential equations. the clinic passed in 1999 from Borrelli to then­ Without possessing extensive background, newly hired faculty member Michael Raugh. an undergraduate can work on solving such systems

APRIL 2006 NOTICES OF THE AMS 459 an d make Study summer mathematics conferences. In 1999 progress on the department began organizing its own annual, u nderstand­ one-day regional research conference. With topics ing them ana­ ranging over analysis, algebra, geometry, mathe­ lytically and matical biology, and scientific computing, the geometri­ conferences typically attract fifty to seventy-five cally. The participants. Although this is a bona fide research proof that the conference, with experts in the field presenting department's their latest work, Harvey Mudd students are en­ Jon jacobsen and Francis Su demonstrate the aPProach couraged to attend just to soak in the atmosphere. fluid dynamics of a vortex cannon (the stable works is in "We always have significant student attendance at toroidal ring will blow out the candle Su is the publica­ the conferences," Moody remarked. holding). Many HMC math faculty use physical tions: Since Hand in hand with the emphasis on research and demonstrations in their lectures. 2002 at least scholarship is the department's strong commit­ twenty pa­ ment to excellence in teaching. As Ward put it, at pers have Harvey Mudd "it's okay to spend a lot of time on been pub­ making your classes great." As at other places, the lished, many college is trying to come to grips with how to doc­ of which ument good teaching. Student evaluations are used; were based also, when an individual comes up for promotion on senior the­ or reappointment, he or she must get other faculty ses and writ­ members to write letters of recommendation ten jointly by focused on teaching. But what really seems to make faculty and the biggest difference in the department's teach­ students, and ing is a highly developed sense of collegiality and many stu­ collective responsibility. "We are really generous dents have and free in sharing teaching ideas," Su noted. He published pa­ recalled that when he was teaching a certain course Weiqing Gu (standing) advising a team of pers on their for the first time, his colleague Art Benjamin lent students on a summer research project own. There his lecture notes to Su. It has become a tradition concerning tumor modeling and are other to hand down one's lecture notes. Su has developed immunotherapy for cancer patients. The proofs as a large collection of what he calls "Fun Facts", project was supervised by Lisette de Pill is. well. Harvey interesting tidbits about a wide variety of mathe­ Muddmathe- matical topics that can be presented in about five matics major Joshua Greene received the 2002 minutes at the beginning of a lecture to awaken AMS-MAA-SIAM Morgan Prize for outstanding re­ students' interest and expand their ideas of what search by an undergraduate, and another Harvey mathematics is. Su started sharing the "Fun Facts" Mudd student, Aaron Archer, was a runner-up for over the Web, and now other professors are using this distinction in 1998. them. On a daily basis the faculty discuss with Of course, for the students to have a meaning­ each other many teaching issues, large and small. ful research experience, the faculty themselves "If we go to lunch, we are as likely to be discussing have to be engaged in research. And at Harvey teaching as research," Ward remarked. The college Mudd, they are. The department's publication out­ has no teaching awards, but Harvey Mudd mathe­ put is impressive, and several faculty are supported maticians have received such awards at the na­ by research grants from the National Science Foun­ tional level: In 2000, Benjamin received the Haimo dation (NSF) . The Keck Foundation provided a Award of the Mathematical Association of Amer­ three-year grant for. the Center for Quantitative ica, and in 2004 Su received the MAA's Alde~ Award. Life Sciences, which is codirected by mathematics The department's efforts to renew it elf have faculty member Lisette de Pillis and a biology fac­ made a big difference in its ability to attract stu­ ulty member. The center has developed new courses dents into mathematics. "The students picked up in mathematical biology, hosted research visitors, a different feel" from the mathematics depart­ and funded two dozen multidisciplinary summer ment, Moody noted. "They could tell we really research projects. The Harvey Mudd mathematics cared about them." The number of mathematics ma­ faculty has also been represented in major research jors increased from a low of ten in 1993 to about conferences: Gu was an invited speaker at the thirty per year in recent years. The class of 2006 International Congress of Mathematicians in Bei­ has forty-one majors, nearly one quarter of Harvey jing in 2002, and four other faculty-Ward, Su, Mudd seniors (this number includes computer sci­ -de Pillis, and Andrew Bernoff-have been course ence/ mathematics and mathematical biology ma­ lecturers at the Park City Institute for Advanced jors). The number of women has also increased, and

460 NOTICES OF THE AMS VOLUME 53, N UMBER 4 women now account for income neighborhood. 1;c. about one third of all math­ This program brings Har- ::>: ematics majors, the same vey Mudd students into the · ~ proportion in which women school's classrooms, which ~ ~ are represented in the Har­ are populated mostly by '5"' vey Mudd student body. Latino students, to lead ac- ~ (With the hiring in fall2005 tivities designed to inspire "§ ::E of Susan Martonosi, the pro­ interest in mathematics, >- portion of women in the science, and engineering. ~ mathematics department In 2004 mathematics fac- ;:0 faculty is also one third.) ulty members Michael Or- ~ Among students who fin­ rison and Jon Jacobsen ~ ished the mathematics Harvey Mudd College students before traveled to Jamaica to lead "' major between 2002 and (above) and after the 2005 Putnam exam. a workshop designed to "'1; 2005, about 60 percent went enrich the mathematical ~ on to graduate school, and background of Jamaican ~ many were accepted in the teachers. They plan to top mathematics graduate offer the workshop again programs in the country. In in 2006. the past five years, nineteen Harvey Mudd recently Harvey Mudd mathematics hired mathematicians in majors were awarded the its top two administrative prestigious NSF Graduate posts: Daniel Goroff came Research Fellowships, and from Harvard University another sixteen were named to become vice president honorable mentions. Majors and dean of faculty in July who do not choose graduate 2005, and Maria Klawe will school are heavily recruited come from Princeton Uni­ by business, industry, and government. versity to take up the post of president in July Not only has the department attracted more 2006. And a couple of years back, the college lost mathematics majors, it has also attracted more a faculty member who went on to an administra­ nonmajors to take part in its programs. For ex­ tive post-namely, Michael Moody, who is now the ample, sixty to seventy students-about 10 percent dean of faculty at Franklin W. Olin College of En­ of the Harvey Mudd student body-take part in the gineering, an innovative engineering school that weekly Putnam Seminar, in which the students opened its doors in 2002. Moody was thrilled when work on practice problems for the Putnam Com­ the department received the AMS Award for Ex­ petition. The seminar is led by Bernoff and Su, who emplary Achievement and also felt a pang of home­ have put the emphasis on having fun solving prob­ sickness. "I'm so proud of the department," he lems rather than on honing an elite team. In the said. "I can't imagine a more deserving group of peo­ process, the department does manage to hone an ple-I am completely biased of course!-but they elite team: Harvey Mudd has placed in the top ten really are incredibly dedicated to mathematics and nationwide in the Putnam Team Competition in four to the community they have created." of the past five years and is the only undergradu­ Where does the department go from here? How ate college to have made it into the tcip five in the does it sustain its success? Moody pointed to the last thirty years. High participation by nonmathe­ need to continue to develop programs that bring matics majors also occurs in the Mathematical faculty together as teams so that they set aside dif­ Contest in Modeling and the Interdisciplinary Con­ ferences and work toward a common good. Su said test in Modeling (both of which are sponsored by he has seen exactly this happening organically in the Consortium for Mathematics and its Applica­ the department: One faculty member gets an idea tions). Harvey Mudd has done very well in both com­ to try something new, and he or she convinces col­ petitions, and the winning teams have often com­ leagues to come on board. If too many ideas pro­ bined mathematics majors and nonmajors alike. liferate, the department will have to prioritize, but Even with all the energy and attention the de­ for now Su is happy to let things develop in this partment puts into its own programs, it neverthe­ organic way. "But whatever we do, our mission is less manages to find ways to reach out beyond the to engage undergraduates in research and discov­ campus.ln 2000 mathematics faculty member Dar­ ery," Su said. "That will always be central." ryl Yong, together with three Harvey Mudd faculty from other departments, started an outreach ·-Allyn jackson program in Pomona High School, which is in a low-

APRIL 2006 NOTICES OF THE AMS 461 GOdel, Inconsistency, Provability, and Truth: An Exchange of letters

On the eve of Godel's centennial, it is distressing assumptions." There is a precise characterization that his incompleteness theorems continue to be (due to Tarski, shortly after the appearance of the so misunderstood. The article "Whither Mathe­ incompleteness theorems) of what it means for a matics?", which appeared in the December 2005 No­ statement in a formal language to be true within a tices, is an egregious example. On page 1350 it is given structure for that language. That character­ stated that Godel "established that the consistency ization is independent of any syntactic considera­ of arithmetic was not provable", a claim reiterated tions, and in the case of formalized arithmetic it again four pages later. But Godel did no such thing. is not expressible within the theory itself (unlike Rather, he proved that if formalized arithmetic is the notion of a proof in that theory). It is true that consistent, a particular numerical encoding of that Godel himself did not demonstrate that inex­ fact is expressible, but not provable, within that the­ pressibility (though there is ample evidence that he ory itself. The theorem does not rule out persua­ was aware of it before Tarski's work). Rather, sive proofs of the consistency of arithmetic that em­ acutely conscious of how controversial the idea of ploy means not formalizable within arithmetic. an objective notion of mathematical truth then The first such proof was given by Gerhard Gentz en was, he invoked the notion of truth only in the in­ in 1936, and Godel himself outlined another in his formal introduction to his incompleteness paper, last published paper, which appeared in 1958. where he "sketch[ed] the main idea of the proof The (second) incompleteness theorem showed ... without any claim to complete precision." In the that Hilbert's proof theory, which aimed to demon­ main part of the paper he eschewed semantic con­ strate the consistency of mathematics by a boot­ siderations altogether, employing only methods strapping process, was incapable of realization. that were acceptable to formalists (whose program But the theorem is irrelevant to those (such as the he had initially set out to advance). article's author) who think that Peano arithmetic might be inconsistent; for if it is, every statement -john W Dawson Jr. expressible within it, including the encoded as­ Pennsylvania State University, York sertion of its consistency, will be provable therein. [email protected] So Hilbert's aim only made sense (even before Godel's theorem) for those who believed that for­ (Received December 30, 2005) malized arithmetic was consistent. Nor do I understand what the author means when he says that "Godel's theorems ... do not Davies Response establish that there is a fundamental distinction Professor's Dawson's criticisms of the small part between truth and provability in mathematics of my article "Whither Mathematics" referring to without the insertion of extra philosophical Godel's work shows how easy it is to annoy and

462 NOTICES OF THE AMS VOLUME 53, NUMBER 4 possibly mislead people by over-abbreviation. He (Gbdel, K., "What is Cantor's Contin­ correctly states that I should have said that "the uum Problem?", 1947, pp. 258-273 in consistency of arithmetic is not provable from Philosophy of Mathematics: Selected within arithmetic" on page 1350 of my article. (I Readings, Second Edition, (Benacerraf, failed to repeat this essential caveat, contained in Paul, and Putnam, Hilary, eds.), Cam­ the previous sentence of my article.) However, if one bridge University Press, New York, NY, proves the consistency of arithmetic by invoking 1983). some other, richer, formal system, one achieves nothing unless one considers that the consistency His use of the word "true" here cannot be that of that new system is less capable of being doubted. of Tarski. Both Schwartz (op. cit.) and the author Gentzen's proof of consistency, for example, uses (Science in the Looking Glass, Oxford Univ. Press, transfinite induction, and was not regarded as per­ 2003, Chap. 1, 2) have argued that Gbdel's argument suasive by Tarski. Angus Macintyre ("Mathemati­ is not tenable, for similar reasons. Paul Cohen re­ jected Godel's Platonist view of set theory in "Com­ cal significance of proof theory", Phil. Trans. Royal. ments on the foundations of set theory", Axiomatic Soc. A 363 (2005), 2419-2435, p. 2426) argues that Set Theory, Proceedings of Symposia in Pure Math­ in Gentzen's work consistency is not really the ematics, (D. Scott, ed.), Vol. 13, Part 1, Amer. Math. main issue at all. Soc., Providence, RI, pp. 9-15, concluding that it "is I am not sure why Professor Dawson states "So our fate, to live with doubts, to pursue a subject Hilbert's aim only made sense for those who be­ whose absoluteness we are not certain of, in short lieved that formalized arithmetic was consistent." to realize that the only 'true' science is itself of the Surely the goal of his research programme should same mortal, perhaps empirical, nature as all other have been even more interesting to someone who human undertakings" (p. 15). In "Skolem and pes­ was prepared to admit some doubt on the matter, simism about proof in mathematics", Phil. Trans. however small, than to the great majority who Royal. Soc. A 363 (2005), 2407-2418, Cohen reit­ thought that it was merely a formal exercise? I erates his doubts, particularly about the relation­ know of nobody who actively believes that Peano's ship between axiom systems involving large car­ axioms are inconsistent, but Jack Schwartz goes dinals and reality (p. 2416). I used the phrase ~ven further than I do in arguing that consistency "philosophical assumptions" to refer to this issue, IS by no means well established; his article "Do the although it seems to provoke some people. I am less integers exist? The unknowability of arithmetic persuaded that Peano arithmetic is necessarily and consistency", Comm. Pure and Appl. Math. 58 obviously consistent than most mathematicians (2005), 1280-1286, is highly relevant to this but I do not lose any sleep over the possibility tha~ matter. my life's work will one day be suddenly rendered The relationship between truth and provability invalid. The ingenuity of mathematicians is enor­ is interesting, but I had not wanted to delve into mous, and any internal contradictions would prob­ such a controversial issue in what was principally ably be overcome fairly quickly with minimal effects a forward-looking article. One can certainly spec­ on the overall body of mathematics. ify a precise technical notion of truth within a for­ mal context, as Tarski did, but I hope that most -E. B. Davies readers did not confuse my use of the word truth Kings College London with that of Tarski, who was careful to avoid mak­ [email protected] ing claims about the philosophical status of the term as he used it; see the polemical section of "The semantic conception of truth and the foundations of semantics", Phil. and Phenom. Res. 4 (1944), 13-47. While Godel may have been acutely aware of how controversial the notion of unconditional or Platonic, truth was, he seemed to believe tha~ there was such an independent notion of truth in 1947. I quote But, despite their remoteness from sense experience, we do have some­ thing like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception.

APRIL 2006 N OTlCES OF THE AMS 463 2006 Steele Prizes

The 2006 Leroy P. Steele Prizes were awarded at Research (this year restricted to the field of applied the 112th Annual Meeting of the AMS in San An­ mathematics); and to FREDERICK W. GEHRING and tonio in January 2006. DENNisP. SULUVANforLifetimeAchievement. The text The Steele Prizes were established in 1970 in that follows presents, for each awardee, the selection honor of George David Birkhoff, William Fogg Os­ committee's citation, a briefbiographical sketch, and good, and William Caspar Graustein. Osgood was the awardee's response upon receiving the prize. president of the AMS during 1905-1906, and Birk­ hoff served in that capacity during 1925-1926. Mathematical Exposition: The prizes are endowed under the terms of a be­ Lars V. Hormander quest from Leroy P. Steele. Up to three prizes are Citation awarded each year in the following categories: (1) The four volumes ofLars Hormander's The Analysis Lifetime Achievement: for the cumulative influ­ of Linear Partial Differential Operators are a com­ ence of the total mathematical work of the recipi­ pendium of practically all of the exciting develop­ ent, high level of research over a period of time, ments that occurred in the theory of linear partial dif­ particular influence on the development of a field, ferential equations and in the area of microlocal and influence on mathematics through Ph.D. stu­ analysis in the period 1960-1985. dents; (2) Mathematical Exposition: for a book or Microlocal analysis emerged as a well-defined substantial survey or expository-research paper; (3) part of modern analysis with the development of Seminal Contribution to Research: for a paper, pseudodifferential operators in the early 1960s. This whether recent or not, that has proved to be of fun­ made possible a "microlocal" way of thinking about damental or lasting importance in its field, or a the basic objects in linear partial differential equa­ model of important research. Each Steele Prize car­ tion theory: the fundamental solutions of these equa­ ries a cash award of US$5,000. tions= and the classes of generalized functions to The Steele Prizes are awarded by the AMS Coun­ which the solutions of these equations belong. cil acting on the recommendation of a selection Thanks to microlocal techniques, one could analyze committee. For the 2006 prizes, the members of the singularities of these functions much more the selection committee were: Rodrigo Banuelos, precisely, and implement much more explicitly than DanielS. Freed, John B. Garnett, Victor W. Guillemin, before, for many different varieties of differential Craig L. Huneke, Tsit-Yuen Lam (chair), Robert D. equations, the "semi-classical limit" of quantum MacPherson, Linda P. Rothschild, and David A. mechanics. Vogan. In these four volumes, Hormander describes The list of previous recipients of the Steele Prizes these developments in a treatment that is seamless may be found in the November 2005 issue of the and self-contained. Moreover, the effort to make Notices, pages 1251-1255, or on the World Wide this treatment self-contained has inspired him to Web, http://www.ams.org/prizes-awards. recast, in much more simple and accessible form, The 2006 Steele Prizes were awarded to LARs V. the approach to much of this material as it origi­ HORMANDER for Mathematical Exposition; to nally appeared in the literature. An example is the CLIFFORD s. GARDNER, JoHN M. GREENE, MARTIN D. KRUSKAL, theory of Fourier integral operators, which was in­ and ROBERT M. MIURA for a Seminal Contribution to vented by him in two seminal papers in the early

464 NOTICES OF THE AMS VOLUME 53, NUMBER 4 19 70s. (These get a completely new and much more Response elegant reworking in volume four.) In brief, these I am very happy and grateful four volumes are far more than a compendium of to receive this award for the random results. They are a profound and master­ activity which has dominated ful "rethinking" of the whole subject of microlocal a great part of my professional analysis. life, and I wish to thank the Hormander's four volumes on partial differen­ members of the Selection tial operators have influenced a whole generation Committee for their consid­ of mathematicians working in the broad area of mi­ eration. crolocal analysis and its applications. In the history My expository writing of mathematics one is hard-pressed to find any started in the 1950s with mod­ comparable "expository" work that covers so much est lecture notes just intended material, and with such depth and understanding, for the students in Stockholm of such a broad area of mathematics. I wanted to introduce to the the­ Another of Hormander's masterpieces in expo­ ory of partial differential equa­ sition is his much shorter book, Complex Analysis tions. I toyed with the idea of in Several Variables, the first edition of which ap­ expanding them to a book but peared in 1973. Like the four volumes cited above, this seemed unrealistic until in Lars V. Hormander it is remarkable in the scope of what it covers. For 1960 I received a letter from instance the first chapter, only 22 pages long, is one one of the.e ditors of the famous Springer "yellow of the best treatments of functions of one complex series" inviting me to write a book for it. This was an variable available anywhere in the literature. Now, enormous encouragement, and as a result I devoted more than 30 years later, this excellent book re­ a great deal of the academic year 1960-1961 to this mains the gold standard in teaching a graduate project, including research on topics which had to be course in several complex variables at many uni­ better understood to make a systematic exposition versities in the U.S. and abroad. This short text of possible. The manuscript of my first book Linear about 200 pages is a "must read" for anyone who Partial Differential Operators was finished in 1962, works or uses the modern theory of analysis of sev­ and it appeared in 1963 in the yellow series. I was then eral complex variables. In particular, it contains the working to understand better the applications of the best treatment available for weighted L 2 estimates theory of functions of several complex variables to for d-bar equations (originally invented by Hor­ the theory of partial differential equations with con­ mander), which continue to be used in other areas stant coefficients which I had not been able to cover of mathematics. in my book, and with the so-called a-Neumann prob­ In conclusion, Lars Hormander's contribution to lem which through work of Morrey, Kohn, and oth­ mathematical exposition is highly unusual and per­ ers had just made it possible to conversely base the haps even unique in modern times. theory of functions of several complex variables on Biographical Sketch the theory of partial differential equations. When I Lars Hormander was born on January 24, 1931, in lectured on these topics at Stanford in 1964 I wrote southern Sweden. He was an undergraduate and a detailed lecture notes. After some additions to round graduate student at the University of Lund, Sweden, them off they were published by van Nostrand in first with Marcel Riesz and then, after Riesz re­ 1966 as An Introduction to Complex Analysis in Sev­ tired, with Lars Garding as advisor. After obtaining eral Variables, which is one of the books mentioned a Ph.D. in 1955 he spent a year in the U.S.-two quar­ in the citation. Expanded editions were published ters at the University of Chicago and a semester at by North Holland in 1973 and 1990. what is now the Courant Institute at New York The rapid development of microlocal analysis in University-before returning as full professor to the the 1960s quickly made the book in the yellow se­ University of Stockholm in 1957. During the acad­ ries obsolete, but the pace was so fast that it seemed J emic year 1960-1961 he was a member of the impossible to make it up to date. However, fifteen Institute for Advanced Study (lAS) in Princeton and years after it had been published I thought that it j a visitor at Stanford University in the summers of was worth trying, and after the year 1977-1978 at 1960 and 1961, where he had a permanent ap­ lAS and Stanford devoted to preparations, I could pointment in 1963-1964 before leaving both make preliminary plans for a replacement in three Stockholm and Stanford to become a professor volumes, again encouraged by Springer Verlag. and permanent member at the lAS. He left Princeton When the manuscript was finished in 1984 the in 1968 to return to Sweden as professor in Lund, third volume had grown so much that it had to be where he remained until retiring in 1996, apart divided in two, the last appeared in 1985. The title from about two more years in the U.S., mainly at lAS, The Analysis of Linear Partial Differential Opera­ Stanford, the Courant Institute, and the University tors was chosen to indicate that the four volumes of California, San Diego. had developed from the 1963 book, which is why

APRIL 2006 NOTICES OF THE AMS 465 I have mentioned it here although it is not included National Laboratory, the Courant Institute, and the in the citation. After two decades they are of course Princeton Plasma Physics Laboratory. He was pro­ no longer up to date but they can still serve as an fessor of mathematics at the University of Texas, introduction to many of the basic techniques in the Austin, from 1967 until his retirement in 1990. field. The first two volumes have been preserved Biographical Sketch: John M. Greene in the Springer Classics in Mathematics series, and John M. Greene received his B.S. degree in physics the last two should soon join them there. from the California Institute of Technology in 1950 In conclusion I would like to thank the many col­ and his Ph.D. in theoretical particle physics from leagues and students whose encouraging interest the University of Rochester in 1956. He worked at has stimulated my expository writing. Without the Princeton Plasma Physics Laboratory (1956- such support and constructive criticism it would 1982) and at General Atomics from 1982 until his have been hard to persevere with that for so many retirement in 1995. He has been a Fellow of the years. American Physical Society and a member of the Seminal Contribution to Research: American Geophysical Union. Clifford S. Gardner, John M. Greene, In 1992 Greene was awarded the James Clerk Martin D. Kruskal, and Robert M. Miura Maxwell Prize from the Division of Plasma Physics of the American Physical Society. The citation reads: Citation "For outstanding contributions to the theory of The prize is awarded for their joint paper "Korteweg­ magnetohydrodynamic equilibria and ideal and re­ deVries equation and generalizations. VI. Methods sistive instabilities, for discovery of the inverse for exact solution", Comm. Pure Appl. Math. 27 scattering transform leading to soliton solutions of (1974), 97-133. many nonlinear partial differential equations, and This is a fundamental paper in the theory of soli­ for the invention of the residue method of deter­ tons, inverse scattering transforms, and nonlinear mining the transition to global chaos." completely integrable systems. Before it, there was Response: John M. Greene no general theory for the exact solution of any im­ portant class of nonlinear differential equations. Ex­ [This response is written by Alice Greene on behalf cept for a few special cases, only approximations to of John Greene.) John was always pleased with the solutions were possible. This paper, in the context of work on the Korteweg-de Vries equations. I recall the Korteweg-deVries equation, introduced the use his triumphal announcement, "It unfolded like a of scattering parameters of an associated linear prob­ lily!" (After much intense work, I imagine.) He would lem to solve a nonlinear equation-effectively gen­ be truly delighted with its recognition by the Amer­ eralizing Fourier series and Fourier transforms to ican Mathematical Society. nonlinear equations. The idea was quickly extended Biographical Sketch: Martin D. Kruskal to other nonlinear evolution equations, triggering Martin D. Kruskal was born in New York City on important work in dynamical systems, inverse scat­ September 28, 1925. He earned his B.S. from the tering, and symplectic geometry, to name a few. In University of Chicago in 1945 and his M.S. and applications of mathematics, solitons and their de- Ph.D. from New York University in 1948 and 1952, scendants (kinks, anti-kinks, in­ respectively. He began his career as an instructor stantons, and breathers) have en­ in the mathematics department at New York tered and changed such diverse University (1946-1951) and then moved to Princeton fields as nonlinear optics, plasma University as a Research Scientist in the Plasma physics, and ocean, atmospheric, Physics Laboratory (formerly Project Matterhorn), and planetary sciences. Nonlin­ becoming Senior Research Associate in 1959. While earity has undergone a revolution: at Princeton, he was a lecturer in astronomy (19 59- from a nuisance to be eliminated, 1961), Director of the Program in Applied (and to a new tool to be exploited. Computational) Mathematics (1968-1986): profes­ Biographical Sketch: Clifford S. sor of astrophysical sciences (1961-1989) profes­ Gardner sor of mathematics (1979-1989), and is professor Clifford S. Gardner was born in emeritus (1989- ). He is currently David Hilbert Fort Smith, Arkansas, in 1924. Professor of Mathematics at Rutgers University. He graduated from Phillips Kruskal has given innumerable invited lectures Academy in 1940 and received at conferences and institutions and has served on his A.B. from Harvard College many advisory committees and editorial boards. He in 1944 and his Ph.D. from New has traveled widely and has held various visiting and York University in 1952. He fellowship positions at the Max Planck Institut fiir worked in applied mathematics Physik und Astrophysik (Munich), the University of at various places including Grenoble (France), the Lebedev Institute (Moscow), the Clifford S. Gardner NASA Langley Field, Livermore Weizmann Institute of Science (Israel), Nagoya

466 NOTICES OF THE AMS VOLUME 53, NUMBER 4 John M. Greene Martin D. Kruskal Robert M. Miura University Uapan), Bharathidasan University (India), degrees in mechanical engineering from the Uni­ Australian National University, the University of New versity of California at Berkeley in 1960 and 1962, South Wales (Australia), the University of Adelaide respectively, and his M.A. and Ph.D. in aerospace (Australia), Los Alamos National Laboratory, the and mechanical sciences from Princeton University University of California at Santa Barbara, and the in 1964 and 1966, respectively. His doctoral re­ University of Chicago. search was in the area of the kinetic theory of Kruskal has been the recipient of numerous gases. His first postdoctoral position in 1965-1967 honors and awards, including the National Medal was at the Princeton Plasma Physics Laboratory, of Science, the National Academy of Sciences Award part of Princeton University, where he started re­ in Applied Mathematics and Numerical Analysis, the search on nonlinear wave propagation. There he von Neumann Prize of the Society for Industrial and worked closely with Martin Kruskal, Clifford Gard­ Applied Mathematics, and the Potts Gold Medal of ner, and John Greene on the Korteweg-de Vries the Franklin Institute (Philadelphia). He has re­ equation, a nonlinear dispersive partial differential ceived an honorary doctorate from Heriot-Watt equation exhibiting soliton solutions and having nu­ University. He is a member of the National Acad­ merous applications. This collaboration led to the emy of Sciences and the American Academy of inverse scattering method for exact solution of the Arts and Sciences, a foreign member of the Royal KdV equation and also to the proof of an infinite Society of London and the Russian Academy of number of conservation laws. His postdoctoral po­ Natural Sciences, and an Honorary Fellow of the sition at the Courant Institute in 196 7-1968 was Royal Society of Edinburgh. in the Magneto-Fluid Dynamics Division headed Response: Martin D. Kruskal by Harold Grad. It is usual for a prize recipient to thank the rele­ Miura has taught at New York University (1968- vant society, the AMS in the present case, and the 1971), Vanderbilt University (1971- 1975), and the committee members who made the selection, for University of British Columbia (1975-2001). In being selected- and I do certainly wish to express 1975, upon his arrival at the University of British those sentiments. However, I also wish warmly to Columbia, his research interests changed to math­ thank my co-recipients, who played such a major ematical neuroscience, specifically excitable cells, role in our joint research, and from whom I learned and mathematical physiology more generally. Since so much in the process. 2001, he has been Professor of Mathematical Among the several functions that such prizes Sciences and of Biomedical Engineering at the New serve, a seldom mentioned one is to validate the Jersey Institute of Technology. He is currently act­ decisions and efforts that the awardees invested ing chair of the Department of Mathematical Sci­ in over, often, years of self-doubt and threatening ences. He is a fellow of the John Simon Guggenheim discouragement. Research success may indeed be Foundation (1980), the Royal Society of Canada its own reward, but it helps nevertheless to receive (1995), and the American Association for the Ad­ the recognition of one's peers. vancement of Science (2005). He has authored So, thanks to all of you! many research papers and served on several edi­ Biographical Sketch: Robert M. Miura torial boards. Presently, he is c o ~ editor -in- chief of Robert M. Mima was born on September 12, 1938, Analysis and Applications and is on the editorial in Selma, California. He received his B.S. and M.S. boards of the Canadian Applied Mathematics

APRIL 2006 NOTICES OF THE AMS 467 Quarterly and Integrative Neuroscience. He is a quasiconformality is an essential ingredient of the member of the American Mathematical Society, Mostow rigidity theorem and of recent work of the Society for Industrial and Applied Mathemat­ Donaldson and Sullivan on gauge theory and four­ ics, the Society for Mathematical Biology, the Cana­ manifolds, and quasiconformality has inspired dian Mathematical Society, and the American As­ much beautiful recent analysis on general metric sociation for the Advancement of Science. spaces by Heinonen, Koskela, and others. Response: Robert M. Miura Gehring's mathematics is characterized by its I am particularly pleased, honored, and humbled elegance and simplicity and by its emphasis on to receive the 2006 Leroy P. Steele Prize along with deceptively elementary questions which later my colleagues, Clifford Gardner, John Greene, and become surprisingly significant. Martin Kruskal, and to be recognized for the work A person of incredible energy and enthusiasm, on the Korteweg-de Vries equation that we did Fred Gehring has trained twenty-nine Ph.D. stu­ forty years ago. As a fresh postdoc, I was very for­ dents, many of whom are now faculty members at tunate to have had the opportunity to work with research universities, and he has mentored more and to have been mentored by three generous and than forty postdoctoral fellows. The list of Gehring's smart guys. The two years at the Princeton Plasma former postdocs at Michigan represents a large Physics Laboratory were the happiest and most ex­ fraction of the present day leaders in complex citing years in my research career. Every day came analysis. · with the time to think deeply about new ideas and Biographical Sketch to produce results. The soliton solutions of the KdV Frederick Gehring was born in Ann Arbor, Michi­ equation, discovered by Kruskal and Zabusky, gan, and his association with the University of showed this equation is special. The initial-value Michigan goes back two generations to his grand­ problem for the KdV equation is fascinating, and father, John Oren Reed, who was a member of the there are many special properties of the equations, physics faculty and Dean of the College of Litera­ e.g., an infinity of conservation laws resulting in ture, Science and the Arts. Gehring joined the U.S. infinitely many conserved integrals of the motion. Navy in 1943 and subsequently earned two de­ A major breakthrough was the development of a grees from Michigan-bachelor of science in math­ method for exact solution of the initial-value prob­ ematics and electrical engineering in 1946, and lem for the KdV equation on the infinite line, which master of science in mathematics in 1949. Here­ we called the "inverse scattering method" since turned to teach mathematics at Michigan in 1955 it utilized the scattering problem for the time­ after completing his Ph.D. at Cambridge and spend­ independent Schrbdinger equation. At the time, we ing three years as a Benjamin Peirce Instructor at thought this method was very special and only Harvard. He became a professor in 1962, was named could be applied to this equation. However, the to a collegiate chair in 1984, and became the T. H. Russians Zakharov and Shabat showed how to Hildebrandt Distinguished University Professor in generalize the method to systems of equations, and 1987, one of the university's highest honors for fac­ the rest is history. ulty. His long and distinguished history of service at Michigan includes three terms as chair of the de­ lifetime Achievement: partment of mathematics. Frederick W. Gehring Gehring has received numerous awards, in­ Citation cluding the Distinguished Faculty Achievement For over fifty years F. W. Gehring has been a leading Award, the Sokol Faculty Award, the Humbolt figure in the theory of quasiconformal mappings. Award, and an Onsager Professorship. He was the The cornerstone of the two-dimensional theory is Henry Russel Lecturer for 1990. In 1989 he was his theorem that the geometric definition of quasi­ elected to the National Academy of Sciences·. He has conformality (infinitesimal discs are mapped to in­ also received honorary degrees from the University finitesimal ellipses with eccentricity bounded) implies of Helsinki, the University of JyvaskyUi., and the Nor­ the more restrictive analytic definition. Gehring cre­ wegian UniverSity of Science and Technology. ated the higher dimensional theory of quasi confor­ Gehring also has a long record of service to tqe mal maps, which is very different from the two­ AMS. He has been a member of the Executive Com­ dimensional case. His work on convergence theo­ mittee (1973-1975, 1980-1982), a Member atLarge rems, Holder exponents, and the LP integrability of of the Council (1980-1982), and a member of the Jacobians forms the foundation of the higher di­ Board of Trustees (1983-1992; chair 1986, 1991). mensional theory. He has served on numerous committees, including Largely because of Gehring's work, the theory of the Committee on Science Policy (1981-1983, 1985- quasiconformal mappings has influenced many 1987), the Committee on Governance (1993; chair), other parts of mathematics, including complex and the Editorial Committees for the Bulletin, Math­ dynamics, function theory, partial differential ematical Reviews, Proceedings, and the Electronic equations, and topology. Higher dimensional journal on Conforrrwl Geometry and Dynamics.

468 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Fulbright and Guggenheim Fellowships in 19 58- manifolds. Later Sullivan de­ 1960 allowed Gehring to study in Helsinki and ZOrich, veloped and applied rational where he began to learn the theory of quasiconfor­ homotopy theory to prob­ mal mappings, a subject that became the focus of lems about closed geodesics, his life's work. He was instrumental in developing the the automorphism group of theory, often in collaboration with Finnish colleagues, a finite complex, the topol­ in bringing it into the mainstream of mathematical ogy of Kahler manifolds, and analysis, and in making contact with potential the­ the classification of smooth ory, partial differential equations, geometric topol­ manifolds. He has reinvented ogy, Riemannian geometry, and complex dynamics, himself several times, playing as well as classical function theory. In particular, he major or dominant roles in pioneered the important extension of the theory to dynamical systems, Kleinian n-dimensional space, emphasizing new tools such as groups, and low dimensional extremal length, and has brought quasiconformal topology. mappings into a broad study of discrete transfor­ These brief remarks do mation groups. He has generously shared his passion not do justice to the scope of for mathematics and research by mentoring over Sullivan's ideas and influ­ seventy Ph.D. students and postdoctoral fellows dur­ ence. Beyond the specific the- Frederick W. Gehring ing his career. ories he has developed and Response the problems he has Some of the earliest memories I had as a child were solved-and there are many music which my father played on a piano and or­ significant ones not men­ chestral pieces which he played on a large victrola. tioned here-his uniform vi­ I was fascinated by what I heard and subsequently sion of mathematics perme­ spent several years learning how to play the piano. ates his work and has Later as I was finishing high school in 1943 I inspired those around him. learned how to build radios and looked forward to For many years he was at the a career in physics. But world events intervened. I center of the mathematical joined the U.S. Navy V-12 program in June 1943 and conversation at IIIES [Institut spent the next thirty-two months as a student in des Hautes Etudes Scien­ the Department of Electrical Engineering at the tifiques). Later he moved to University of Michigan. New York where his weekly This was a fascinating but somewhat frustrating seminar remains an impor­ experience since I would have preferred to see more tant feature of mathemati­ rigorous proofs for the things I had learned. Hence cal life in the City. after the war I changed my major and studied math­ Biographical Sketch ematics at Michigan and then at Cambridge Univer­ Dennis Sullivan was born sity in England. February 12, 1941, in Port Dennis P. Sullivan I never regretted that decision, and I consider Huron, Michigan, but he grew the ensuing years of teaching and research as the up in Houston, Texas. He happiest possible. The opportunity to guide my graduated from Rice University in 1963 and went Ph.D. students and the postdoctoral fellows with to Princeton University; his Ph.D. thesis (1966) on whom I have worked was educational, rewarding, geometric topology was written under the direction and fulfilling. of William Browder. After graduation Indeed I would feel quite remiss in accepting this he held a NATO Fellowship at Warwick, where he award without acknowledging how. much I owe to continued work in the general area of his thesis j them. So now I thank you for this award which I (Hauptvermutung for manifolds, 1967), and a Miller I accept in their names also. Fellowship at Berkeley (work on the Adams con­ jecture, K-theory, and etale homotopy). He spent Lifetime Achievement: Dennis P. Sullivan 1969 to 1973 as a Sloan Fellow of Mathematics at Citation the Massachusetts Institute of Technology, study­ Dennis Sullivan has made fundamental contribu­ ing localization in homotopy theory (in particular, tions to many branches of mathematics. Sullivan's Galois symmetry), etale theory, and the construc­ theory of localization and Galois symmetry, tion of minimal models for the rational homotopy propagated in his famous 19 70 MIT [Massachusetts type of manifolds, using differential forms. Institute of Technology) notes, has been at the heart He shared the AMS Veblen Prize with Rob Kirby of many subsequent developments in homotopy the­ in 1971. In 1973-1974, Sullivan visited the ory. Sullivan used it to solve the Adams Conjecture University of Paris-Orsay. He remained in France as and the Hauptvermutung for combinatorial professeur permanent at the Institut des Hautes

APRIL 2006 NOTICES OF THE AMS 469 Etudes Scientifiques, full-tim'e until1981, when he was named Einstein Professor at the City Univer­ Photo Index to Pages 410-411 sity of New York, and half-time after that until 1996, when he joined the Mathematics Department and the Institute for Mathematical Sciences at SUNY, Stony Brook. During his years in France, his interests expanded first towards dynamical sys­ tems, including ergodic theory, foliations, Kleinian groups, and renormalization, and then, motivated originally by problems in conformal dynamics, to­ wards Teichmilller theory (No Wandering Domains Theorem, 1982). He was awarded the Prix Elie Car­ tan by the Academie des Sciences de Paris in 1981, the King Faisal Prize in Science in 1994, the Ordem Scientifico Nacional by the Brazilian Academy of Sciences in 1998, and the United States National Medal of Science in 2005. He was elected to the United States National Academy of Sciences in 1983 and to the Brazilian National Academy of Sci­ 1. Welcome to the Joint Mathematics Meetings, San ences in 1984. He was awarded honorary degrees Antonio, TX, Henry B. Gonzalez Convention Center. by the University of Warwick in 1983 and the Ecole Normale Superieure de Lyon in 2001. His most 2. MAA Booth, Exhibits area. 3. Moving about the Convention Center. recent work centers on quasiconfonnal analysis, 4. Ribbon-cutting ceremony for the Exhibits area (left to holomorphic dynamics, and the relation between algebraic topology, quantum theory, and fluid dy­ right: James Tattersall (MAA), Martha Siegel (MAA), John Ewing (AMS), James Arthur (AMS), Carl Cowen namics. Dennis Sullivan has three daughters, three (MAA), Robert Daverman (AMS), Tina Straley (MAA)). sons, and two grandchildren. 5. Sharing ideas between sessions. Response 6. AMS Booth, Exhibits area. I am very honored and pleased to receive the Steele · 7. Opening day of the Exhibits. Prize-with a small nuance that it is awarded for 8. Birkhoff Prize winner Cathleen S. Morawetz. work done up to now. I am still trying to understand 9. Steele Prize winner Dennis P. Sullivan. the correct algebraic structure of an algebraic model 10. Steele Prize winner Clifford S. Gardner with AMS for manifold or spacetime. My thesis advisor's orig­ president James Arthur. inal emphasis on Poincare duality is still the guide, 11. AMS Colloquium Lecturer Hendrik W. Lenstra Jr. but now expressed in new algebraic data related to 12. Audience in large lecture. the physicist's correlations, or multilinear func­ 13. Who Wants To Be a Mathematician game contestant. tions on a space of states. I hope to apply this to 14. "Hands-on" at the Math in Art exhibit. write down finite dimensional computationally ef­ 15. Interviewing in the Employment Center. fective algorithms in nonlinear problems like fluid 16. Math on the Web demonstration. dynamics with applications to problems like help­ 17. Left to right, AMS executive director John Ewing, AMS ing out the 48 hour more precise advance predic­ senior editor Ina Lindemann, former AMS president tion of the landfall of hurricanes like Katrina and David Eisenbud. Rita. 18. Mathematical artwork. 19. New Orleans (site of 2007 JMM) information booth. 20. Interviewing in the Employment Center. 21. Steele Prize winner Frederick W. Gehring and Mrs. Gehring. 22. ]MM registration booth. 23. Who Wants To Be a Mathematician contestants during the game. 24. Chess between sessions. 25. Networking area. 26. "Glass Geometry" booth. 27. Email center. 28. Dusa McDuff, AMS Invited Address speaker. 29. Message Board. 30. Who Wants To Be a Mathematician winners Susan Zhan and David Neville with host Mike Breen.

470 NOTICES OF THE AMS VOLUME 53, NUMBER 4 The Open Computer Algebra System

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r,- ~ JMacKichan SOFTWARE, INC. Tools for Scientific Creativity Since 1981 Toll-free : 877-724-9673 • E-mail: info@ mackichan.com 2006 Cole Prize in Algebra

The 2006 Frank Nelson Cole Citation Prize in Algebra was awarded The 2006 Cole Prize in Algebra is awarded to Jimos at the 112th Arumal Meeting of Kollar of Princeton University for his outstanding the AMS in San Antonio in Jan­ achievements in the theory of rationally connected uary 2006. varieties and for his illuminating work on a con­ The Cole Prize in Algebra is jecture of Nash. awarded every three years for The notion of a rational variety has long played a notable research memoir in an important role in algebraic geometry. An alge­ algebra that has appeared braic variety X is rationally connected if there are during the previous five years enough rational curves to connect points in X. A (until 2000, the prize was usu­ pioneer of the notion of rationally connected va­ ally awarded every five years). rieties, Kollar extended the theory from the com­ The awarding of this prize plex numbers to local fields. His papers (Annals of alternates with the awarding Math. 150 (1999), 357-367, and Michigan Math.]. Janos Kollar of the Cole Prize in Number 48 (2000), 359-368) and his joint work with Endre Theory, also given every three Szabo (Duke Math.]. 120 (2003), 251-267) are rec­ years. These prizes were established in 1928 to ognized as significant advancements in the theory honor Frank Nelson Cole on the occasion of his re­ of rationally connected varieties. tirement as secretary of the AMS after twenty-five In 1952, after proving that a compact differen­ years of service. He also served as editor-in-chief tiable manifold M is diffeomorphic to the zero set of the Bulletin for twenty-one years. The Cole Prize of real polynomials, John Nash conjectured that carries a cash award of US$5,000. there exists a smooth real algebraic variety, The Cole Prize in Algebra is awarded by the birational to projective space, whose real points are AMS Council acting on the recommendation of a diffeomorphic to M. Although known to be false selection committee. For the 2006 prize, the mem­ in dimension two, evidence suggested a positive so­ bers of the selection committee were: Georgia lution in higher dimensions until Kollar provided Benkart (chair), Eric M. Friedlander, and Craig L. counterexamples by classifying the diffeomor­ Huneke. phism types of smooth threefolds birational to Previous recipients of the Cole Prize in Algebra projective space whose real points are orientable. are: L. E. Dickson (1928), A. Adrian Albert (1939), This work is explained in a series of remarkable Oscar Zariski (1944), Richard Brauer (1949), Harish­ papers, notably his paper in]. Amer. Math. Soc. 12 Chandra (1954), Serge Lang (1960), Maxwell A. (1999), 33-83. Rosenlicht (1960), Walter Feit and John G. Thompson (1965), John R. Stallings (1970), Richard G. Swan Biographical Sketch (1970), Hyman Bass (1975), Daniel G. Quillen (1975), Janos Kollar was born in Budapest, Hungary, in Michael Aschbacher (1980), Melvin Hochster (1980), 1956. He did his undergraduate studies at Eotvos George Lusztig (1985), Shigefumi Mori (1990), Michel University in Budapest and his graduate studies at Raynaud and David Harbater (1995), Andrei Suslin Brandeis University with Teruhisa Matsusaka. After (2000), Aise Johan de Jong (2000), and Hiraku receiving his doctorate in 1984 he was a Junior Nakajima (2003). Fellow at Harvard University (1984-87) and then a The 2006 Cole Prize in Algebra was awarded to faculty member at the University of Utah (198 7-99). JANos KOLLAR. The text that follows presents the se­ Since 1999 he has been a professor at Princeton lection committee's citation, a brief biographical University. sketch, and the awardee's response upon receiving Kollar was elected to the Hungarian Academy of the prize. Sciences in 1995 and to the National Academy of

472 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Happy 1OOth Birthday, Godel! Sciences in 2005. He gave the AMS Colloquium Logic would be incomplete without you ... Lectures at the New Orleans Annual Meeting in 2001. Kollar's main research area is the birational geometry of higher dimensional algebraic varieties, Godel's Theorem and he also likes to explore the various applications An Incomplete Guide to Its Use and Abuse of algebraic geometry to algebra, combinatorics, complex analysis, differential geometry, and num­ Torkel Franzen ber theory. $24.95; Paperback; 182 pp.

Response "This unique exposition ofKurt Godel's The most basic algebraic variety is affine n-space stunning incompleteness theorems for a en' and it has been a long-standing problem to general audience manages to do what understand which varieties behave like en. For none other has accomplished: explain '--~""-----'~~_., surfaces the problem was settled by Castelnuovo clearly and thoroughly just what the theorems really say and in the 1890s: these are the surfaces which are imply and correct their diverse misapplications." birational to e2 . It took nearly a century to un­ -Solomon Feferman, editor of The Collected Works ofKurt GOdel derstand that the correct higher dimensional concept is not so global. Instead, we should focus on rational curves on varieties. There are plenty of Save 20% using discount code AMS rational curves in en: lines, conics, etc. Roughly speaking, a variety is rationally connected if it contains rational curves in similar abundance. It took some time to establish that rationally con­ nected varieties are indeed the right class, but by Logical Dilemmas now it is firmly settled that, at least in character­ The Life and Work of Kurt Godel istic zero, we have the right definition. I am very glad that the committee recognized the John W. Dawson, Jr. significance of this field and I feel deeply honored $34.00; Paperback; 376 pp. that they chose me to represent a whole area. This was truly a joint effort over the past fifteen years. "The tension between Godel's Much of the foundational work was done with Cam­ scientific rationalism and his personal pana, Miyaoka, and Mori, and the last piece of the instability is ably explored in this '""-'.___ ,..- basic theory was completed by Graber, Harris, solid biography." de ]ong, and Starr. Arithmetic questions over finite -Publishers Weekly and p-adic fields were explored with Colliot­ Thelene, Esnault, Kim, and Szabo, but the theory over global fields consists mostly of questions. Order online at Joint work with Bien, Borel, Corti, Schreyer, and www.akpeters.com Smith touched other aspects of rational connect­ edness. The Nash conjecture on the topology of rationally connected varieties over R turned out to be beau­ From Trotsky to Godel tiful algebraic geometry in dimension three, and the The Life of Jean van Heijenoort higher dimensional versions by Eliashberg and Viterbo use techniques from symplectic geometry. Anita Burdman Feferman The theory of rationally connected varieties is $34.95; Paperback; 432 pp. rapidly growing, with recent major results by Hacon, Hassett, McKernan, Tschinkel, and Zhang. I hope "F eferman introduces an unlikely hero .. . that the recognition by the Cole Prize will spur [in this] moving, original book." further activity. -The New Yorker Finally, I would like to thank three mathemati­ cians who had a great influence on my work: my "Many of Feferrnan's informants are now her eager thesis advisor Teruhisa Matsusaka, who taught readers ... sitting down with her book to fill in the puzzle me to look for the big picture; my collaborator of Van Heijenoort's life." Shigefumi Mori, with whom many of these ideas -Chicago Tribune were developed; and my former colleague Herb Clemens and the University of Utah for providing a wonderful environment to accomplish most of ~ www.akpeters.com this research.

APRIL 2006 NOTICES OF THE AMS 473 2006 BirkhoffPrize

The 2006 George David educated John L. Synge, was a professor of math­ Birkhoff Prize in Applied ematics. The family returned to Ireland from 192 5 Mathematics was awarded at to 1930. From 1930 to 1945 Morawetz received her the 112th Annual Meeting of education in the public schools of Toronto and the AMS in San Antonio in later her B.A. at the University of Toronto. She January 2006. started graduate school at the Massachusetts In­ The Birkhoff Prize recog­ stitute of Technology, receiving an M.S. in 1946. In nizes outstanding contribu­ October 1945 she married Herbert Morawetz, who tions to applied mathematics became a professor of polymer chemistry at Brook­ in the highest and broadest lyn Polytechnic. In 1946 Morawetz began working sense and is awarded every at New York University with Courant and Friedrichs, three years (until 2003, it was awarded usually every editing their book on compressible flow. In 1950 five years). Established in she completed a Ph.D. thesis on imploding shocks. 1967, the prize was endowed From 1950 to 1951 she worked at MIT with C. C. by the family of George Lin on fluid dynamic stability. In 1951 she returned David Birkhoff (1884- 1944), to NYU on a part-time basis and worked with who served as AMS president Friedrichs and Bers, mainly on the problems of Cathleen S. Morawetz during 1925-1926. The prize transonic flow and mixed equations. In the late is given jointly by the AMS 1950s, at Courant's suggestion, she began working and the Society for Industrial and Applied with Harold Grad on the mathematical problems Mathematics (SIAM). The recipient must be a mem­ of plasma physics, where she showed how a colli­ ber of one of these societies and a resident of the sionless shock could exist without invoking tur­ United States, Canada, or Mexico. The prize carries bulence. In 1957 she was appointed to the faculty a cash award of US$5,000. of the Courant Institute. She continued to work in The recipient of the Birkhoff Prize is chosen by partial differential equations, mainly on problems a joint AMS-SIAM selection committee. For the 2006 of mixed type but also on the wave equation. There prize, the members of the selection committee she solved problems of decay by new conservation were: Barbara L. Keyfitz, Charles S. Peskin, and laws and later used the same type of estimates Gunther Uhlmann (chair). Previous recipients of the Birkhoff Prize are: with Ludwig to justify geometrical optics in the lit Jurgen K. Moser (1968), Fritz John (1973), James B. region of a star shaped object. She continued to con­ Serrin (1973), Garrett Birkhoff (1978), Mark Kac centrate on these topics for the rest of her career. (1978), Clifford A. Truesdell (1978), Paul R. Garabe­ She retired in 1993 and became president of the dian (1983), Elliott H. Lieb (1988), Iva Babuska AMS in 1995 (she had also served as anAMS trustee (1994), S. R. S. Varadhan (1994), Paul H. Rabinowitz in the 1980s). Morawetz was awarded the National (1998), John N. Mather (2003), and Charles S. Medal of Science in 1998. Peskin (2003). The 2006 Birkhoff Prize was awarded to Response CATHLEEN S. MORAWETZ. The text that follows presents It is a totally unthought of and a wonderful surprise the selection committee's citation, a brief bio­ to receive the Birkhoff Prize. I am very, very grate­ graphical sketch, and the awardee's response ful to the two societies, AMS and SIAM, for choos­ upon receiving the prize. ing me. There are many, many people whom I would have liked to thank for helping me over the Citation years, but I would not have room for their names To Cathleen S. Morawetz for her deep and influ­ on this page. But one person stands out for sup­ ential work in partial differential equations, most porting and encouraging me when I was between notably in the study of shock waves, transonic the crucial professional ages of twenty-three and flow, scattering theory, and conformally invariant estimates for the wave equation. thirty-five. I worked part-time on my Ph.D., part­ time as a postdoc, and I had four children. That Biographical Sketch person was Richard Courant, the creator of the Cathleen Synge Morawetz was born in Toronto, Courant Institute at New York University, where I Canada, in 1923, where her father, Irish-born and have been a professor ever since.

474 NOTICES OF THE AMS VOLUME 53, NUMBER 4 2006 Conant Prize

The 2006 Levi L. Conant Prize was awarded at the makes connections with other 112th Annual Meeting of the AMS in San Antonio aspects of group theory so in January 2006. that the subject becomes The Conant Prize is awarded annually to recog­ more than just taxonomy. nize an outstanding expository paper published in Thus, he provides a glimpse either the Notices of the AMS or the Bulletin of the into a broad panorama of fi­ AMS in the preceding five years. Established in nite group theory. The article 2001, the prize honors the memory of Levi L. Co­ gives an unusual insider's look nant (1857-1916), who was a mathematician at at the process of mathemati­ Worcester Polytechnic University. The prize carries cal research, with its false a cash award of US$1,000. starts, insightful conjectures, The Conant Prize is awarded by the AMS Coun­ and dogged determination. cil acting on the recommendation of a selection One sees different approaches committee. For the 2006 prize, the members of the go in and out of fashion and selection committee were: Noam D. Elkies, Carl R. sometimes return with re- Ronald M. Solomon Riehm, and M. B. Ruskai (chair). newed vigor. Finally, he ar- Previous recipients of the Conant Prize are: Carl gues convincingly that even if the classification is Pomerance (2001), Elliott Lieb and Jakob Yngvason complete, many avenues remain open for further (2002), Nicholas Katz and Peter Sarnak (2003), investigation. The exposition is enhanced by de­ Noam D. Elkies (2004), and Allen Knutson and scriptions of the personalities of the many con­ Terence Tao (2005). tributors and their interactions. The 2006 Conant Prize was awarded to RoNALD M. Solomon has written a valuable survey, accessible SOLOMON. The text that follows presents the commit­ to a broad spectrum of mathematicians, that is tee's citation, a brief biographical sketch, and the both engaging and enlightening. awardee's response upon receiving the prize. Biographical Sketch Citation Ron Solomon was turned on to mathematics by his The Levi L. Conant Prize in 2006 is awarded to high school geometry teacher, Blossom Backal. He Ronald Solomon for his article "A Brief History of fell in love with group theory as an undergraduate the Classification of the Finite Simple Groups", at Queens College and had the great good fortune Bulletin oftheAMS38 (2001), no. 3, 315-352. to study with the masters-Walter Feit, David Gold­ Solomon gives a remarkable overview of the schmidt, Richard Lyons, and Leonard Scott-while work on the classification problem, from its in­ earning a Ph.D. at Yale University in 1971. The Na­ ception in an 1893 paper by Otto Holder to there­ tional Science Foundation Summer Institute in 1970 cent two-volume proof of the final theorem by was an unforgettable interlude. In the summer of Michael Ashbacher and Stephen Smith. Solomon's 1972, he heard Danny Gorenstein propose his vi­ article stresses key developments in a way that sionary sixteen-step program for the classification

APRIL 2006 NOTICES OF THE AMS 475 of the finite simple groups and spent two years as Call for Original Research Articles a Dickson Instructor at the University of Chicago, Emphasis should be in a) combinatorial results and analytic methods, learning with Jon Alperin and George Glauberman, b) analytic results and combinatorial methods, or c) a mixture and climbing one of Danny's steps. In 1974-1975, of combinatorics and analysis in the methods or in their applications. he made the first of several fruitful pilgrimages to Rutgers University, and then began thirty years We also encourage submission of high qua lily expository papm on 1opics (and counting) on the faculty of the Ohio State Uni­ of signilicant interest and short leHers (1-3 pages) describing open research problems versity. His sons, Ari and Michael, were born in 1980 of cunent interest that fall witlin the scope of the journal. and 1982, and have filled his life with love, joy, in­ tellectual sparring, and periodic tsurus. In 1982, he For mare information on submissions visit http:/ /IIWoiW.ojac.org/info.html began an ongoing collaboration with Gorenstein and Lyons to write a series of monographs presenting a substantial portion of the proof of the classifi­ Oolioe c;fouroAl of cation theorem. Since 2004, he has been blessed oZ\oAl!Jtie CombioAtories with the love of his wife, Rose. Department of Mathematics Response University of Missouri-Columbia Columbia, Missouri 65211 It is a great honor to receive the Levi L. Conant Prize FAX (573) 882-1869 from the Society. I am saddened that neither my [email protected] mother nor Walter Feit nor Danny Gorenstein are ..._____ --! http://www.ojac.org alive to share the joy of this occasion. My mother deserves double credit. I learned my writing skills from her, and my teenage rebellion against her Managing Editors authority drove me into mathematics. Walter and Alex losevich (University of Missouri) Sidnie Feit have always been most complimentary lzabella Laba (University of British Columbia) of my skills at group theory exposition, and of Sinai Robins (Temple University) course I learned much at the knee of that master expositor, Danny Gorenstein. A work of historical narrative can only be as good as its subject, and I had the advantage of a won­ 2006 AMS Sectional Meetings derful theme. The saga of the taming of the finite simple groups is a great one, shaped by titans of Aprill-2, 2006 the imagination from Lagrange, Gauss, and Galois Florida International University, Miami, FL to Thompson, Gorenstein and Aschbacher, with April 8-9, 2006 many other illustrious participants. It has been a University of Notre Dame, Notre Dame, IN rare privilege to be a friend and collaborator of the (features the Erdos Memorial Lecture by latter-day titans, and to tell a bit of their story. My Bela Bollobas) thanks to you all for reading and enjoying the tale. April 22-23, 2006 University of New Hampshire, Durham, NH April 29-30, 2006 San Francisco State University, San Francisco, CA (features the Einstein Public Lecture in Mathematics by Benoft Mandelbrot) October 7-8, 2006 University of Utah, Salt Lake City, UT October 21-22, 2006 University of Cincinnati, Cincinnati, OH October 28-29, 2006 University of Connecticut, Storrs, CT November 3-4, 2006 University of Arkansas, Fayetteville, AR For more information, see http://www.ams.org/amsmtgs/sectional.html

476 NOTICES OF THE AMS VOLUME 53, NUMBER 4 2006Awardfor Distinguished Public Service

The 2006 Award for Distinguished Public Service Yale University. This award rec­ was presented at the 112th Annual Meeting of the ognizes Dr. Howe for his multi­ AMS in San Antonio in January 2006. faceted contributions to math­ The Award for Distinguished Public Service is ematics and to mathematics presented every two years to a research mathe­ education. Not only is Dr. Howe matician who has made a distinguished contribu­ recognized for his mathemati­ tion to the mathematics profession during the pre­ cal research but he has also ceding five years. The purpose of the award is to taken a leadership role in na­ encourage and recognize those individuals who tional initiatives focused on the contribute their time to public service activities in teaching of mathematics and in support of mathematics. The award carries a cash the education of teachers. For prize of US$4,000. several years he served as Chair of the American Mathematical The Award for Distinguished Public Service is Society's Committee made by the AMS Council, acting on the recom­ on Educa­ mendation of a selection committee. For the 2006 tion, and he was a member of Roger Howe the National Research Council's award, the members of the selection committee Mathematical Sciences Education Board. He served were: William Lewis, Carolyn R. Mahoney, Paul J. J. as chair of the American Mathematical Society's Sally Jr., William Y. Velez (chair), and Margaret H. Consultative Committee involved in a revision of Wright. national mathematics standards in 1998. For many Previous recipients of the award are: Kenneth M. years he was on the board of directors of the Hoffman (1990), Harvey B. Keynes (1992), I. M. Connecticut Academy for Education in Mathemat­ Singer (1993), D. J. Lewis (1995), Kenneth C. Millett ics, Science and Technology. Moreover, he has (1998), Paul J. Sally Jr. (2000), Margaret H. Wright served on several national panels and study com­ (2002), and Richard Tapia (2004). mittees that have resulted in influential publica­ The 2006 Award for Distinguished Public Service tions, including the National Research Council's was presented to ROGER HoWE. The text that follows Mathematics Learning Study Committee (Adding presents the selection committee's citation, a brief It Up), the RAND Mathematics Study Panel (Math­ biographical sketch, and the recipient's response ematical Proficiency for All Students: Toward a upon receiving the award. Strategic Research and Development Program in Mathematics Education), and the Conference Board Citation of the Mathematical Sciences steering committee The 2006 Award for Distinguished Public Service (The Mathematical Education of Teachers). Dr. Howe is presented to Professor Roger Howe. Dr. Howe, a is currently chair of the Mathematics Standards member of the National Academy of Sciences, is the Study Group, a group of mathematicians who are William R. Kenan Jr. Professor of Mathematics at analyzing the mathematics standards in each state.

APRIL2006 NOTICES OF THE AMS 477 Dr. Howe has worked diligently over the years to in the mathematics curriculum from a high-level broaden and professionalize the involvement of a mathematical viewpoint. This project would in­ research mathematician in educational reform, to volve mathematicians and mathematics educators lead us towards the goal where involvement of working together to improve understanding of how mathematicians in education is viewed as a well­ these topics do and should play out in a produc­ informed professional activity by mathematicians tive curriculum. The results of this project would and educators alike. be a series of essays distilling our best current knowledge of these topics. Biographical Sketch It took me several years of working on mathe­ Roger Howe earned his Ph.D. in 1969 from the matics education before I began to feel I had a University of California at Berkeley, under the di­ perspective which to some extent integrated edu­ rection of Calvin C. Moore. He spent 1969 to 1974 cational and mathematical concerns in a sensible at SUNY Stony Brook, and has been at Yale since way. If all mathematicians who are to work in ed­ 1974. His research has been mainly in the repre­ ucation require a comparable initiation period, the sentation theory of groups and harmonic analysis, barriers to entry into educational work will always and its applications to the theory of automorphic remain too high. I have been discussing how to con­ forms, invariant theory, geometry, ergodic theory, struct a workshop which would enable interested partial differential equations, and mathematical mathematicians to learn, in a few weeks, much of physics. He is a member of the National Academy which it took me and others of my generation of of Sciences, the American Academy of Arts and Sci­ mathematicians-in-education years to absorb. ences, and the Connecticut Academy of Science Finally, one of the most important publications and Engineering. He has served as the editor of in mathematics education in the last ten years was Research Announcements for the Bulletin of the the book Knowing and Teaching Elementary Math­ American Mathematical Society, and as chair and ematics, by Liping Ma. I wrote a review of this book member of the Committee on Education. He has for the Notices in 1999, and I have continued to also served on many non-AMS committees devoted think about it since. This book presents responses to issues of mathematics education. He currently of Chinese teachers of elementary mathematics to is visiting Stony Brook University in hopes of ini­ several questions about teaching important math­ tiating a long-term project for improvement of ematics topics. I have become convinced that the K-12 mathematics teaching and curriculum in the level of understanding of teaching and curriculum United States. revealed by their answers is something that is very much needed, but very rare, in the United States. Response I am currently working with the Mathematics for I thank the Society for this distinction. I am grate­ America foundation and Stony Brook University to ful in many ways and for many reasons. initiate a project to develop and disseminate this I have been working on issues in mathematics kind of understanding. The thrust of the project education for ten to fifteen years. As the citation would be to produce and work with teams of math­ says, I have been on lots of committees! Mathe­ ematics specialist teachers who know mathemat­ matics education is an area with few proofs and ics very well, and who teach all grades one to six. I feel even fewer theorems. Therefore, it is immensely en­ that there should be many projects of this couraging to have one's efforts applauded in this sort, and each such project would need a dedi­ cated research mathematician at its core. official and striking way. At the same time, I am mindful that awards like this can never recognize all who may merit them. I know several colleagues whose work in education deserves commendation as much or more than mine, but I have been the lucky one this time. I have become convinced that it is vital for the health of U.S. mathematics education that in the future more mathematicians contribute their time, knowledge, and insights to improve it. This cannot happen to the extent it needs to unless work on ed­ ucation no longer makes one a candidate for the Public Service Award! It must become a somewhat normal thing to do and consistent with maintain­ ing a research program. My current projects aim at making this possible. With Alan Tucker, I am working to design a pro­ ject intended to inspect critical issues and topics

478 NOTICES OF THE AMS VOLUME 53, NUMBER 4 2005 Morgan Prize

The 2005 AMS-MAA-SIAM Frank and Brennie Mor­ Citation gan Prize for Outstanding Research in Mathemat­ The winner of the 2005 Mor­ ics by an Undergraduate Student was awarded at gan Prize for Outstanding Re­ the Joint Mathematics Meetings in San Antonio in search in Mathematics by an January 2006. Undergraduate is Jacob Fox. The Morgan Prize is awarded annually for out­ Jacob Fox is now in his fourth standing research in mathematics by an under­ year of undergraduate stud­ graduate student (or students having submitted ies at MIT [Massachusetts In­ joint work). Students in Canada, Mexico, or the stitute of Technology]. The United States or its possessions are eligible for award is based on a most as­ tounding collection of re- consideration for the prize. Established in 199 5, the search papers by any under- prize was endowed by Mrs. Frank Morgan of Al­ graduate mathematician. lentown, Pennsylvania, and carries the name of her Jacob Fox's research is in three late husband. The prize is given jointly by the AMS, areas: Ramsey-type problems, the Mathematical Association of America (MAA), rainbow patterns in colwings Jacob Fox and the Society for Industrial and Applied Mathe­ of the integers or Z /mZ, and matics (SIAM) and carries a cash award ofUS$1,000. other problems in graph the- Recipients of the Morgan Prize are chosen by a ory (namely on discrepancy, clique number, em­ joint AMS-MAA-SIAM selection committee. For the bedding, and diameter). Jacob Fox is an excellent 2005 prize, the members of the selection commit­ problem solver, passionately interested in these tee were: Kelly J Black, James H. Curry, Herbert A. subjects, driven by his love of mathematics, his tal­ Medina, Philippe M. Tondeur (chair), Judy L. Walker, ents, and his originality. He communicates easily and Paul Zorn. and frequently collaborates with a variety of dis­ tinguished researchers. He also frequently pub­ Previous recipients of the Morgan Prize are: Kan­ lishes alone. Jacob Fox's research exhibits a for­ nan Soundararajan (1995), Manjul Bhargava (1996), midable ability to get to the heart of the issues in Jade Vinson (1997), Daniel Biss (1998), Sean the problems at hand, and the ability to develop McLaughlin (1999), Jacob Lurie (2000), Ciprian extremely ingenious and novel techniques. In ad­ Manolescu (2001), Joshua Greene (2002), Melanie dition to being able to solve problems posed by oth­ Wood (2003), and Reid Barton (2004). ers, Fox has also excelled at finding topics all by The 2005 Morgan Prize was awarded to JACOB Fox. himself, formulating novel conjectures and ap­ The text that follows presents the selection com­ proaches to solutions. His accomplishments are mittee's citation, a brief biographical sketch, and shaping his areas of research and are of extraor­ the awardee's response upon receiving the prize. dinary promise for the future.

APRIL 2006 NOTICES OF THE AMS 479 I

NATIONAL I SECURITV AGENCY Biographical Sketch Jacob Fox (previously Jacob Licht) is a senior ma­ joring in theoretical mathematics at the Massa­ chusetts Institute of Technology. He first studied advanced mathematics as an "epsilon" at the Ross Program at Ohio State University. His love for math­ ematics was further developed through the Re­ search Science Institute, which laid the foundation for work that earned him his first publication, second place in the Intel Science Talent Search, and fourth place in the Siemens Westinghouse Competition. In college Jacob's interest in combi­ natorics research was strengthened through un­ dergraduate research supervised by Daniel J. Kleitman, Lucent summer internships at Bell Labs, and, most recently, Joe Gallian's summer Research Experiences for Undergraduates program at the University of Minnesota, Duluth. In a paper in the journal of Combinatorial Theory Series A, Fox and Put Your Math Kleitman proved the first nontrivial case of Richard Rado's 1933 Boundedness Conjecture. Extending Intelligence to Work earlier work of Erdos, Kakutani, Komjath, and Rado, Jacob proved an infinite color analogue of Rado's When you join NSA. you join a highly theorem on partition regularity of systems of lin­ talented group of Mathematicians who deduce ear equations. At the Duluth program, he proved a bipartite analogue of Dilworth's theorem on structure where it is not apparent, find patterns partially ordered sets, which will appear in the in seemingly random sets, and create order out journal Order. His research interests are in Hun­ of chaos. They apply Number Theory, Group garian-style combinatorics, particularly Ramsey Theory, Finite Field Theory, Linear Algebra, theory, extremal graph theory, combinatorial number theory, and probabilistic methods in Probability Theory, Mathematical Statistics, combinatorics. Combinatorics, and more to a world of challenges. They exchange ideas and work with Response some of the finest minds and most powerful I am honored.to be the recipient of this prize. I would like to thank Mrs. Frank Morgan for en­ computers in the country. And you can too, dowing the prize and the AMS, MAA, and SIAM for when you put your math intelligence to sponsoring it. Daniel J. Kleitman and Rados work at NSA. Radoicic deserve special thanks for the many years they have mentored my research. I would also like to thank Yuliy Baryshnikov, Joe Gallian, Mohammad Mahdian, Janos Pach, Igor Pak, and numerous oth­ NSA: Securing Tomorrow Today I ._ ers for helping my development as a research math­ ematician. I thank my family for their love and For more information and to apply support. online, visit our Web site.

www.NSA.gov/Careers WHERE INTELLIGENCE GO ES TO WORK

U.S. citizenship is required. NSA is an equal opportunity employer. All applicants for employment are considered without regard to race, color, religion, sex, national origin, age, marital status, handicap, sexual orientation, or status as a parent.

480 N OTICES OF THE AMS VOLUME 53, N UMBER 4 2006Awardfor an Exemplary Program or Achievement in a Mathematics Department

The 2006 Award for an Exemplary Program or The Harvey Mudd College Mathematics Clinic has Achievement in a Mathematics Department was served as a trailblazer and a model for other pro­ presented at the 112thAnnual Meeting of the AMS grams for more than thirty years. This innovative in San Antonio in January 2006. program connects teams of math majors with This award, established by the AMS Council in real-world problems, giving students a terrific re­ 2004, was given for the first time in 2006. The search experience as well as a glimpse at possible purpose is to recognize a department that has future careers. Undergraduate research is a theme distinguished itself by undertaking an unusual throughout the mathematics program at Harvey or particularly effective program of value to the Mudd College, as exemplified by the over twenty mathematics community, internally or in relation papers published in the last three years by Harvey to the rest of society. Departments of mathemati­ Mudd College mathematics faculty with student cal sciences in North America that offer at least a co-authors. bachelor's degree in mathematical sciences are The Harvey Mudd College Mathematics Depart­ eligible. The award carries a cash prize of US$1 ,200 ment promotes the pleasures of mathematics to and is to be given annually. nonmajors so well that many nonmajors participate The award is presented by the AMS Council in the weekly Putnam Seminar on problem solving, acting on the recommendation of a selection com­ leading to an unusually large number of Harvey mittee. For the 2006 award, the members of the Mudd students taking the Putnam Exam each year. selection committee were: Sheldon Axler (chair), The Putnam Seminar's work has produced consis­ Joel V. Brawley, James H. Curry, Karl Knight, and tently outstanding performances in the Putnam Donal B. O'Shea. Exam, with Harvey Mudd ranking in the top ten The recipient of the 2006 Award for an Exem­ nationwide in 2001, 2002, and 2003 (and just plary Program or Achievement in a Mathematics missing in 2004 with an eleventh-place finish). Department is the MATHEMATICS DEPARTMENT AT Amazingly, Harvey Mudd mathematics students HARVEY MUDD COLLEGE. have won nineteen NSF [National Science Founda­ tion] fellowships over the last six years. Citation The Harvey Mudd College Mathematics Depart­ The first Award for an Exemplary Program or ment also devotes serious effort toward outreach Achievement in a Mathematics Department is to low-income and underrepresented minority com­ presented to Harvey Mudd College in Claremont, munities. This work includes programs aimed at California. The Mathematics Department at Harvey stimulating interest in mathematics and science in Mudd College excels in numerous dimensions. a local high school in a low-income area. The Its exciting programs have led to a doubling of the department also runs a workshop in Jamaica for number of math majors over the last decade. Jamaican high school mathematics teachers, Currently more than one out of every six graduat­ focusing on creative methods for teaching mathe­ ing seniors at Harvey Mudd College majors in math­ matics. ematics or in new joint majors of mathematics The mathematics community is fortunate to with computer science or mathematical biology. have Harvey Mudd College present such an out­ Furthermore, about 60 percent of these math ma­ standing example of an exemplary program in a jors continue their education at the graduate level. mathematics department.

APRIL 2006 NOTICES OF THE AMS 481 MathSciNet Matters

references for a review or a paper or other bibli­ The "MathSciNet Matters" column appears in ography. Simply click the "Add to Clipboard" but­ the Notices several times a year. It includes ton from a headline list or full item. If you are information on new features of MathSciNet signed in, the Clipboard can be saved indefinitely. and on the underlying Mathematical ,Reviews Otherwise it is saved for 24 hours. The contents of Database, together with tips on how to use the Clipboard can be viewed by clicking the Clip­ MathSciNet to make the most of its richness of board link on the side panel of search screens, or structure and content. the View clipboard link at the top of results screens. To download to your computer, select a format and Champion Reviewers. The review forms an im­ then click Text next to the format menu. The re­ portant core of the information available in Math­ sulting page can be saved to your computer using SciNet. It is too bad there isn't Reviews in the Math­ the browser File-Save Page As . The Clipboard can SciNet name somewhere, as there is in the paper be viewed and saved in a vanilla format (Citations publication Mathematical Reviews. The assembly of (ASCII)), in BibTeX, in AMSRefs or in EndNote. more than 12,000 reviewers worldwide performs Factoid. On January 6, 2006, there were 121,856 an invaluable service to the mathematical com­ reference lists in MathSciNet. These lists contained munity by reading mathematical papers and books, almost 1.9 million individual references. The num­ and writing thoughtful reviews of their contents. ber of reference lists grows daily. These reviews help guide researchers through the Reviewers Corner. Have you ever wondered literature. Here we salute those reviewers who have why you get the items that you are selected to re­ written reviews consistently over a very long span view? This match-up process is called assignment of years. Tom Apostol wrote his first review in in the argot of MR. At the MR offices in Ann Arbor 1951, and is still an active reviewer. That's 54 years there is a very cool computer application that aids of service. Other active reviewers who have been the process by which a Mathematical Reviews ed­ reviewing for at least 50 years include: R. Finn itor assigns a particular paper or book to a re­ (first review in 195 5), R. G. Langebartel (1949), viewer and that item is put into the hands of the B. N. Moyls (1948), and F. Ursell (1955). Albert]. chosen reviewer. However, the heart of this as­ Coleman wrote many reviews. The first was pub­ signment is what you have told us, possibly a long lished in 1946 and the last in 2001: fifty-five years time in the past, about what your mathemati­ later. Adolph W. Goodman (1915-2004) wrote re­ cal/reviewing interests are. The information that views from 1946 to 1995: forty-nine years. Then you gave us was boiled down to a list of classifi­ there are the long-term reviewers whose reviews ap­ cations from the MSC (Mathematics Subject Clas­ peared in Volume 1 of MR (1940). Dirk Struik wrote sification, 2000) and to a set of comments with re­ reviews up to 1999. H. S. M. Coxeter wrote reviews gard to your interests. It is a good idea to review until1998. There is John Todd, who wrote reviews these classes and interests from time to time. A new through 1997; Paul Erdos, through 1996; A. H. interface to reviewer data and to review submission Taub, through 1996;].]. Burckhardt, through 1995; was released at the end of 2005. It makes it easy F.]. Murray, through 1993; R. P. Boas Jr., through to review the data we have for you on MSC classes 1992; Saunders Mac Lane, through 1991; Z. W. Birn­ and interests and to make changes, if needed. You baum, through 1991; D. H. Lehmer, through 1990; may also review and update name, postal address, M. Marden, through 1990; and]. C. Oxtoby, through and email information. Even if you don't have are­ 1990. view to submit, you will find it interesting to go to Long-time reviewers can take a long-term view. www. ams. org/mresubs, Sign In, and check out the ]. S. Frame's review of a 1940 part I paper of Cox­ new interface. All that is needed is an AMS Web Ac­ eter appeared in Volume 2 of MR (MR0002181). count, which most reviewers already have, and Forty-six years later, he reviewed Coxeter's part II your reviewer number in your Web Account pro­ paper (MR0774558). file. Give it a try. Have you tried the Clipboard? The Clipboard -Norman Richert in MathSciNet is an excellent way to collect verified Mathematical Reviews

482 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Mathematics People

of the Banff International Research Station (BIRS). She has Tomczak-Jaegermann Awarded also served as the University of Alberta site director of PIMS CRM-Fields-PIMS Prize and as associate editor of the Canadian journal of Mathe­ matics and the Canadian Mathematical Bulletin. NICOLE TOMCZAK-JAEGERMANN of the University of Alberta has The CRM and the Fields Institute established the CRM­ been awarded the 2006 CRM-Fields-PIMS Prize. The prize, Fields Prize in 1994 to recognize exceptional research in awarded annually by the Centre de Recherches Mathema­ the mathematical sciences. In 2005 PIMS became an equal tiques (CRM), the Fields Institute, and the Pacific Institute partner, and the name was changed to the CRM-Fields­ for the Mathematical Sciences (PIMS), recognizes excep­ PIMS Prize. Previous recipients of the prize are H. S. M. tional contributions by a mathematician working in Canada. (Donald) Coxeter, George A Elliott, James Arthur, Robert V. The prize carries a cash award of 10,000 Canadian dollars Moody, Stephen A. Cook, Israel Michael Sigal, William T. (approximately US$8,600) and an invitation to give alec­ Tutte, John B. Friedlander, John McKay, Edwin Perkins, ture at each institute. Donald A Dawson, and David Boyd. Tomczak-Jaegermann was honored "in recognition of her exceptional achievements in functional analysis and geometric analysis." According to the prize citation, she -From a Fields Institute announcement "is one of the world's leading mathematicians working in functional analysis. She has made outstanding contribu­ tions to infinite dimensional Banach space theory, as­ AWM Essay Contest Winners ymptotic geometric analysis, and the interaction between these two streams of modern functional analysis. She is Announced one of the few mathematicians who have contributed im­ The Association for Women in Mathematics (AWM) has an­ portant results to both areas. In particular, her work con­ nounced the winners of its 2005 essay contest, "Biographies stitutes an essential ingredient in a solution by the 1998 of Contemporary Women in Mathematics". The grand prize Fields Medallist W. T. Gowers of the homogeneous space went to ARICA FoNG, a student at Bucknell University, for problem raised by Banach in 1932." her essay "Discovering Mathematics in Nature: Dr. Linda Tomczak-Jaegermann received her master's (1968) and Smolka". This essay won first place in the college category Ph.D. (1974) degrees from WarsawUniversityinPoland. She and, as grand prize winner, will be published in the AWM taught at Warsaw University from 1975 to 1983 and held a Newsletter. The first-place winner in the grade 9-12 cate­ visiting professorship at Texas A&M University from 1981 gory was TYLER.WOTIRICH, a student at Roseville Area High to 1983. She has been teaching at the University of Alberta since 1983 and currently holds a Canada Research Chair School in Roseville, Minnesota, for an essay titled "The Per­ in Geometric Analysis. She was an invited lecturer at the In­ severance of a Woman in Actuarial Science: Nancy Myers". ternational Congress of Mathematicians in 1998. Her awards First place in the middle school category (grades 6-8) was include a Killam Research Fellowship and the Krieger-Nelson awarded to NINA KAJVIATH of Joaquin Miller Middle School, Prize Lectureship of the Canadian Mathematical Society. Saratoga, California, for an essay titled "The Beauty of She is a Fellow of the Royal Society of Canada. She has served Mathematics". A complete list of the winners, as well as on committees of the Natural Sciences and Engineering copies of their essays, can be found on the AWM website, Research Council of Canada (NSERC) and the Canadian http://www.awm-m ath.org/biographies/contest/ Mathematical Society (CMS), as well as on the Canada Coun­ 200 5. ht ml. cil Killam Research Fellowship Committee, the Canada Re­ search Chairs College of Reviewers, and the scientific board -From an A WM announcement

APRIL 2006 . NOTICES OF THE AMS 483 Mathematics Opportunities

Division of Mathematical Sciences (DMS) invite submis­ NSF Program in Informal sion of research proposals for projects that advance the Science Education mathematical and statistical foundations of research in the social, behavioral, or economic sciences. Proposals The Informal Science Education (ISE) Program of the for workshops or symposia that foster the interaction of National Science Foundation (NSF) supports projects that social, behavioral, and economic scientists with mathe­ develop and implement informal learning experiences maticians and statisticians also are welcome. designed to increase interest and engagement in and It is estimated that nine to eighteen awards will be understanding of science, technology, engineering, made, ranging in duration from one to four years and and mathematics (STEM) among individuals of all ages carrying award amounts of US$150,000 to US$650,000. and backgrounds, as well as projects that advance knowl­ The deadline date for full proposals is April 20, 2006. edge and practice of informal science education. Projects For more information, see http://www.nsf.gov/ intended to target public audiences may do so through publications/pub_summ.jsp?ods_key=nsf06531&org= such means as permanent and traveling exhibitions; films, NSF. television, and radio series; Web-based projects; citizen science programs; and youth and community programs. - From an NSF announcement In addition, the program supports projects that target ISE professionals to increase knowledge and the implemen­ tation of practice, such as through research studies, AP Calculus Readers Sought conferences, formation of networks, and professional development. These projects should strengthen the The Educational Testing Service and the College Board in­ infrastructure for informal science learning by the public. vite interested college faculty to apply to be readers for Projects are expected to demonstrate strategic impact, the Advanced Placement Calculus Exam. In June, AP high innovation, and collaboration. school and college faculty members from around the world The deadline for required preliminary proposals is gather in the United States for the annual AP Reading. There March 21, 2006. The deadline for full proposals is June 22, they evaluate and score the free-response sections of 2006. For more detailed information, see the website the AP Exams. AP Exam readers are led by a chief reader, http://www.nsf.gov/publications/pub_summ. a college professor who has the responsibility of ensur­ jsp?ods_key=nsf06520. ing that students receive grades that accurately reflect college-level achievement. Readers find the experience an -From an NSF announcement intensive collegial exchange in which they can receive professional support and training. To learn more about this opportunity or to apply for a position as a reader, see the website http: I /apcent ra l . Call for Proposals for NSF collegeboard.com/article/0, ,153-176-0-4137,00. Program in Mathematical, html or send email to apreader@ets. org. Sodal, and Behavioral Sdences - Caren L. Diefenderfer, Hollins University The National Science Foundation's (NSF) Directorate for Social, Behavioral, and Economic Sciences (SBE) and the

484 NOTICES OF THE AMS VOLUME 53, NUMBER 4 For Your Information

Oberwolfach Photo Collection Now Available Online Most mathematicians who have been visiting the Mathe­ matisches Forschungsinstitut Oberwolfach (MFO) over the years might remember the "White Box", a filing cabinet containing photographs of mathematicians from all over the world. This well-known collection has been digitized, kindly supported by Springer-Verlag Heidelberg, and is now available online at http: I I owpdb. mfo. de. The Oberwolfach Photo Collection is based on Komad Jacobs's (Erlangen) large photo collection. In the 1950s Jacobs started to make copies of the photographs he had taken and donated these copies to the MFO. In 2005 he transferred his own collection completely with all rights to the MFO. The collection also contains many copies of pictures taken by George M. Bergman (Berkeley). Since about 1998 we have taken photos of every single A sample from the Oberwolfach Photo Collection: group visiting Oberwolfach and so try to continue this Henri Cartan at Oberwolfach, September 1971. excellent collection. (Photograph by Klaus Peters) We are interested in further enlarging our collection and would very much appreciate receiving additional for his generous support in building up our photo collec­ photographs, such as, for example, photographs of well­ tion; George M. Bergman, Berkeley; and Ludwig Danzer, known mathematicians or of events interesting to the Dortmund. Special thanks go to Springer-Verlag Heidelberg mathematical community. Since we want our collection to for helping to organize the digitization of the photos. be as complete as possible, a relation to Oberwolfach is not compulsory. For that reason we would like to invite -Gert-Martin Greuel, Director, those who are interested in supporting us in our efforts Mathematisches Institut Oberwolfach to provide us with photographs (either conventional or dig­ itized) or to inform us about potential sources. Appropriate photographs will then be added to our collection. We do Correction: NSF Graduate not claim a copyright, but we need the authorization that allows the publication of the photographs and the charge­ Fellowships free transmission to third parties for scientific needs only. Additional information and corrections concerning the photographs we have published are most welcome. The February 2006 issue of the Notices carried an The online version of the collection is presented in a announcement about the most recent recipients of minimized format. Publishing houses or researchers Graduate Research Fellowships from the National wishing to use some of the photos in a publication are re­ Science Foundation. The announcement gave the wrong monetary amount of these fellowships; they provide quested to contact the MFO to get the original version, pro­ vided that there are no outstanding copyright problems. US$30,000 per year. In case of publication we would appreciate receiving a -Allyn jackson charge-free copy of the respective book or issue for our library. We would like to express our sincere thanks to all per­ sons who have been providing photographs to the MFO. In particular we wish to thank: Komad Jacobs, Erlangen,

APRIL 2006 NOTICES OF THE AMS 485 Inside the AMS

News about Notices Website Early Career Profile Network At its meeting in January 2006, the AMS Council decided The AMS recruits and supports a network of mathemati­ to abolish the requirement that users who want to access cal sciences departments that systematically provide job Notices articles must log on to the AMS website. The Notices profiles of their recent bachelor's-level alumni. This net­ is available on the Web free of charge to AMS members and work is called the Early Career Profile Network. Participating nonmembers alike thanks to the support of dues-paying departments place alumni profiles on their individual AMS members. departmental websites, and the AMS links to the profiles The search function on the Notices website was recently from the AMS Early Career Profile webpage. The project upgraded through the use of the Google search function. was initiated with partial support from the Alfred P. Users who previously found it difficult to locate Notices Sloan Foundation under the auspices of the Sloan Career materials using the search function will find a significant Cornerstone Series. improvement. Many undergraduates have only a vague idea about the Questions and comments about the Notices may be utility of a major in the mathematical sciences. Even expe­ directed to noti ces@ams. o rg. rienced counselors are sometimes hard-pressed to provide specific details to inquiring students, and the information that is provided - Allyn jackson tends to be anecdotal and incomplete. And yet, most faculty know that a student with a major in the mathematical sciences has excellent opportunities in busi­ ness, government, and further education in virtually any AMS Presidents: A Timeline field. In general, departments haven't collected the sup­ The AMS has posted on its website a timeline that presents porting evidence, the information about what jobs their graduates enter when biographical information about AMS presidents. The roster they first leave the university or college. of AMS presidents includes some of the most outstanding Career profiles are a valuable source of information and influential mathematicians in North America, so the for prospective mathematical sciences majors. The timeline provides an interesting view of the history of goal of the Early Career Profile Network is to collect relevant and mathematics inNorth America. AMS presidents have played timely information about career opportunities for under­ a key role in leading the Society in its publications, meet­ graduate mathematical sciences majors that can be broadly ings, professional visibility, and support for research. Start­ disseminated to hig)l school and college students. This net­ ing with the very first president, John H. van Amringe, who work is a straightforward and efficient way to continue the served from 1888 untill890, the timeline includes biogra­ flow of timely career information needed by students, phies of such individuals as Oswald Veblen, G. D. Birkhoff, counselors, teachers, and faculty. Marston Morse, John von Neumann, Oscar Zariski, Saunders The URL for the Early Career Profile Network is http: I I Mac Lane, Julia Robinson, and Michael Artin, as well as the www.ams.orglearly-careersl. College math depart­ current president, James Arthur. ments wishing to become involved in posting profiles of AMS Presidents: A Timeline is available at http: I lwww. their recent graduates should contact Ellen Maycock, AMS ams.orglamslamspresidents.html. associate executive director, at ejm@ams. org.

-Allyn jackson - AMS Membership and Programs Department

486 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Reference and Book List

The Reference section of the Notices Upcoming Deadlines . March 31, 2006: Nominations for is intended to provide the reader with March 21,2006: Preliminary proposals the Prize for Achievement in Informa­ frequently sought information in for NSF Program in Informal Science tion-Based Complexity. Contact Joseph an easily accessible manner. New Education. See "Mathematics Oppor­ Traub,[email protected]. information is printed as it becomes tunities" in this issue. March 31, 2006: Nominations for available and is referenced after the Third World Academy of Sciences March 31, 2006: Applications for first printing. As soon as information Prizes. See http : I lwww. twas. orgl. AMS Congressional Fellowships. See is updated or otherwise changed, it April 7, 2006: Proposals for 2007 will be noted in this section. http:llwww.ams.orglgovernmentl NSF-CBMS Regional Conferences. See cong ressfe ll ow ann. htm l or con­ the CBMS website, http: I l www. Contacting the Notices tact the AMS Washington office at cbmsweb.orgiNSFI2007_call.htm, The preferred method for contacting 202-588-1100; email: amsdc@ams. org. or contact: Conference Board of the the Notices is electronic mail. The editor is the person to whom to send Where to Find It articles and letters for consideration. A brief index to information that appears in this and previous issues of the Notices. Articles include feature articles, AMS Bylaws-November 2005, p. 1239 memorial articles, communications, AMS Email Addresses-February 2006, p. 251 opinion pieces, and book reviews. The AMS Ethical Guidelines- ]une/]uly 2004, p. 675 editor is also the person to whom to AMS Officers 2004 and 2005 (Council, Executive Committee, send news of unusual interest about Publications Committees, Board of Trustees)-May 2005, p. 564 other people's mathematics research. AMS Officers and Committee Members-October 2005, p. 1073 The managing editor is the person Conference Board of the Mathematical Sciences-September 2005,, to whom to send items for "Mathe­ ~~2 . matics People", "Mathematics Op­ Information for Notices Authors- ]une/]uly 2005, p. 660 portunities", "For Your Information", Mathematics Research Institutes Contact Information-August; 2005, "Reference and Book list", and "Math­ p. 770 . ematics Calendar". Requests for National Science Board- january 2006, p. 62 J permissions, as well as all other New Journals for 2004-june/July 2005, p. 662 inquiries, go to the managing editor. NRC Board on MathematiCal Sciences and Their Applications- March 2006, The electronic-mail addresses are p. 369 noti ces@math. ou. edu in the case of NRC Mathematical Sciences Education Board-April 2006, p. 488 the editor and noti ces@ams. org in NSF Mathematical and Physical Sciences Advisory Committee- February the case of the managing editor. The 2006,p.255 fax numbers are 405-325-7484 for Program Officers for Federal Funding Agencies-October 2005, the editor and 401-331-3842 for the p. 1069 (DoD, DoE); November 2005, p. 122~ (NSF) managipg edi!or. Postal addresses Stipends for Study and Travei-September.2005, p. 900 may be found in the masthead.

APRIL 2006 NOTICES OF THE AMS 487 Reference and Book List

Mathematical Sciences, 1529 Eigh­ Keisha M. Ferguson, Pattengill Ele­ may be sent to noti ces-bookl i st@ teenth Street, NW, Washington, DC mentary School, Ann Arbor, Michigan ams .org. 20036; telephone: 202-293-1170; fax: Louis Gomez, Northwestern Uni­ ''Added to "Book List" since the 202-293-3412; email: l ko l be@maa. versity list's last appearance. org or rosi er@georgetown. edu. Javier Gonzalez, Pioneer High April 20, 2006: Proposals for NSF School, Whittier, California A 3 & His Algebra: How a Boy from Program in Mathematical, Social, and Sharon Griffin, Clark University Chicago's West Side Became a Force in Behavioral Sciences. See "Mathemat­ Phillip A. Griffiths (chair), Institute American Mathematics, by Nancy E. ics Opportunities" in this issue. for Advanced Study Albert. iUniverse, Inc., January 2005. May 1, 2006: Applications for AWM Arthur Jaffe, Harvard University ISBN 0-595-32817-2. (Reviewed De­ Travel Grants. See http: I lwww. awm­ Jeremy Kilpatrick, University of cember 2005.) math. orgltravel grants. html; tele­ Georgia Action This Day, edited by Michael phone 703-934-0163; email: awm@ math. julie Legler, St. Olaf College Smith and Ralph Erskine. Random umd. edu; or contact Association for Jim Lewis, University of Nebraska; House of Canada, February 2003. WomeninMathematics, 11240Waples Lincoln ISBN 0-593-04910-1. Mill Road, Suite 200,Fairfax, VA22030. Kevin F. Miller, University of Michi­ Beyond Coincidence: Amazing Sto­ May 31,2006: Registration for the gan, Ann Arbor ries of Coincidence and the Mystery Thirteenth International Mathemat­ Marge Petit (vice chair), Consultant, and Mathematics behind Them, by ics Competition for University Stu­ Fayston, Vermont Martin Plimmer and Brian King. dents (IMC). See the website http: I I Donald Saari, University of Cali­ Thomas Dunne Books, December www. i me-math. org or contact John E. fornia, Irvine 2005. ISBN 0-312-34036-2. Com­ Jayne, Department of Mathematics, Nancy]. Sattler, Terra State The Book of Presidents. London Freemont, Ohio University College London, Gower munity College, Mathematical Society, 2005. ISBN 0- Richard ]. Schaar, Texas Instru­ Street, London WClE 6BT, United 950-27341-4. ments Kingdom; telephone +44-20-7679- A Brief History of Infinity, by Paolo Frank Wang, Oklahoma School of 7322; fax +44-20-7419-2812; email: Zellini. Penguin Books (paperback), Science and Mathematics [email protected]. March 2005. ISBN 0-141-00762-1. june 1, 2006: Applications for fall MSEBStaff The Calculus Gallery: Masterpieces Christine Mirzayan from Newton to Lebesgue, by William program of the David R. Mandel, Director Grad­ Dunham. Princeton University Press, Science and Technology Policy Mary Ann Kasper, Senior Program December 2004. ISBN 0-691-09565-5. uate Fellowship Program of the Assistant Chance: A Guide to Gambling, Love, National Academies. See the website The contact information is: Math- · the Stock Market and just About Every­ http:llwww7.nationalacademies. ematical Sciences Education Board, thing Else, by Amir D. Aczel. Thun­ orglpol i cyfell ows, or contact The National Academy of Sciences, 500 der's Mouth Press, October 2004. ISBN National Academies Christine Mirza­ Fifth Street, NW, 11th Floor, Wash­ August yan Science and Technology Policy ington, DC 20001; telephone 202-334- 1-56858-316-8. (Reviewed Graduate Fellowship Program, 500 3294; fax 202-344-1453; email: mseb@ 2005.) Women Fifth Street, NW, Room 508, Wash­ nas. edu; World Wide Web http: I I Change Is Possible: Stories of ington, DC 20001; telephone: 202- www7.nationalacademies.orgl and Minorities in Mathematics, by Pa­ 334-2455; fax: 202-334-1667. msebllMSEB_Membership.html. tricia Clark Kenschaft. AMS, Septem­ june 22, 2006: Full proposals for ber 2005. ISBN 0-8218-3748-6. NSF Program in Informal Science Book List Coincidences, Chaos, and All That Education. See "Mathematics Oppor­ The Book List highlights books that Math jazz: Making Light of Weight¥ tunities" in this issue. have mathematical themes and are Ideas, by Edward B. Burger and Michael October 1, 2006: Applications for aimed at a broad audience potentially Starbird. W. W. Norton, August 2005. 0-393-05945-6. AWM Travel Grants. See http: I lwww. including mathematicians, students, ISBN I awm-math.orgltravelgrants.html; and the general public. When a book The Colours of Infinity: Th~ Beauty telephone 703-934-0163; email: awm@ has been reviewed in the Notices, a and Power of Fractals, by Michael math. umd. edu; or contact Associa­ reference is given to the review. Gen­ Barnsley, Nigel Lesmoir-Gordon, tion for Women in Mathematics, erally the list will contain only books Benoit B. Mandelbrot, Ian Stewart, 11240 Waples Mill Road, Suite 200, published within the last two years, Gary Flake, Robert Prechter, and Fairfax, VA 22030. though exceptions may be made in Arthur C. Clarke. Clear Press, March cases where current events (e.g., the 2004. ISBN 1-904-55505-5. Mathematical Sciences death of a prominent mathematician, Complexities: Women in Mathe­ Education Board, National coverage of a certain piece of mathe­ matics, edited by Bettye Anne Case Research Council matics in the news) warrant drawing and Anne M. Leggett. Princeton Uni­ ]an tje Lange, Freudenthal Insti­ readers' attention to older books. Sug­ versity Press, January 2005. ISBN tute, The Netherlands gestions for books to include on the list 0-691-11462-5.

488 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Reference and Book List

Converging Realities: Toward a God Created the Integers, by Stephen 1-56881-150-0. (Reviewed December Common Philosophy of Physics and Hawking. Running Press, October 2005. 2005.) Mathematics, by Roland Omnes. ISBN 0-762-41922-9. ''The Man Who Knew Too Much: Princeton University Press, November Code/'s Theorem: An Incomplete Alan Turing and the Invention of the 2004. ISBN 0-691-11530-3. Guide to Its Use and Abuse, by Torkel Computer, by David Leavitt. Great The Curious Incident of the Dog in Franzen. A K Peters, May 2005. ISBN Discoveries series, W. W. Norton, the Night-time, by Mark Haddon. Vin­ 1-56881-238-8. December 2005. ISBN 0-393-05236-2. tage, May 2004. ISBN 1-400-03271-7. Graphic Discovery: A Trout in the Math and the Mona Lisa: The Art (Reviewed March 2006.) Milk and Other Visual Adventures, by and Science of Leonardo da Vinci, by Dark Hero of the Information Age: Howard Wainer. Princeton University Bulent Atalay. Smithsonian Books, In Search of Norbert Wiener, by Flo Press, October 2004. ISBN 0-691- April 2004. ISBN 1-588-34171-2. Conway and Jim Siegelman. Basic 10301-1. The Math Instinct: Why You're a Math­ Books, December 2004. ISBN 0-738- ,., Hiding in the Mirror: The Mysteri­ ematical Genius (Along with Lobsters, 20368-8. ous Allure of Extra Dimensions, from Birds, Cats, and Dogs), by Keith Devlin. ''Decoding the Universe: How the Plato to String Theory and Beyond, by Thunder's Mouth Press, March 2005. New Science ofInformation Is Explain­ Lawrence M. Krauss. Viking Adult, ISBN 1-56025-672-9. ing Everything in the Cosmos, from Our October 2005. ISBN 0-670-03395-2. Mathematical Adventures for Stu­ Brains to Black Holes, by Charles Seife. Incompleteness: The Proof and dents and Amateurs, David F. Hayes Viking Adult, February 2006. ISBN 0- Paradox of Kurt Code/, by Rebecca and Tatiana Shubin, editors. Mathe­ 670-03441-X. Goldstein. W. W. Norton, February maticalAssociation of America, 2004. ISBN 0-88385-548-8. Piero della Francesca: A Mathe­ 2005. ISBN 0-393-05169-2. (Reviewed Mathematical fl/ustrations: A Manual matician's Art, by]. V. Field. Yale in this issue.) ofGeometry and PostScript, by Bill Cas­ University Press, August 2005. ISBN The Infinite Book: A Short Guide to selman. Cambridge University Press, 0- 300-10342-5. the Boundless, Timeless and Endless, December 2004. ISBN 0-521-54788-1. Divine Proportions: Rational by John D. Barrow. Pantheon, August Mathematical Musings: A Collection Trigonometry to Universal Geometry, 2005. ISBN 0-375-42227-7. ofQuotes, edited by Dan Sonnenschein. by N. ]. Wildberger. Wild Egg Books, Introducing Game Theory and Its Clarium Press, November 2005. ISBN Applications, by Elliott Mendelson. September 2005. ISBN 0-9757492-0-X. 0-9697688-8-5. CRC Press, July 2004. ISBN 1-584- The Equation That Couldn't Be Mathematics by Experiment: Plau­ 88300-6. Solved (How Mathematical Genius Dis­ sible Reasoning in the 21st Century, by janos Bolyai, Euclid, and the Nature covered the Language of Symmetry), Jonathan Borwein and David Bailey. ofSpace, by Jeremy]. Gray. MIT Press, by Mario Livio. Simon and Schuster, A K Peters, December 2003. ISBN September 2005. ISBN 0-743-25820-7. May 2003. ISBN 0-262-57174-9. (Re­ 1-56881-211-6. (Reviewed September M. C. Escher's Legacy: A Centen­ viewed October 2005.) 2005.) nial Celebration, edited by Doris john Pel/ (1611-1685) and His Mathematics in Nature: Modeling Schattschneider and Michele Emmer. Correspondence with Sir Charles Patterns in the Natural World, by Springer, September 2005 (paperback Cavendish: The Mental World of an John A. Adam. Princeton University edition). ISBN 3-540-20100-9. Early Modern Mathematician, by Noel Press, November 2003. ISBN 0-691- The Essential Turing, edited by Malcolm and Jacqueline Stedall. 11429-3. (Reviewed June/July 2005.) B. Jack Copeland. Oxford University Oxford University Press, second Meta Math! The Quest for Omega, by Press, September 2004. ISBN 0-198- edition, January 2005. ISBN 0-198- Gregory Chaitin. Pantheon, October 25080-0. 56484-8. 2005. ISBN 0-375-42313-3. Experimentation in Mathematics: The Knot Book: An Elementary R. L. Moore: Mathematician and Computational Paths to Discovery, by Introduction to the Mathematical Teacher, by John Parker. Mathemati­ Jonathan Borwein, David Bailey, and Theory ofKnots, Colin C. Adams. AMS, cal Association of America, 2004. ISBN Roland Girgensohn. A K Peters, March September 2004. ISBN 0-8218-3678-1. o-88385-s so-x. 2004. ISBN 1-56881-136-5. (Reviewed (Reviewed September 2005.) More Mathematical Astronomy September 2005.) Knots and Links, by Peter R. Morsels, by Jean Meeus. Willmann­ The Fermat Diary, by C.]. Mozzochi. Cromwell. Cambridge University Press, Bell, 2002. ISBN 0-943396-743. AMS, August 2000. ISBN 0-8218- October 2004. ISBN 0-521-83947-5. Musings of the Masters: An An­ 2670-0. Luck, Logic, and White Lies: The thology of Miscellaneous Reflections, The Fermat Proof, by C.]. Mozzochi. Mathematics of Games, by Jorg edited by Raymond G. Ayoub. Trafford Publishing, Inc., February Bewersdorff. Translated by David Mathematical Association of Amer­ 2004. ISBN 1-412-02203-7. Kramer. A K Peters, November 2004. ica, June 2004. ISBN 0-88385-549-6. Geometry and Meaning, by Dominic ISBN 1-56881-210-8. New Mexico Mathematics Contest Widdows. Center for the Study of Lan­ Saunders Mac Lane: A Mathemati­ Problem Book, by Liong-shin Hahn. guage and Information, November cal Autobiography, by Saunders University of New Mexico Press, No­ 2004. ISBN 1-575-86448-7. Mac Lane. A K Peters, May 2005. ISBN vember 2005. ISBN 0-8263-3534-9.

APRIL2006 NOTICES OF THE AMS 489 Reference and Book List

The Newtonian Moment: Isaac New­ Association of America, January 2005. ton and the Making ofModem Culture, ISBN 0-88385-036-2. by Mordechai Feingold. New York The Transformation of Mathemat­ Library and Oxford University Press, ics in the Early Mediterranean World: December 2004. ISBN 0-195-17735-5. From Problems to Equations, by Reviel Numbers, the Language of Science, Netz. Cambridge University Press, by Tobias Dantzig. Pi Press, fifth edi­ June 2004. ISBN 0-521-82996-8. tion, March 2005. ISBN 0-131-85627-8. Using the Mathematics Literature, The Oxford Murders, by Guillermo by Kristine K. Fowler. Marcel Dekker, Martinez. Abacus, January 2005. ISBN June 2004. ISBN 0-824-75035-7. 0-349-11721-7. (Reviewed November The Visual Mind II, edited by 2005.) Michele Emmer. MIT Press, May 2005. The Pea and the Sun: A Mathe­ ISBN 0-262-05076-5. The Works of Archimedes: Trans­ matical Paradox, by Leonard M. lation and Commentary. Volume I: Wapner. A K Peters, April2005. ISBN The Two Books On the Sphere and 1-56881-213-2. The Cylinder. Edited and translated by PopCo, by Scarlett Thomas. Har­ Reviel Netz. Cambridge University vest Books, October 2005. ISBN 0-156- Press, April 2004. ISBN 0-5 21-66160- 03137-X. (Reviewed February 2006.) 9. (Reviewed May 2005.) Probability Theory: The Logic of Science, by E. T. Jaynes. Edited by G. Larry Bretthorst. Cambridge Univer­ sity Press, April 2003. ISBN 0-521- 59271-2. (Reviewed January 2006.) Reality Conditions: Short Mathe­ matical Fiction, by Alex Kasman. Math­ ematical Association of America, May 2005. ISBN 0-88385-552-6. 1'Reflections: V I. Arnold's Remi­ niscences, by V. I. Arnold. Springer, April 2006. ISBN 3-540-28734-5. The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose. Knopf, February 2005. ISBN 0-679-45443-8. Science in the Looking Glass, by E. Brian Davies. Oxford University Press, August 2003. ISBN 0-198- 52543-5. (Reviewed December 2005.) - Sneaking a Look at God's Cards: -- Unraveling the Mysteries of Quantum Mechanics, by Giancarlo Ghirardi, translated by Gerald Malsbary. Prince­ ton University Press, revised edition, January 2005. ISBN 0-691-12139-7. Space/and, by Rudy Rucker. Tor Books, June 2002. ISBN 0-765-30366- 3. (Reviewed August 2005.) Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers, by Dan Rockmore. Pantheon, April 2005. ISBN 0-375- 42136-X.

1' The Three Body Problem, by Catherine Shaw. Allison and Busby, March 2005. ISBN 0-749-08347-6. A Tour through Mathematical Logic, by Robert S. Wolf. Mathematical

490 NOTICES OF THE AMS VOLUME 53, NUMBER 4 AMERICAN MATHEMATICAL SOCIETY

Call for Nominations AMERICAN MATHEMATICAL SOCIETY

The selection committees for these prizes request nominations for consideration for the 2007 awards, which will be presented at the Joint Mathematics Meetings in New Orleans, LA, in January 2007. Information about these prizes may be found in the November 2005 Notices, pp. 1243-1260, and at http://www.ams.org/prizes-awards.)

Oswald Veblen Prize Ruth Lyttle Satter Prize in Geometry

The Ruth Lyttle Satter Prize, first awarded The Oswald Veblen Prize in Geometry, in 1991, is presented every two years in first awarded in 1964, is presented every recognition of an outstanding contribu­ three years in recognition of a notable tion to mathematics research by a woman research memoir in geometry or topolo­ in the past six years. gy published in the preceding six years. To be considered, either the nominee should be a member of the Society or the memoir should have been published in a Norbert Wiener Prize in recognized North American journal. Applied Mathematics

The Norbert Wiener Prize, first awarded in 1970, is awarded jointly by the AMS Levi L. Conant Prize and SIAM every three years for an out­ standing contribution to applied mathe­ The Levi L. Conant Prize, first awarded in matics in its highest and broadest sense. January 2001, is presented annually for an outstanding expository paper published in either the Notices or the Bulletin of the American Mathematical Society during the preceeding five years.

Nominations should be submitted to the secretary, Robert J. Daverman, American Mathematical Society, 312D Ayres Hall, University of Tennessee, Knoxville, TN 37996-1330. Include a short description of the work that is the basis of the nomination, with complete bibliographic citations where appropriate. A brief curriculum vitae should be included for the nominee. The nominations will be forwarded by the secretary to the appropriate prize selection committee, which, in effect, will make final decisions on the awarding of these prizes.

Deadline for nominations is June 30,2006. I AMERICAN MATHEMATICAL SOCIETY Uf E~ H~ MOORE Research Article Prize

First awarded in 2004, the E. H. Moore Research Article Prize is awarded every three years for an outstanding research article that has appeared in one of the AMS primary research journals (namely, the journal of the AMS, Proceedings of the AMS, Transactions of the AMS, Memoirs of the AMS, Mathematics of Computation, Electronic journal of Conformal Geometry and Dynamics, and the Electronic journal of Representation Theory) during the six calendar years ending a full year before the meeting in which the prize is awarded.

Among other activities, E. H. Moore founded the Chicago branch of the AMS, served as the Society's sixth president (1901-2), delivered the Colloquium Lectures in 1906, and founded and nurtured the Transactions of the American Mathematical Society. The name of the prize honors his extensive contributions to the discipline and to the Society.

The Moore Prize Selection Committee requests nominations for the 2007 award, which will be presented at the Joint Mathematics Meetings in New Orleans, LA, in January 2007. To be specific, papers published in one of the journals named in the first paragraph during the years 2000-2005 are consid­ ered eligible for the 2007 award.

Nominations should be submitted to the secretary, Robert J. Daverman, American Mathematical Society, 3120 Ayres Hall, University ofTennessee, Knoxville, TN 37996-1330. Include a short description of the work that is the basis of the nomination, with complete bibliographic citations. A brief curriculum vitae should be included for all nominees. The nominations will be forwarded by the secretary to the prize selection committee, which will make final decisions on the awarding of this prize.

Deadline for nominations is june 30, 2006. 2006 Frank and Brennie Morgan AMS-MAA-SIAM Prize for Outstanding Research in Mathematics by an Undergraduate Student

he prize is awarded each year to an Tundergraduate student (or students having submitted joint work) for out­ standing research in mathematics. Any stu­ dent who is an undergraduate in a college or university in the United States or its posses­ sions, or Canada or Mexico, is eligible to be considered for this prize. he recipients of the prize are to be Tselected by a standing joint committee he prize recipient's research need not of the AMS, MAA, and SIAM. The deci­ be confined to a single paper; it may T sions of this committee are final. The 2006 be contained in several papers. prize will be awarded for papers submitted However, the paper (or papers) to be consid­ for consideration no later than June 30, ered for the prize must be submitted while 2006, by (or on behalf of) students who were the student is an undergraduate; they can­ undergraduates in December 2005. not be submitted after the student's gradua­ tion. The research paper (or papers) may be submitted for consideration by the student or a nominator. All submissions for the prize must include at least one letter of sup­ Nominations and submissions should be sent to: port from a person, usually a faculty mem­ Morgan Prize Committee ber, familiar with the student's research. c/o Robert J. Daverman, Secretary Publication of research is not required. American Mathematical Society 312D Ayres Hall University of Tennessee • • • • Knoxville, TN 37996-1330 Questions may be directed: Dr. Martha J. Siegel, MAA Secretary Mathematics Department Stephens Hall 302 Towson University 8000 York Road · Towson, MD 21252-0001 telephone: 410-704-2980 e-mail: siegel @t:owson. edu Mathematics Calendar

The most comprehensive and up-to-date Mathematics Calendar information is available one-MATH at http://www.ams .org/mathcal/.

April 2006 Geometry and Operator Algebras: Noncommutative Geome­ try, Quantum Field Theory and Motives, Vanderbilt University, '' 20-21 Workshop in Mathematical Physics, Stevens Institute of Nashville, Tennessee. Technology, Hoboken, New Jesey. Description: A combination of spring school and international Aim: Bringing together the researchers in mathematical physics conference. The conference will comprise a number of invited m order to understand the state of the art in mathematical research talks and short contributions. We strongly encourage physics and discuss promising research directions in: Conservation students and postdocs to attend this conference. Laws, Kinetic Equations, Nonlinear POE's, Variational Principles; Topics: During Riemann Boundary-Value Problems and Singular Integral Equations; the school part of the meeting several mini-courses on a variety Applications in Hydrodynamics, Theory of Elasticity, Fiber Optics, of topics from noncommutative geometry, operator algebras and related Magnetohydrodynamics (MHO), Contact Problems in Mechanics. topics will be given by leading experts. Organizers: P. Dubovski and M. Zabarankin. Minicourse speakers: Alain Cannes, College de France, IHES & Vanderbilt University; Katia Consani, Information: http: I /personal. stevens. edu;-pdubovsk/ Johns Hopkins University & University of Toronto; Herbert Gangl, mathphysics. html;email: pdubovsk@stevens. edu. Max-Planck-Institute Bonn; Li Guo, Rutgers University; Dirk Kreimer, IHES & Boston University; '' 20-21 Workshop in Mathematical Physics, Stevens Institute of Yuri Marrin, Northwestern University; Matilde Marcolli, Max-Planck­ Technology, Hoboken, New Jersey. Institute, Bonn; Niranjan Ramachandran, University of Maryland; Description: The workshop in Mathematical Physics aims at bring­ Laura Reina, Florida State University. mg together the researchers in mathematical physics in order Organizing committee: Dietmar Bisch (Vanderbilt University); to understand the state of the art in mathematical physics and Alain Cannes (College de France, IHES & Vanderbilt University; discuss promising research directions in: Conservation Laws, Ki­ D1rector of the Fourth Annual Spring Institute); Bruce Hughes netic Equations, Nonlinear POE's, Variational Principles; Riemann (Vanderbilt University); Gennadi Kasparov (Vanderbilt University); Boundary-Value Problems and Singular Integral Equations; Ap­ Matilde Marcolli (Max-Planck-Institute Bonn); Guoliang Yu (Vander­ plications in Hydrodynamics, Theory of Elasticity, Fiber Optics, bilt University). Magnetohydrodynamics (MHD), Contact Problems in Mechanics. Sponsors: National Science Foundation through a Research Training Organizers: P.Dubovski and M. Zabarankin. Group grant, Vanderbilt University Department of Mathematics, Information: email: pdubovsk@stevens. edu; http : I /personal . Vanderbilt University College of Arts and Science. stevens.edu/-pdubovsk/mathphysics.html. Information: http: I /www .math. vanderbilt . edu;-ncgoa06/.

May 2006 '' 2 2-2 5 Ecole de Printemps d' Analyse Fonctionnelle, Faculte des Sciences, Rabat, Morocco. "8-1 7 The Fourth Annual Spring Institute on Noncom mutative Program: John Conway (University of Tennessee Kr10xville), Gilles

This section contains announcements of meetings and conferences respect to participation in the meeting, this fact should be noted. of interest to some segment of the mathematical public, including ad All communications on meetings and conferences in the mathematical hoc, local, or regional meetings, and meetings and symposia devoted sciences should be sent to the Editor of the Notices in care of the American to specialized topics, as well as announcements of regularly scheduled Mathematical Society in Providence or electronically to netices@ams . erg meetings of national or international mathematical organizations. A or mathcal@ams . erg. complete list of meetings of the Society can be found on the last page of In or~er to allow participants to arrange their travel plans, organizers of each Issue. meetmgs are urged to submit information for these listings early enough to allow them to appear An announcement will be published in the No tices if it contains a call in more than one issue of the NoHces prior to for papers and specifies the place, date, subject (when applicable), and the meeting in question. To achieve this, listings should be received in Providence the speakers; a second announcement will be published only if there eight months prior to the scheduled date of the meeting. The are changes or necessary additional information. Once an announcement complete listing of the Mathematics Calendar will be published has appeared, the event will be briefly noted in every third issue until only in the September issue of the No tices. The March, June/July, and December issues 1t has been held and a reference will be given in parentheses to the will include, along with new announcements, references month, year, and page of the issue in which the complete information to any previously announced meetings and conferences occurring within appeared. Asterisks (") mark those announcements containing new or the twelve-month period following the month of those issues. New revised information. information about meetings and conferences that will occur later than In general, announcements of meetings and conferences held in North the twelve-month period will be announced once in full and will not be America carry only the date, title of meeting, place of meeting, names of repeated until the date of the conference or meeting falls within the speakers (or sometimes a general statement on the program), deadlines twelve-month period. for abstracts or contributed papers, and source of further information. The Mathematics Calendar, as well as Meetings and Conferences of the AMS, is now Meetings held outside the North American area may carry more detailed available electronically through the AMS website on the information. In any case, if there is any application deadline with World Wide Web. To access the AMS website, use the URL: http: 1 /www. ams. erg/.

494 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Mathematics Calendar

Pisier (Universite Pierre et Marie Curie, Paris, France), Thomas Organizers: Ozgur Ceyhan (MPI, Bonn), Kobi Kremnitzer (MIT, Ransford (Universite Laval, Quebec, Canada). Cambridge), Altug Ozpineci (METU, Ankara), Muhammed Uludag Information: email: elfallah@fsr. ac .ma. (GSU, Istanbul). Speakers: S. Akbulut (Michigan State University), P. Aluffi (Flor­ ,., 22-26 Integrable Systems, Random Matrices, and Applications ida State University), D. Bar-Natan (University of Toronto), R. Conference in honor of Percy Deift's 60th Birthday, Courant Bezrukavnikov (MIT), A. Braverman (Brown University)'", T. Dereli Institute of Mathematics Sciences, New York University, New York (Koc University), K. Fukaya (Kyoto University), D. Gaitsgory (Har­ City, New York. vard University), E. Getzler (Northwestern University), S. Gurevich Deadlines: March 1, 2006 for those who want to be considered (Tel Aviv University), R. Hadani (Tel Aviv University),]. Kamnitzer for a talk or poster or who want to apply for financial support. (MIT)'', A. Klyachko (Bilkent University), M. Lapidus (University of Otherwise the deadline for registration is May 1, 2006. California, Riverside)'', Yu-l. Marrin (Northwestern University), M. Information: http: I /math. arizona. ecturmcl/ISRMA. html. Marcolli (Max-Planck-Institute). Information: Limited funding is available for graduate students * 22- 26 Knots, Groups and 3-Manifolds in Marseille 2006, Univer­ and postdocs. For registration procedure and other information, sity of Provence, Marseille, France. please see the web page: http: I /guests .mpim-bonn .mpg. de/agaq. Description: The University of Marseille-Provence is organizing a Colloquium on Knot Theory, Low dimensional Topology and Group '' 8-10 Lehigh University Geometry/Topology Conference, Lehigh Geometry from May 22 to June 2, 2006 in Marseille (France). The University, Bethlehem, Pennsylvania. conference is dedicated to the 60th birthday of Michel Domergue Program: Invited lectures and 40-minute contributed talks. Banquet and to the memory of Yves Mathieu. The conference will be held honors C.C. Hsiung's 90th birthday, and 40 years of Journal of during the first week, whereas the second is devoted to a Summer Differential Geumetry. School on Heegaard-Floer homology and Khovanov homology, Confirmed speakers: Fedor Bogomolov (Courant), Ralph Cohen mainly intended for students. (Stanford), Lisa Jeffrey (Toronto), Frank Morgan (Williams), Dennis Organizing Committee: P. Derbez & D. Matignon and secretaries Sullivan (Stony Brook). A. Blanc & M.C. Tort. Support and Deadline: NSF grant provides limited travel sup­ Scientific Committee: M. Boileau (Univ. Paul Sabatier), C. Lescop port (up to $300) for recent Ph.D.'s, students, and members of (Inst. Fourier), ]. Los (Univ. de Provence), M. Lustig (Univ. de Paul underrepresented groups. Register by May 15. cezanne), K. Millett (Univ. de Santa Barbara), H. Short (Univ. de Information: http: I /www .lehigh. edu;-dljO/geotop. htmlorcon­ Provence). tact David Johnson, [email protected]; tel: 610-758-3759?. Deadline to register: February 26, 2006. Information: LATP, Universite de Marseille-Provence, CMI, 39 rue '' 1 0-16 32nd International Conference "Applications of Math­ Joliot-Curie, 13453 Marseille Cedex 13, France; http : I /www. cmi. ematics in Engineering and Economics" (AMEE'06), Town of univ-mrs .fr/%7Ematignon/colloquium2006.html. Sozopol, Bulgaria. Topics: Potential theory and partial differential equations, math­ '' 29-June 2 Analytic Function Spaces, University of Joensuu, Joen­ ematical analysis and applications, differential equations and suu, Finland. differential geometry, theoretical and applied mechanics, opera­ Topics: QP classes and function spaces of several complex variables. tion research and statistics, numerical methods and mathematical Scientific committee: Rauno Aulaskari (Joensuu), Miroslav Englis modeling, computer science. (Prague) and Jari Taskinen (Helsinki). Main lectures will be given Organizer: The faculty of mathematics and informatics by the by Mats Andersson, Jonathan Arazy, David Bekolle, Eo Berndtsson, Technical University of Sofia, Bulgaria, Manager and Scientific Aline Bonami, Huaihui Chen, Lewis Coburn, Konstantin Dyakonov, Secretary of AMEE'06: Assoc. Prof. Michail Todorov. Daniel Girela, Kang-Tae Kim, Steven Krantz, Marco Peloso, Richard Information: http: I /www. tu-sof ia. bg/fpmi/amee; email: mtod@ Rochberg, Harald Upmeier, Zhijian Wu, Jie Xiao, Ruhan Zhao, Kehe tu-sof ia . bg. Zhu. Deadline: Deadline of registration is April 18th. '' 1 3-1 5 2nd IMT-GT Conference on Mathematics, Statistics and Information: http : I /www. math. helsinki. fi/research/ Their Applications, Gurney Hotel, Penang, Malaysia. FMSvisitor0506/severalcv-conf/. Description: Progress in the mathematical sciences within the IMT­ GT region will benefit greatly from regular meetings and interaction june 2006 amongst mathematicians and scientists in the region. Organizer: Universiti Sains Malaysia in collaboration with The '' 1-3 Carleton Applied Probability Workshop, Carleton University, Malaysian Mathematical Sciences Society. Ottawa, Canada. Topics: Mathematics, Applications of Mathematics, Statistics, Op­ Organizers: A. Bose, Z. Gao, A. Jarai, D. Panario, and Y. Zhao erations Research, Mathematics Education and Computer Science. (Carleton). Plenary Speakers: Andonowati (Indonesia): Extreme Waves; Gan Invited speakers: Fan Chung Graham, Jim Dai, John Dixon, Jim Eng Hui (Malaysia): Quality Control in Manufacturing Industries; E. Fill, Ingemar Kaj, Boris Pittel, Nick Wormald and Susan Xu. Van Groesen (Netherlands): Mathematical Modelling of Tsunamis; Sponsor: The Fields Institute, Toronto, Ontario, Canada. Wolfgang Hardie (Germany): A Topic in Statistics; Kamel Ariffin Deadline: To submit an abstract for contributed talks: May 1, 2006. Mohd. Atan (Malaysia): On the Explicit Estimation of Exponential Information: http: I /www. fields . utoronto. ca/programs/ Sums; I-Ming Tang (Thailand): Mathematica Modelling of Trans­ scientific/05-06/applied_probability/. mission of Avian Influenza Virus; Vu Kim Tuan (USA): Irregular Sampling of Bandlimited & Positive Frequency Signals. There will '' 5-1 5 Arithmetic and Geometry Around Quantization, European also be workshops on Bioinformatics, Graphing Calculators, Quality Mathematical Society Summer School, Galatasaray University, Control and Generation of Tsunamis. Istanbul, Turkey. Information: VISA. Please visit the Immigration Department of Purpose: Bringing the leading experts in quantum groups, mirror Malaysia web site for Visa regulations and forms. symmetry, quantum knot invariants, noncommutative geometry, quantum chaos and related areas. The aim is to introduce 'quantum '' 1 3-1 7 CBMS conference on Cluster Algebras and Applications, mathematics' to young researcher as well as to discuss the recent Raleigh, North Carolina. advances and open problems in these areas. Principal speaker: Andrei Zelevinsky (Northeastern University).

APRIL 2006 NOTICES OF THE AMS 495 Mathematics Calendar

Other speakers include: M. Okado, N. Reading, H. Thomas, R. Program: Week 1: School; 1. Introduction to Valuations, 2. Rees Schiffler etc. Valuations and Integral Closure of Ideals, Week 2: School; 1. Com­ Support: Available for graduate students and young researchers. binatorial and Computational Techniques for Computing Integral Information: email: j ing@math. ncsu . edu. Closures, 2. Further Developments and Applications of Valuation Theory, Week 3: Conference. '' 1 5-22 Operator Theory, Analysis and Mathematical Physics: Deadlines: Feb. 1: Application for funding. Feb. 1: Submission of OTAMP2006, Lund Institute of Technology, Lund, Sweden. abstracts to be considered for conference. Mar. 1: Notification of Purpose: To bring together researchers working in operator theory, funding. Mar. 30: Application for summer school (no funding). May analysis and their applications in mathematical physics. 15: Conference registration. Special sessions: I. Spectral analysis of Schrodinger operators. II. Information: http : I /www .mathstat. da1 . carfaridi/integra1- Jacobi matrices and CMV matrices and orthogonal polynomials. III. c1osure .html. Quasi-periodic and random Schrodinger operators. IV. Quantum graphs. '' 1 0-1 3 Markov Processes and Related Topics, University of Organizing Committee: Jan Janas (Krakow), Pavel Kurasov (Lund), Wisconsin-Madison, Madison, Wisconsin. Ari Laptev (Stockholm), Serguei Naboko (St.Petersburg), Gunter Description: To honor Tom Kurtz on his 65th birthday for his Stolz (Birmingham, USA). many fundamental contributions to probability theory. Lecturers: M. Aizenman (USA), S. Albeverio (Germany), A. Avilla (France), ]. Bellissard (France-USA), Ch. Bennewitz (Sweden), M. Information: For a list of Invited Speakers, go to http: I /www. math. utah. edu/-ethier/kurtzfest . html. There will also be con­ Brown (UK), Y. Colin de Verdiere (France), M. Combescure (France), D. Damanik (USA), ]. Derezinski (Poland), ]. Geronimo (USA), F. tributed talks. Gesztesy (USA), L. Golinskii (Ukraine), H. Hedenmalm (Sweden), S. '' 1 0- 1 5 Workshop on Stochastic Eigen-Analysis (Random Matri­ Jitomirskaya (USA), Yu. Karpeshina (USA), ]. P. Keating (UK), A. ces) and its Applications, MIT, Cambridge, Massachusetts. Kiselev (USA), Y. Last (Israel), A. Magnus (Belgium), B. Pavlov (New Zealand), B. Simon (USA),]. Stochel (Poland), R. Weder (Mexico), and Description: To bring together mathematicians, statisticians, sci­ R. Weikard (Birmingham, USA). entists and engineers to share recent work related to random Deadlines: April 1, 2006: Titles and abstracts of the proposed matrices and discuss future directions. talks; May 1, 2006: Reduced registration fee. Information: http : I /web .mit. edu/sea06/. Information: http: I /www. maths .1 th. serkurasov /OTAMP2006/ OTAMP2006. htm1;email: otamp2006@maths .1 th. se. '' 12- 1 5 Geometric Analysis and Applications, University of Illinois, Urbana-Champaign, Illinois. '' 19-23 La Pietra Week in Probability, Firenze 2006: Stochastic Focus: On recent developments in the study of geometry and analy­ Processes in Mathematical Physics, Villa La Pietra, Firenze, Italy. sis in metric measure spaces (with an emphasis on sub-Riemannian Description: La Pietra Week in Probability 2006 will be a school geometry) and several applications to problems "outside" of math­ and conference presenting relevant topics in the probabilistic ematics (in particular: robotics, control theory, geometry of the approach to Mathematical Physics, including some of the most visual cortex, and digital image reconstruction). recent advances and some of the prospective challenges of the Invited Speakers: Zoltan M. Balogh (University of Bern), Anthony field. During the meeting we take the occasion to celebrate Michael Bloch (University of Michigan), Mario Bonk (University of Michigan), Aizenman's and Chuck Newman's 60th birthdays. Roger Brockett (Harvard University), Monique Chyba (University of Scientific Committee: ]. Lebowitz (Rutgers University), A. Klein Hawaii), Giovanna Citti (University of Bologna), Michael Cowling (University of California, Irvine), A. Gandolfi (Universita di Firenze), (University of New South Wales), Selim Esedoglu (University of V. Sidoravicius (IMPA, Rio de Janeiro). Michigan), Bruce Kleiner (Yale University), Adam Koranyi (City Minicourses: The school will feature three mini-courses, planned University of New York, Lehman College), Pertti Mattila (University at advanced Ph.D. level, held at the beginning of the week by: J. of Helsinki), Jean Petitot (Centre d'Analyse et de Mathematiques Chayes (Microsoft Research) G. Giacomin (Universite Paris 7) W. Sbciales, Paris), Alessandro Sarti (University of Bologna). While Werner (Universite Paris-Sud). some of the invited lecturers may fall into purely one category Conference speakers: G. Ben Arous (EPF Lausanne) F. Camia ("pure" or "applied"), a substantial number of lecturers will speak (Vrije Universiteit Amsterdam) P. Contucci (Universita di Bologna) on aspects of both pure and applied topics. B. Derrida (Ecole Normale, Paris) B. Duplantier (CEA, Saclay) L. R. Organizers: Luca Capogna, capogna@1agrange. uark. edu; Scott Fontes (IME, Sao Paulo) G. M. Graf (ETH, Zurich) A. Klein (University of Pauls, [email protected]; Jeremy Tyson, tyson@math . California, Irvine)]. Lebowitz (Rutgers University) E. Lieb (Princeton uiuc. edu. University) F. Martinelli (Universita di Roma 3) E. Presutti (Tor Information: http: I /www. math. uiuc. edu;-tyson/UIUC06. html. Vergata, Roma) ]. Schenker (lAS, Princeton) S. Sheffield (Courant Institute, New York) V. Sidoravicius (IMPA, Rio de Janeiro) S. R. '' 14-31 The ninth International Diffiety School, Santo Stefano del S. Varadhan (Courant Institute, New York) S. Warzel (Princeton Sole, Avellino, Italy. University) E Waymire (Oregon State University). Aim: To introduce undergraduate and Ph. D. students in Mathemat­ Organizing committee: D. Stein(New York University), B. Duplantier ics and Physics as well as post-doctoral researchers in a recently (CEA, Sa clay) A. Gandolfi (Universita diFirenze), M. Isopi (La Sapienza, emerged area of Mathematics and Theoretical Physics: Secondary Roma), P. Contucci (Universita di Bologna), M. Romito (Universita Calculus. A diffiety is a new geometrical object that properly di Firenze). formalizes the concept of the solution space of a given system of Deadline: April 15th, 2006. (nonlinear) PDEs, much as an algebraic variety does with respect Information: http: I /www. math. unif i . i t/LaPietra. to solutions of a given system of algebraic equations. Secondary Calculus is a natural diffiety analogue of the standard Calculus on July 2006 smooth manifolds, and as such leads to a very rich general theory '' 3-22 Summer School & Conference: Valuation Theory and of nonlinear PDEs. Moreover, it appears to be the unique natural Integral Closures in Commutative Algebra, University of Ottawa, language for quantum physics, just as the standard Calculus is the Ottawa, Ontario, Canada. natural language for classical physics. Organizers: StevenDale Cutkosky (University ofMissouri-Columbia). Deadline: June 30, 2006. Sara Faridi (Dalhousie University), Franz-Viktor Kuhlmann (Univer­ Information: A. M. Vinogradov; email: schoo106@diff iety. org; sity of Saskatchewan), Irena Swanson (New Mexico State University). http://diffiety .ac.ru.

496 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Mathematics Calendar

August 2006 Information: Contact: SatGupta(sngupta@uncg . edu), Carlos Coelho (cmac@fct .unl.pt) or Satya Mishra ([email protected]. '' 16-1 9 Satellite conference on Algebraic Geometry, Segovia edu); http: I /scra2006. southalabama. edu/. Campus of the Universidad de Valladolid, Segovia, Spain. Description: Satellite Conferences are relevant scientific activities ,., 1-5 International Summer School and Workshop of Operator organized around ICM Madrid 2006. This conference will deal with Algebras, Operator Theory and its Applications 2006, Instituto recent trends in Algebraic Geometry. Superior Tecnico, Universidade Tecnica de Lis boa, Lisbon, Portugal. Main Speakers:]. I. Burgos, G. M. Greuel, P. Griffiths,]. Lipman, Z. Information: http: I /woat2006 . ist . utl. pt/. Mebkhout, T. Pantev, M. Reid, A. Sommese Sessions: Interactions with physics, Classical geometry and Singu­ '' 4-6 Optimal Discrete Structures and Algorithms (ODSA 2006), larities. University of Rostock, Rostock, Germany. Speakers (with '' to be confirmed): R. Donagi, C. Hertling, R. Information: http: I /www. math. uni -rostock. de/ odsa/. Kaufmann, A. Kuznetsov, Orlov'', M. Popa, B. Totaro, D. Van Straten'',]. Wahl",]. A. Wisniewski. ,., 8-12 1st Dolomites Workshop on Constructive Approximation Information: http: I /www. escet. urj c. es/satelli te/. and Applications: Dedicated to Walter Gautschi for his 50 years of professional activity, Alba di Canazei, Trento, Italy. '' 16-19 VII Workshop on Symplectic and Contact Topology, Satel­ Topics: Approximation by Multivariate Polynomials (Interpola­ lite Conference oft he International Congress of Mathematicians tion, Orthogonal Polynomials, ... ), Approximation by Radial Basis (ICM2006), Universidad Carlos III (Campus at Getafe), Madrid, Spain. Functions and other Meshfree Methods, Cubature Methods, Compu­ Topics: Symplectic topology, Contact topology Gauge theories, tational Tools, Applications to Scientific Computing, Applications Topology of low dimension manifolds, Mirror symmetry. to Numerical Modelling in Engineering and Finance. Speakers: Bojanov B. (Sofia, BG), Bos L. (Calgary, CA), Bozzini M. Scientific Committee: Simon Donaldson, Imperial College, London, (Milan, I), Brezinski C. (Lille, F), Buhmann M. (Giessen, D), Fasshauer UK; Yakov Eliashberg, Stanford University, USA; Jose M. Fernandez G. (Chicago, IL, USA), Iske A. (Hamburg, D), Levesley ]. (Leicester, de Labastida, CSIC, Madrid, Spain; Kenji Fukaya, Kyoto University, UK), Montefusco L. (Bologna, I), Sauer T. (Giessen, D), Schaback R. Japan; Robert E. Gompf, University of Texas at Austin, USA; Helmut (Goettingen, D), Sloan I. (Sydney, AU), Wendland H. (Dresden, D) , Hofer, Courant Institute, New York University, USA; Dusa McDuff, Xu Y. (Eugene, OR) Stony Brook University, USA; Gang Tian, Princeton University, USA. Registration: Since the meeting is limited to at most 80 participants, Invited Speakers: Denis Auroux, MIT, Cambridge, USA; Paul Biran, people interested in participating in this workshop are invited to Tel-Aviv University, Israel; Ko Honda, University of Southern pre-register. California, USA; Robert Gompf, University of Texas at Austin, USA; Information: http: I /www . sci. univr . i trdwcaa06. Dusa McDuff, Stony Brook University, USA; Grigory Mikhalkin, University of Toronto, Canada; William P. Minnicozzi, The Johns '' 11-1 5 Complexity of mappings in CR geometry, AIM Research Hopkins University, Maryland, USA; Tom Mr6wka, MIT, Cambridge, Conference Center, Palo Alto, California. USA; PeterS. Oszvath, Columbia University, New York, USA; D.H. Topics: Study mappings in CR geometry. Participants will focus on Phong, Columbia University, New York, USA; Bernd Siebert, Albert­ a quite specific part of this general subject, namely the relationship Ludwigs-Universitat, Freiburg, Germany. between the complexity of a CR mapping and the complexity of Deadlines: For grant applications is June 1, 2006. The deadline the CR structures on the domain and target manifolds. for registration, including reception of the workshop fee, is July Goals: To determine the fundamental notions of CR complexity 1, 2006. Abstracts submission: May 1, 2006 Applying for financial and to prove sharp results about these notions; to organize CR support: June 1, 2006. Registration (including reception of fee complexity theory into a broad framework that will be useful in CR payment): July 1, 2006. geometry and also apply to other parts of mathematics; to bring Information: For further information and queries, please contact active senior researchers and young mathematicians together to the conference secretariat: email: gesta@vilma. upc. edu; http: I I work in a focused manner that will forge interactions and guide www. mal . upc . edu/ gesta. future research. Organizers: Peter Ebenfelt and John P. D'Angelo. * 31-September 2 Geometry and Topology of Low Dimensional Sponsors: AIM and the NSF. Manifolds, Burgo de Osma, Spain. Deadline: June 11, 2006. Description: A satellite conference to ICM 2006, Madrid, Spain. Information: http: I I aimath. org/ ARCC/workshops/ Plenary Speakers: Michel Boileau (Univ. Paul Sabatier, Toulouse); complexitycrmap.html. Cameron Gordon (Univ. of Texas, Austin); William Harvey (Kings College, Univ. of London); Gareth A. Jones (Univ. of Southampton); '' 1 5-1 9 International Conference of Numerical Analysis and Louis H. Kauffman (Univ. of Illinois, Chicago); Alexander Mednykh Applied Mathematics 2006 (ICNAAM 2006), Hotel Belvedere (Sobolev Inst. of Math., Novosibirsk); Hugh Morton (Univ. of Liv­ Imperial, Hersonnisos, Crete, Greece. erpool); Sergei Natanzon (Independent Univ. of Moscow); Jozef Aim: To bring together leading scientists of the International Przytycki (The George Washington Univ., Washington DC); ]iirgen Numerical & Applied Mathematics community and to attract Wolfart (Univ. of Frankfurt). original research papers of very high quality. Information: http: I /www .mai .liu. se/LowDim. Topics: The topics to be covered include (but are not limited to): All the research areas of Numerical Analysis and Computational September 2006 Mathematics and all the research areas of Applied Mathematics: (see http: I / www. icnaam . org/topics. htm). '' 1-4 SCRA 2006-FIM XIII, Thirteenth International Conference Scientific committee: G. Vanden Berghe, Belgium; S.C. Brenner, of Forum for Interdisciplinary Mathematics, New University of USA;]. R. Cash, UK; R. Cools, Belgium; A. Cuyt, Belgium; B. Fischer, Lisbon-Tomar Polytechnic Institute, Lisbon, Portugal. Germany; R. W. Freund, USA; I. Gladwell, USA; B. Hendrickson, Theme: Interdisciplinary Mathematical and Statistical Techniques. USA; W. F. Mitchell, USA; G. Psihoyios, UK; T. E. Simos, Greece; W. Speakers: C. R. Rao (Penn State), Barry Arnold (University of Sproessig, Germany; Ch. Tsitouras, Greece; G. Alistair Watson, UK. California, USA), Carlos Brauman (Evora, Portugal), Tadeusz Calinsky Invited Speakers: Peter R. Graves-Morris, University of Bradford, (Poznan, Poland), Angela Dean (Ohio State, USA), Malay Ghosh UK; Gene H. Golub, Fletcher Jones Professor of Computer Science, (Florida, USA), Ivete Gomez (Lisbon, Portugal), Benjamin Kedem Stanford University, USA; Bernhard Beckermann, Universite des (Maryland, USA), and John Stufken (Georgia, USA). Sciences et Technologies de Lille, France; Gerard L. G. Sleijpen,

APRIL 2006 NOTICES OF THE AMS 497 Mathematics Calendar

Utrecht University, The Netherlands; Mourad E. H. Ismail, University Beirut, Beirut, Lebanon. of Central Florida, USA; Ronald Hoppe, University of Augsburg, Goals: The major goals of the International Symposium are: (a) Germany, University of Houston, USA; Guido Vanden Berghe, to share innovative, unique and creative solutions for enacting Universiteit Gent, Belgium; Yang Chen, Imperial College London, reform in the areas of elementary mathematics and science teacher UK and Center for Combinatorics, Nankai University, P. R. China; preparation and development, school organization, policy, and Vladislav V. Kravchenko, Instituto Politecnico Nacional, Mexico. classroom practices; (b) to document and widely disseminate ideas Information: SecretaryiCNAAM; email: icnaam@uop . gr, 10 Konitsis presented at the symposium; (c) to initiate new and creative Street, Amfithea-Paleon Faliron, GR-175 64, Athens, Greece; fax: solutions to endemic problems; and (d) to initiate discussion of a +30210 94 20 091 or+ 302710 237 397. grant proposal to enact and study the enactment in International House School settings of some of the innovative ideas presented * 2 5-26 Conference on Mathematics and its Applications, The in the Symposium. University of the West Indies, St. Augustine, Trinidad. Deadlines: Proposals are sought in the following areas: Teacher Organizing Committee: B. Bhatt (chairman), bbhatt@fsa. uwi. tt, preparation and ongoing development, Policy initiatives, School D. Owen, dowen@carib-link . net, G. Shrivastava, shri v@eng. uwi . organization, Classroom practices. Completed proposals are due tt, C. Ward, cward@f sa. uwi. tt, R. Dow, rdow@eng. uwi . tt, L. M. no later than May 15, 2006. Pinto Pereira, lexelyp©gmail. com. Information: email: arogerson@inetia. pl. Invited Speakers: T.]. Pedley (Cambridge, UK),]. C. R. Hunt (LIMS & UCL, UK), K. R. Sreenivasan (ICTP, Italy), L. Moseley (UWI, CaveHill). '' 8- 1 2 Policy and Practice in Mathematics and Science Teaching Conference Themes: Medicine, Environment, Information Pro­ and Learning in the Elementary Grades, Beirut, Lebanon. cessing, Biology, Epidemiology, Petroleum Engineering, Hydrology, Organizers: Marj Henningsen and Madeleine Long, in cooperation Meteorology & Seismic Phenomena, Physiological flows, General. with our project. Important Deadlines: Due date for abstracts: March 31, 2006; Information: email: arogerson@inetia. pl. Acceptance Notification: April 30, 2006; Full paper: July 31, 2006; Registration fees: July 31, 2006. Information: Instructions and how to participate and more at: http: //sta . uwi. edu/ conferences/ cmaia/; email: cmaia@fsa. uwi. tt. The following new announcements will not be repeated until the criteria in the next to the last paragraph at the bottom of October 2006 the first page of this section are met. '' 4-6 Second Announcement: International Conference on Multi· September 2007 field Problems, Universitat Stuttgart, Stuttgart, Germany. '' 7-1 39th International Conference ofThe Mathematics Education Topics: Numerical analysis and efficient algorithms, Volume cou­ into the 21st Century Project, Charlotte, North Carolina. pling in suspensions and porous media, Surface coupled problems, Preliminary Announcement and Call for Papers: The Mathematics Material modeling and multiscale problems. Education into the 21st Century Project has just completed its Plenary Speakers: F. Baaijens (Eindhoven); M. Celia (Princeton); eighth successful international conference in Malaysia, following ]. Delfs (Braunschweig); P. Deuflhard (Berlin); R. Klein (Berlin); conferences in Egypt, Jordan, Poland, Australia, Sicily, Czech T. Laursen (Durham); M. Ortiz (Pasadena); A. Quarteroni (Lau­ Republic and Poland. Our project was founded in 1986 and is sanne/ Milano). dedicated to the planning, writing and disseminating of innovative Deadlines: Each participant is invited to give a contributed talk. ideas and materials in Mathematics and Statistics Education. Prospective speakers are asked to submit a one-page absract by Organizer: David K. Pugalee (chairman), of the University of North April 15, 2006. Deadline for conference early registration is May Carolina Charlotte. 31, 2006. Information: email: arogerson@vsg. edu . au. Information: For up-to-date information on the conference and online-registration, please visit the web-site: http: //www. icmp. September 2008 uni-stuttgart .de. * 12-1 8 Models in Developing Mathematics Education, Dresden '' 23-2 7 Spectra of families of matrices described by graphs, University of Applied Sciences, Dresden, Germany. digraphs, and sign patterns, AIM Research Conference Center, Description: lOth International Conference of The Mathematics Palo Alto, California. Education into the 21st Century Project Our project was founded in Aim: During the workshop we hope to resolve the 2n-conjecture 1986 and is dedicated to the planning, writing and disseminating and develop new approaches to the minimum rank problem that of innovative ideas and materials in Mathematics and Statistics will lead to significant progress in the future. We hope to get a Education. clearer picture of how energy depends on graph structure, and in Program: Papers are invited on all innovative aspects of mathe­ particular, to understand the structure of graphs with maximal or matics education. There will be an additional social programme for minimal energy. accompanying persons. Our conferences are renowned for their Topics: This workshop, sponsored by AIM and the NSF, will bring friendly and productive working atmosphere. They are attended together people interested in combinatorial matrix theory and spec­ by innovative teachers and mathematics educators from all over tral graph theory for investigation of the following problems: The the world, 2 5 countries were represented at our last conference for 2n-conjecture for spectrally arbitrary sign patterns. Determination example! of the minimum rank of symmetric matrices described by a graph. Information: email: arogerson@inetia. pl. The energy of graphs. Organizers: Leslie Hogben, Richard Brualdi, and Bryan Shader. December 2008 Deadline: July 23, 2006. * 9-1 8 Models in Developing Mathematics Education, Dresden, Information: http:// aimath. org/ ARCC/workshops/ Germany. matrixspectrum.html. Description: Just to alert you to the dates of our lOth Project Conference. November 2006 Organizer: Ludwig Paditz (chairman). Information: email: arogerson@vsg. edu. au. '' 8-1 0 Policy and Practice in Mathematics and Science Teaching and Learning in the Elementary Grades, American University of

498 NOTICES OF THE AMS VOLUME 53, NUMBER 4 New Publications Offered by the AMS

quantitative estimate of unique continuation and the cost of Analysis approximate controllability for coupled parabolic systems; H. S. Luk, S. S.-T. Yau, and W. Zang, Complete invariant of a family of strongly pseudoconvex domain in A1-singularity: Bergman function; G. A. Mendoza, Anisotropic blowup and Recent Progress on compactification; A. Meziani, Planar complex vector fields and CONTEMPORARY MATHEMATICS Some Problems in infinitesimal bendings of surfaces with nonnegative curvature; S.-K. Yeung, Bergman metric on Teichmuller spaces and Recent Progress on Some Several Complex moduli spaces of curves. Problems in Several Complex Variables and Partial Contemporary Mathematics, Volume 400 Differential Equations Variables and Partial May 2006, approximately 215 pages, Softcover, ISBN 0-8218- Shiferaw Berhanu. Hua Chen. Differential Jorge Hounie, Xiaojun Huang, Sheng-U Ton, and Stephen S.-T. You, 3921-7, 2000 Mathematics Subject Classification: 32-XX, 35-XX, Editors Equations All AMS members US$47, List US$ 59, Order code CONM/ 400 Shiferaw Berhanu, Hua Chen,

Arnli~Mathf!maf!UllSnafety Jorge Hounie, Xiaojun Huang, Sheng-Li Tan, and StephenS.­ The Cauchy T. Yau, Editors The Cauchy Transform The papers in this volume cover many important topics of Transform Joseph A. Cima, University of current interest in partial differential equations and several Joseph A. Cima North Carolina, Chapel Hill, complex variables. An international group of well-known Alec Matheson NC, Alec L. Matheson, Lamar mathematicians has contributed original research articles on William T. Ross diverse topics such as the geometry of complex manifolds, the University, Beaumont, TX, and mean curvature equation, formal solutions of singular partial William T. Ross, University of differential equations, and complex vector fields. Richmond, VA The material in this volume is useful for graduate students The Cauchy transform of a measure and researchers interested in partial differential equations and on the circle is a subject of both several complex variables. classical and current interest with a sizable literature. This This item will also be of interest to those working in differential book is a thorough, well-documented, and readable survey of equations. this literature and includes full proofs of the main results of the subject. This book also covers more recent perturbation Contents: C. Anedda and G. Porru, Problems on the Mange­ theory as covered by Clark, Poltoratski, and Aleksandrov and Ampere equation in the plane; A. P. Bergamasco and contains an in-depth treatment of Clark measures. P. L. D. da Silva, Global solvability for a special class of vector fields on the torus; F. Catanese and P. Frediani, Deformation Contents: Overview; Preliminaries; The Cauchy transform as a in the large of some complex manifolds, II; A. Chau and L.- function; The Cauchy transform as an operator; Topologies on F. Tam, Gradient Kahler-Ricci solitons and complex dynamical the space of Cauchy transforms; Which functions are Cauchy systems; H. Chen, Z. Luo, and C. Zhang, On the summability integrals?; Multipliers and divisors; The distribution function of formal solutions for a class of nonlinear singular PDEs with for Cauchy transforms; The backward shift on H 2 ; Clark irregular singularity; Z. Chen and S.-L. Tan, Upper bounds on measures; The normalized Cauchy transform; Other operators the slope of a genus 3 fibration; j.-H. Cheng, The mean on the Cauchy transforms; List of symbols; Bibliography; curvature equation in pseudohermitian geometry; W. M. Eby, Index. Moment results for the Heisenberg group interpreted using Mathematical Surveys and Monographs, Volume 125 the Weyl calculus; N. Gan and X.-Y. Zhou, The cohomology of vector bundles on general non-primary Hopf manifolds; April2006, 272 pages, Hardcover, ISBN 0-8218-3871-7, LC H. Hannah, A. A. Himonas, and G. Petronilho, Gevrey 2005055587, 2000 Mathematics Subject Classification: 30£20, regularity in time for generalized KdV type equations; 30£10, 30H05, 32A35, 32A40, 32A37, 32A60, 47B35,47B37, j. Hounie and E. Lanconelli, An Alexandrov type theorem for 46£27, All AMS members US$60, List US$75, Order code Reinhardt domains of 1(2; L. Lei, G. Wang, and L. Zhang, The SURV/ 125

APRIL 2006 NOTICES OF THE AMS 499 New Publications Offered by the AMS

The Ubiquitous Heat ~ ;~0~~~~ CONTEMPORARY MATHEMATICS Function Theory ------'------Kernel of One Complex The Ubiquitous jay jorgenson, The City Heat Kernel College of New York, NY, and Variable Lynne Walling, University of Jay Jorgenson Lynne Walling Colorado at Boulder, CO, Third Edition Editors Editors Robert E. Greene, University of The aim of this volume is to bring -­ California, Los Angeles, CA, ------~ ------·- AtnSricartM.ili!etrmbOiliS!Ictt1ly together research ideas from various and Steven G. Krantz, fields of mathematics which utilize -.---~ Washington University, the heat kernel or heat kernel techniques in their research. St. Louis, MO The intention of this collection of papers is to broaden productive communication across mathematical sub­ Complex analysis is one of the most central subjects in disciplines and to provide a vehicle which would allow experts mathematics. It is compelling and rich in its own right, but it in one field to initiate research with individuals in another is also remarkably useful in a wide variety of other field, as well as to give non-experts a resource which can mathematical subjects, both pure and applied. This book is facilitate expanding their research and connecting with others. different from others in that it treats complex variables as a direct development from multivariable real calculus. As each This item will also be of interest to those working in differential new idea is introduced, it is related to the corresponding idea equations and mathematical physics. from real analysis and calculus. The text is rich with examples Contents: L. Barchini, M. Sepanski, and R. Zierau, Positivity of and exercises that illustrate this point. zeta distributions and small unitary representations; The authors have systematically separated the analysis from R. Berndt, The heat equation and representations of the Jacobi the topology, as can be seen in their proof of the Cauchy group; J. Dodziuk and V. Mathai, Kato's inequality and theorem. The book concludes with several chapters on special asymptotic spectral properties for discrete magnetic topics, including full treatments of special functions, the Laplacians; D. S. Fine, The heat kernel in low-dimensional prime number theorem, and the Bergman kernel. The authors quantum theories; A. Grigor'yan, Heat kernels on weighted also treat HP spaces and Painleve's theorem on smoothness to manifolds and applications; J, F. Grotowski, Heat kernels in the boundary for conformal maps. geometric evolution equations; B. C. Hall, The range of the heat operator; B. Harris, Heat kernels and cycles; G. Hein, This book is a text for a first-year graduate course in complex Green currents on Kahler manifolds; S. Hofmann, Heat kernels analysis. It is an engaging and modern introduction to the and Riesz transforms; M. D. Horton, D. B. Newland, and subject, reflecting the authors' expertise both as A. A. Terras, The contest between the kernels in the Selberg mathematicians and as expositors. trace formula for the (q + 1) -regular tree; J. jorgenson and Contents: Fundamental concepts; Complex line integrals; J. Kramer, Expressing Arakelov invariants using hyperbolic Applications of the Cauchy integral; Meromorphic functions heat kernels; M. H. Lee and E. Previato, Grassmannians of and residues; The zeros of a holomorphic function; higher local fields and multivariable tau functions; V. Mathai Holomorphic functions as geometric mappings; Harmonic and I. Chatterji, Heat kernels and the range of the trace on functions; Infinite series and products; Applications of infinite completions of twisted group algebras; E. Previato, Theta sums and products; Analytic continuation; Topology; Rational functions, old and new; P. Sawyer, The heat kernel on the approximation theory; Special classes of holomorphic symmetric space SL(n,F) /SU(n,F); B. Wang, Incidence functions; Hilbert spaces of holomorphic functions, the structure. Bergman kernel, and biholomorphic mappings; Special Contemporary Mathematics, Volume 398 functions; The prime number theorem; Appendix A: Real analysis; Appendix B: The statement and proof of Goursat's April 2006, 402 pages, Softcover, ISBN 0-8218-3698-6, LC theorem; References; Index. 2005057186, 2000 Mathematics Subject Classification: 35K05, 22E45, llFSO, 14C99, 47N50, 58]35, 53C44, 32W3 0, 11F72, Graduate Studies in Mathematics, Volume 40 53Cl5, All AMS members US$79, List US$99, Order code April 2006, 504 pages, Hardcover, ISBN 0-8218-3962-4, LC CONM/398 2005057188, 2000 Mathematics Subject Classification: 30-01; 30-02, 30-03, All AMS members US$63, List US$79, Order code GSM/40.R

500 NOTICES OF THE AMS VOLUME 53, NUMBER 4 New Publications Offered by the AMS

in France decided to write a fundamental treatise on analysis Differential Equations to replace the standard texts of the time. They ended up writing the most influential and sweeping mathematical treatise of the twentieth century, Les elements de Gradient Inequalities mathematique. Maurice Mashaallifts the veil from this secret society, showing Gradient with Applications to us how heated debates, schoolboy humor, and the devotion Asymptotic Behavior and hard work of the members produced the ten books that Inequalities took them over sixty years to write. The book has many first­ wHh Applications and Stability of to Asymptotic Behavior hand accounts of the origins of Bourbaki, their meetings, their and Stability of Gradient-like Systems seminars, and the members themselves. He also discusses the Gradient-like Systems lasting influence that Bourbaki has had on mathematics, Sen-Zhong Huang Sen-Zhong Huang, Universitat through both the Elements and the Seminaires. The book is Rostock, Germany illustrated with numerous remarkable photographs.

\® ...... w .... --... ISocllly This book presents a survey of the Contents: A group forms; The story of a name; Young Turks relatively new research field of against stubborn priests; Bourbaki's Elements de gradient inequalities and their Mathematique; Towards axioms and structures; A snapshot of applications. The exposition emphasizes the powerful Bourbaki's work: Filters; The Bourbaki seminar; Subtle and applications of gradient inequalities in studying asymptotic austere schoolboys; "For the honor of the human spirit"; New behavior and stability of gradient-like dynamical systems. It math in the classroom; An immortal mathematician?; explains in-depth how gradient inequalities are established Acknowledgments; Bibliography and how they can be used to prove convergence and stability June 2006, approximately 260 pages, Softcover, ISBN 0-8218- of solutions to gradient-like systems. This book will serve as 3967-5, 2000 Mathematics Subject Classification: 01A70, an introduction for further studies of gradient inequalities and 01A60, All AMS members US$23, List US$29, Order code their applications in other fields, such as geometry and BOURBAKI computer sciences. ~·>; COURSE This book is written for advanced graduate students, . ADOPTION researchers and applied mathematicians interested in dynamical systems and mathematical modeling. ~"~~ Euler through Time: lt'nlla•''l'loo~.,,ftlo Contents: Introduction and overview of the results; Gradient A New Look at Old inequality; Abstract convergence results; Applications to semilinear gradient-like systems in Hilbert spaces; Themes Applications to the stability problem; Bibliography; Index. V. S. Varadarajan, University Mathematical Surveys and Monographs, Volume 126 of California, Los Angeles, CA June 2006, approximately 190 pages, Hardcover, ISBN 0-8218- Euler is one of the greatest and most 4070-3, LC 2005058916, 2000 Mathematics Subject prolific mathematicians of all time. He Classification: 35A15, 35Bxx, 35Kxx, 37Ll5, 47J35 ; 35Q80, wrote the first accessible books on 47Hxx, All AMS members US$47, List US$ 59, Order code calculus, created the theory of circular SURV/126 functions, and discovered new areas· of research such as elliptic integrals, the calculus of variations, graph theory, divergent series, and so on. It took hundreds of years for his successors to develop in full the theories he began, and some of his themes are still at the center of today's mathematics. It is of great interest therefore General and to examine his work and its relation to current mathematics. Interdisciplinary This book attempts to do that. In number theory the discoveries he made empirically would ~~ COURSE require for their eventual understanding such sophisticated • ADOPTION developments as the reciprocity laws and class field theory. His pioneering work on elliptic integrals is the precursor of Bourbaki the modern theory of abelian functions and abelian integrals. A Secret Society of His evaluation of zeta and multizeta values is not only a fantastic and exciting story but very relevant to us, because Mathematicians they are at the confluence of much research in algebraic Maurice Mashaal, Pour la geometry and number theory today (Chapters 2 and 3 of the Science, Paris, France book). Anticipating his successors by more than a century, Euler The name Bourbaki is known to every created a theory of summation of series that do not converge mathematician. Many also know in the traditional manner. Chapter 5 of the book treats the something of the origins of Bourbaki, progression of ideas regarding divergent series from Euler to yet few know the full story. In 1935, a many parts of modern analysis and quantum physics. small group of young mathematicians

APRIL 2006 NOTICES OF THE AMS 501 New Publications Offered by the AMS

The last chapter contains a brief treatment of Euler products. May 2006, 209 pages, Softcover, ISBN 0-8218-3696-X, LC Euler discovered the product formula over the primes for the 2005057189, 2000 Mathematics Subject Classification: 11S70, zeta function as well as for a small number of what are now 19D55, 20D20,46L80, 55N91, 55P35, 55P60, 55R80, 55U35, called Dirichlet L-functions. Here the book goes into the 57T25, All AMS members US$ 55, List US$69, Order code development of the theory of such Euler products and the role CONM/399 they play in number theory, thus offering the reader a glimpse of current developments (the Langlands program). This item will also be of interest to those working in number ------The Geometry of theory and analysis. CONTEMPORARY MATHEMATICS Contents: Leonhard Euler (1707-1783); The universal ------Riemann Surfaces "' mathematician; Zeta values; Euler-Maclaurin sum formula; The Geometry and Abelian Varieties Divergent series and integrals; Euler products; Gallery. of Riemann Surfaces Jose M. Munoz Porras, June 2006, 296 pages, Hardcover, ISBN 0-8218-3580-7, and Abelian Varieties LC Universidad de Salamanca, 2005057177, 2000 Mathematics Subject Jose M. Mui'ioz Porras Classification: 01A70; Sarin Popescu 01A50, 11-03, 40-03, All AMS members US$47, List US$ 59, Rubi E. Rodriguez Spain, Sorin Popescu, State Editors Order code EULER 12\ University of New York at ------\0' Stony Brook, NY, and Rubi E. Amenean MathemaloeijlSC>cOel;' Rodriguez, Pontificia Universidad Cat6lica de Chile, Geometry and Topology Santiago, Chile, Editors Most of the papers in this book deal with the theory of Riemann surfaces (moduli problems, automorphisms, etc.), CONTEMPORARY An Alpine Anthology abelian varieties, theta functions, and modular forms. Some of MATHEMATICS of Homotopy Theory the papers contain surveys on the recent results in the topics ------of current interest to mathematicians, whereas others contain An Alpine Dominique Arlettaz, new research results. Anthology of Universite de Lausanne, Contents: A. Basmajian and M. Zeinalian, Mobius Homotopy Theory Switzerland, and Kathryn transformations of the circle form a maximal convergence Dominique Arlettaz group; A. Campillo and J. Olivares, On the polar linear system Kathryn Hess Hess, Ecole Polytechnique Editors Federale de Lausanne, of a foliation by curves in a projective space; F. J. Cirre, Switzerland, Editors Birational classification of hyperelliptic real algebraic curves; C. j. Earle, The genus two Jacobians that are isomorphic to a The second Arolla conference on product of elliptic curves; H. M. Farkas, Vanishing thetanulls algebraic topology brought together and Jacobians; C. Florentino, J. Mourao, and J. P. Nunes, Theta specialists covering a wide range of homotopy theory and K­ functions, geometric quantization and unitary Schottky bundles; theory. These proceedings reflect both the variety of talks Y. Fuertes and G. Gonziilez-Diez, Smooth double coverings of given at the conference and the diversity of promising hyperelliptic curves; J. Gihnan and L. Keen, Planar families of research directions in homotopy theory. The articles contained discrete groups; E. Gomez Gonziilez and C. Gonziilez-Martinez, in this volume include significant contributions to classical Generalized addition formulae for theta functions; E. Gomez unstable homotopy theory, model category theory, equivariant Gonziilez and F. J. P. Martin, Curves with a group action and homotopy theory, and the homotopy theory of fusion systems, Galois covers via infinite Grassmannians; V. Gonziilez-Aguilera, as well as to K-theory of both local fields and C* -algebras. J. M. Mufioz·Porras, and A. G. Zamora, Some recent results on the irreducible components of the singular locus of A 9 ; Contents: G. Arone, A note on the homology of ~n, the M. R. Gonzalez-Dorrego, A note on the arithmetic genus of Schwartz genus, and solving polynomial equations; C. Broto, reducible plane curves; F. Herrlich, Teichmilller curves defined R. Levi, and B. Oliver, A geometric construction of saturated by characteristic origamis; R. A. Hidalgo and B. Maskit, Lowest fusion systems; C. Casacuberta and B. Chorny, The orthogonal uniformizations of compact real surfaces; H. Lange, Principal subcategory problem in homotopy theory; W. Chacholski, polarizations on products of elliptic curves; F. P. Romo, An W. Pitsch, and J. Scherer, Homotopy pull-back squares up to approach to a 2-dirnensional Contou-Carrere symbol; S. Recillas localization; I. Chatterji and G. Mislin, Traces and reduced and R. E. Rodriguez, Prym varieties and fourfold covers II: The group C* -algebras; A. Clement, Integral cohomology of 2-local dihedral case; G. Schrnithiisen, Examples for Veech groups of Hopf spaces with at most two non-trivial finite homotopy origarnis; R. Silhol, Genus 2 translation surfaces with an order 4 groups; B. Gray and S. Theriault, On the double suspension automorphism; R. Smith and R. Varley, The Pfaffian structure and the mod-p Moore space; J. P. C. Greenlees and ].Ph. defining a Prym theta divisor. Hoffmann, Rational extended Mackey functors for the circle group; L. Hesselholt, On the topological cyclic homology of the Contemporary Mathematics, Volume 39 7 algebraic closure of a local field; M. Joachim and April2006, 236 pages, Softcover, ISBN 0-8218-3855-5, LC M. W. Johnson, Realizing Kasparov's KK -theory groups as the 2005057090, 2000 Mathematics Subject Classification: 30Fl0, homotopy classes of maps of a Quillen model category; J. Lin, 14H30, 14H40; 14H10, 14H15, 14H37, 14K25, 14K05, 30F40, Homology commutators and T 1 actions. All AMS members US$ 55, List US$69, Order code CONM/397 Contemporary Mathematics, Volume 399

502 NOTICES OF THE AMS VOLUME 53, NUMBER 4 New AMS-Distributed Publications

Analysis New AMS-Distributed Publications Current Trends in Potential Theory Dominique Bakry, University of Toulouse, France, Algebra and Algebraic Lucian Beznea, Romanian Academy, Bucharest, Romania, Gheorghe Bucur, University of Geometry Bucharest, Romania, and Michael Rockner, Bielefeld University, Germany, Editors This is the proceedings volume of two mathematical meetings Linear Algebra and Group Theory on Potential Theory organized in Bucharest, Romania, in September 2002 and September 2003. It includes six survey for Physicists articles and seven selected research papers, covering the main Second Edition topics of the conferences: geometric aspects in potential theory, Dirichlet structures, stochastic analysis, potential K. N. Srinivasa Rao, University of Mysore, India theory, and Markov processes. A publication of the Theta Foundation. Distributed worldwide, except Professor Srinivasa Rao's text on Linear Algebra and Group in Romania, by the AMS. Theory is directed to undergraduate and graduate students who wish to acquire a solid theoretical foundation in these Contents: Survey articles: N. Arcozzi, E. C. Tarabusi, F. Di mathematical topics which find extensive use in physics. Biasse, and M. Picardello, A potential theoretic approach to Based on courses delivered during Professor Srinivasa Rao's twisting; D. Feyel, A survey of the Monge transport problem; long career at the University of Mysore, this text is remarkable B. Fuglede, Harmonic maps from Riemann polyhedra to for its clear exposition of the subject. spaces of nonpositive curvature; F. Hirsch, Measurable metrics, intrinsic metrics and Lipschitz functions; A. Lejay Advanced students will find a range of topics such as the and T. Lyons, On the importance of the Levy area for studying Representation theory of Linear Associative Algebras, a the limits of functions of stochastic processes. Application to complete analysis of Dirac and Kemmer algebras, homogenization; V. Metz, Superadditive Perron-Frobenius Representations of the Symmetric group via Young Tableaux, a theory; Research papers: I. Bachar, Estimates for the Green systematic derivation of the Crystallographic point groups, a function and existence of positive solutions of nonlinear comprehensive and unified discussion of the Rotation and equations with Navier boundary conditions; D. Bakry and Lorentz groups and their representations, and an introduction Z. Qian, Volume comparison theorems without Jacobi fields; to Dynkin diagrams in the classification of Lie groups. In N. B. Rhouma and M. Bezzarga, On a singular value problem addition, the first few chapters on Elementary Group Theory and the boundary Harnack principle for fractional Laplacian; and Vector Spaces also provide useful instructional material M. Biroli and P. G. Vernole, Brelot property for the sheaf of even at an introductory level. harmonics relative to a Dirichlet form; K. Janssen, An authority on diverse aspects of mathematical physics, Factorization of excessive kernels; E. Popescu, Pseudo Professor Srinivasa Rao taught at the University of Mysore differential operators in the context of Feller sernigroups and until1982 and was subsequently at the Indian Institute of Dirichlet forms; C. Udrea, Resolvent and nonlinear potential Science, Bangalore. He has authored a number of texts, among theory. them The Rotation and Lorentz Groups and Their International Book Series of Mathematical Texts Representations for Physicists (Wiley, 1988) and Classical Mechanics (Universities Press, 2003). The first edition of Linear january 2006, 174 pages, Hardcover, ISBN 973-85432-6-6, Algebra and Group Theory for Physicists was co-published in 2000 Mathematics Subject Classification: 31-06, 60-06, All 1996 by New Age International and Wiley, New York. AMS members US$22, List US$28, Order code THETA/7 A publication of Hindus tan Book Agency. Distributed on an exclusive basis by the AMS in North America. Online bookstore rights worldwide. Contents: Elements of group theory; Some related algebraic Advances in Operator Algebras structures; Linear vector space; Elements of representation and Mathematical Physics theory; Representations of finite groups; Representations of linear associative algebras; Representations of the symmetric Florin-Petre Boca, University of Illinois, Urbana, IL, group; The rotation group and its representations; The Ola Bratteli, University of Oslo, Norway, Roberto crystallographic point groups; The Lorentz group and its Longo, University of Rome Tor Vergata, Italy, and representations; Introduction to the classification of Lie groups- Dynkin diagram; Index. Heinz Siedentop, University of Regensburg, Germany, Editors Hindustan Book Agency January 2006, 608 pages, Hardcover, ISBN 81-85931-64-X, A series of international conferences in operator algebras and 2000 Mathematics Subject Classification: 15-XX, 15-01, 20-XX, mathematical physics was initiated by the Institute of 20-01, 20-02, All AMS members US$42, List US$ 52, Order Mathematics of Bucharest in 2001. The second meeting was code HIN/30 held in Sinaia from june 26 to july 4, 2003. The volume

APRIL 2006 NOTICES OF THE AMS 503 New AMS-Distributed Publications contains the proceedings of this conference. It consists of less central topics) can be taught in two quarters of twenty­ eighteen refereed papers, pointing out the interrelation of five to thirty lectures each. these two important domains of research. Among the subjects The course material is deeply intertwined with the exercises, covered are structure and classification of C'-algebras, as it is intended for the student to actively learn the material invariants for subfactors and connections with quantum field (and to practice thinking and writing rigorously) by proving theory, quantum systems, Weyl pseudo-differential calculus several of the key results in the theory. and twisted crossed products, Schrodinger operators, trace class approach for scattering, coherent states, quantum gauge A publication of Hindus tan Book Agency. Distributed on an exclusive basis by the AMS in North America. Online bookstore rights worldwide. theories, fractals and spectral triples, invariant subspaces and reflexivity in operator algebras, and amenability for groupoids. Contents: Volume 1: Introduction; The natural numbers; Set theory; Integers and rationals; The real numbers; Limits of This item will also be of interest to those working in sequences; Series; Infinite sets; Continuous functions on R; mathematical physics. Differentiation of functions; The Riemann integral; A A publication of the Theta Foundation. Distributed worldwide, except Appendix: the basics of mathematical logic; B Appendix: the in Romania, by the AMS. decimal system; Index. Contents: S. Berceanu, Realization of coherent state Lie Hindustan Book Agency algebras by differential operators; M. R. Buneci, Amenable equivariant maps defined on a groupoid; S. Carpi and January 2006, 420 pages, Softcover, ISBN 81-85931-62-3, 2000 M. Weiner, Uniqueness of the Diff+ (S 1 ) symmetry for local Mathematics Subject Classification: 26A03, 26A42, 26B05, nets of von Neumann algebras; V. Deaconu, C* -algebras of 26B10, All AMS members US$29, List US$36, Order code commuting endomorphisms; P. Duclos, E. Soccorsi, HIN/ 28 P. Stovicek, and M. Vittot, Dynamical localization in periodically driven quantum systems; D. E. Evans and P. R. Pinto, Modular invariants and the double of the Haagerup subfactor; D. Guido and T. Isola, Dimensions and Analysis II spectral triples for fractals in RN; C. Ivanescu, On the Terence Tao, University of California, classification of continuous trace C* -algebras with spectrum Los Angeles, CA homeomorphic to the closed interval [0, 1]; M. Mantoiu, R. Purice, and S. Richard, Twisted corssed products and This is part two of a two-volume introduction to real analysis magnetic pseudodifferential operators; G. Nenciu, On the and is intended for honours undergraduates, who have smoothness of gap boundaries for generalized Harper already been exposed to calculus. The emphasis is on rigour operators; G. K. Pedersen, A note on fixed points of and on foundations. The material starts at the very completely positive maps; B. Prunaru, Dual algebras and beginning-the construction of the number systems and set approximate reflexivity; M. Rordam, The real rank of certain theory, then to the basics of analysis (limits, series, continuity, simple C* -algebras; G. Scharf, Supersymmetric quantum differentiation, Riemann integration), through to power series, gauge theories; H. Siedentop, The relativistic electron-positron several variable calculus and Fourier analysis, and finally to field; A. L. Svendsen, Outer automorphisms of a series of non­ the Lebesgue integral; these are almost entirely set in the amenable subfactors; M. Vittot, A Lie algebra point of view on concrete setting of the real line and Euclidean spaces, global perturbation theory; D. R. Yafaev, Trace-class approach although there is some material on abstract metric and in scattering problems for perturbations of media. topological spaces. There are also appendices on mathematical International Book Series of Mathematical Texts logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of twenty­ January 2006, 286 pages, Hardcover, ISBN 973-85432-7-4, five to thirty lectures each. 2000 Mathematics Subject Classification: 46-06, 47-06, 81-06, 35-06, All AMS members US$34, List US$42, Order code The course material is deeply intertwined with the exercises, THETA/8 as it is intended for the student to actively learn the material (and to practice thinking and writing rigorously) by proving several of the key results in the theory. A publication of Hindustan Book Agency. Distributed on an exclusive Analysis I basis by the AMS in North America. Online bookstore rights worldwide. Terence Tao, University of California, Contents: Volume 2: Metric spaces; Continuous functions on Los Angeles, CA metric spaces; Uniform convergence; Power series; Fourier series; Several variable differential calculus; Lebesgue This is part one in a two-volume introduction to real analysis measure; Lebesgue integration; Index. and is intended for honours undergraduates, who have Hindustan Book Agency already been exposed to calculus. The emphasis is on rig our and on foundations. The material starts at the very January 2006, 272 pages, Softcover, ISBN 81-85931-63-l, 2000 beginning- the construction of the number systems and set Mathematics Subject Classification: 26A03, 26A42, 26B05, theory, then to the basics of analysis (limits, series, continuity, 26B10, All AMS members US$24, List US$ 30, Order code differentiation, Riemann integration), through to power series, HIN/29 several variable calculus and Fourier analysis, and finally to the Lebesgue integral; these are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. There are also appendices on mathematical logic and the decimal system. The entire text (omitting some

504 NOTICES OF THE AMS VOLUME 53, NUMBER 4 New AMS-Distributed Publications

Memoires de Ia Societe Mathematique de France, Number Differential Equations 101-102 December 2005, 208 pages, Softcover, ISBN 2-85629-180-5, 2000 Mathematics Subject Classification: 35A17, 35A22, Systemes differentiels involutifs 35Q40, 35Q55, Individual member US$53, List US$ 59, Order code SMFMEM/101/102 Bernard Malgrange, University of Grenoble I, St. Martin d'Heres, France The first part of this volume is an exposition of the theory of "systemes en involution" of E. Cartan, from the homological point of view of Spencer, Sternburg eta/. The point of view of Mathematical Physics Cartan himself is also recalled, and compared to the preceding one, in Appendix B. The second part proves the generic involutiveness of analytic differential systems, which is a precise version of an assertion of Cartan saying roughly that, New Trends in Continuum "by prolongation, a differential system becomes eventually involutive". Mechanics This item will also be of interest to those working in geometry Mihaela Mih

APRIL 2006 NOTICES OF THE AMS 505 New AMS-Distributed Publications decode the microstructure characteristics from divided into four independent chapters; although the most macroexperience; G. Marinoschi, On a nonlinear boundary recent developments are studied, it remains mostly accessible value problem of infiltration in unsaturated media; B. Matei, to non-specialists. Nonlinear multiscales representations for homogenization A publication of the Societe Mathematique de France, Marseilles (SMF), problems: A case study; M. Negreanu, Discrete inequalities; distributed by the AMS in the U.S., Canada, and Mexico. Orders from G. Pasa, Secondary oil recovery and variable permeability; other countries should be sent to the SMF. Members of the SMF re~:eive E. Perez, Correcting terms from local problems for vibrating a 30% discount from list. systems with concentrated masses; K. Piechor, On the Contents: F. Martin and E. Royer, Formes modulaires et hydrodynamic limit of the Enskog equation with weak square­ periodes; V. Bosser, Inctependance algebrique de valeurs de well potential; D. Polisevski and R. Schiltz-Bunoiu, Heat series d'Eisenstein (theoreme de Nesterenko); Ph. Graftieaux, conduction through a first-order jump interface; M. Popescu, Theoreme stephanois et methode des pentes; F. Pellarin, On the optimal control of bilinear systems; R. Raducanu, On Introduction aux formes modulaires de Hilbert et a leurs the mortar finite element method in linear elasticity; proprietes differentielles; Annexe. Liste des participants. S. Sburlan and C. Sburlan, A coincidence degree for bifurcation problems with applications in mechanics of Seminaires et Congres, Number 12 continua; N. Simian, Models of heat propagation in solid December 2005, 271 pages, Softcover, ISBN 2-85629-176-7, bodies; 0. Simionescu·Panait, Propagation of attenuated 2000 Mathematics Subject Classification: 11F03, 11F06, 11Fll, waves in isotropic solids subject to initial electro-mechanical 11F25, 11F30, 11F37, 11F41, 11F60, 11F67, llGxx, 11G35, fields; D. Socolescu, On the Leray problems for the stationary 11G50, 11]85, 11]91, 14G40, Individual member US$53, List and non-stationary-Navier-Stokes equations; R. Stavre, US$ 59, Order code SEC0/ 12 Boundary control of a non-stationary magnetohydrodynamic flow; P. P. Teodorescu, T. Badea, L. Munteanu, and J. Onisoru, On the wave propatagion in materials with negative stiffness components; P. P. Teodorescu, L. Munteanu, and V. Chiroiu, On the wave propagation in a chiral medium; V. Tigoiu and C. Cipu, Flow of some viscoelastic fluids in a falling cylinder viscometer and the evaluation of shear viscosity; A. Ursescu, Influence of the electrode ends on the channel flow of electrorheological fluids; A. Ursescu and C. Dascalu, Thermally conductive elliptic hole in an anisotropic solid; C. Vallee, C. Lerintiu, D. Fortnne, M. Ban, and G. de Saxce, Hill's bipotential. International Book Series of Mathematical Texts November 2005, 352 pages, Hardcover, ISBN 973-85432-5-8, 2000 Mathematics Subject Classification: 74-06, 76-06, All AMS members US$38, List US$48, Order code THETA/6

Number Theory

Formes modulaires et transcendance Colloque JEUNES Stephane Fischler, Universite Paris-Sud, Orsay, France, Eric Gaudron, Institut Fourier, Saint­ Martin-d'Heres, France, and Samy Khemira, Institut de Mathematiques de ]ussieu, Paris, France, Editors The present volume arises from a conference on the links between modular forms and transcendence held at the C.I.R.M (Marseille) from May 26 to 30, 2003. It includes an overview of the few existing proofs of transcendence or algebraic independence of numbers coming from modular forms theory as well as more general techniques offering new perspectives (periods, Rankin-Cohen brackets, slope method, Hilbert modular forms). The book is

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CALIFORNIA completion of a Ph.D. in Mathematics and GREECE present evidence of superb teaching skills University of California in Berkeley in various mathematics courses across the UNIVERSITY OF CYPRUS Mathematical Sciences Research undergraduate curriculum. Expectations Department of Mathematics and Institute include active involvement in committees Statistics Director and scholarly activity. The ideal candidate The Department invites applications for will demonstrate a commitment to schol­ one position in Partial Differential Equa­ Applications are invited for the position of arship and provide evidence of research ac­ Director at the Mathematical Sciences Re­ tions (Theory) at the rank of Lecturer or search Institute (MSRI), an independent tivity in support of the institution's tran­ Assistant Professor. The official languages research organization located on the cam­ sition to a doctoral level university. of the University are Greek and/or Turk­ pus of the University of California in Berke­ Review of applications will begin im­ ish. For the above position knowledge of Greek is necessary. The deadline for ap­ ley. The appointment will be for a five­ mediately. The filling of all positions is year term starting July 2007. For more plications is May 5, 2006. For more infor­ dependent on available funding. Appli­ mation, see http: mas. ucy. ac. cy /; information, see http: I /www . ms ri . o rg/ I /www. cants should submit a letter of interest University of Cyprus, P.O. Box 20537, 1678 about/jobs/director. Applications will Nic-osia, Cyprus; tel: 00357-22892606; fax: be considered starting March 1, 2006. addressing qualifications for the position, a curriculum vita, graduate transcripts, 00357-22892601. MSRI is an Equal Opportunity Employer. 000234 000236 and three letters of reference to Mrs. Libby Whitaker, Director of Human Resources, Lincoln Memorial University, 6965 Cum­ · TENNESSEE berland Gap Parkway, Harrogate, TN 37752. LMU's hiring policies are in accor­ LINCOLN MEMORIAL UNIVERSITY (LMU) dance with EEO regulations and policies. Assistant Professor of Mathematics LMU is committed to diversity and is an The department of Mathematics and Nat­ Equal Opportunity Employer. Women and ural Sciences anticipates filling a full-time minorities are strongly encouraged to faculty position in mathematics to start apply. August 15, 2006. Candidates must verify 000235

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508 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Meetings & Conferences oftheAMS

IMPORTANTINFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information with links to the abstract for each talk can be found on theAMS website. See http: I /www. ams. org/meeti ngs/. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL and in an electronic issue of the Notices as noted below for each meeting.

Tulane University, and Aron Simis, University Federal de Miami, Florida Pernambuco. Composition Operators and Complex Dynamical Systems, Florida International University Brian P. Kelly, University of Louisiana, Monroe, and Christo­ April1-2, 2006 pher N. B. Hammond, Connecticut College. Saturday - Sunday Financial Mathematics, Alec N. Kercheval and Craig A. Nolder, Florida State University. Meeting #1 015 Geometry of Banach Spaces and Connections with Other Southeastern Section Areas, Edward W. Odell, University of Texas at Austin, Associate secretary: Matthew Miller Thomas B. Schlumprecht, Texas A&M University, and Announcement issue of Notices: January 2006 Stephen Dilworth, University of South Carolina. Program first available on AMS website: February 16, 2006 Geometry of Riemannian Manifolds with Additional Struc­ Program issue of electronic April 2006 Notices: tures, Tedi C. Draghici, Gueo V. Grantcharov, and Philippe Issue of Abstracts: Volume 27, Issue 2 Rukimbira, Florida International University. Deadlines Harmonic Analysis and Partial Differential Equations, Mario For organizers: Expired Milman, Florida Atlantic University, and Marius Mitrea, Uni­ For consideration of contributed papers in Special Sessions: versity of Missouri. Expired History of Mathematics, Karen H. Parshall, University of For abstracts: Expired Virginia. Imaging, Homogenization, and Shape Optimization, Michael Invited Addresses S. Vogelius, Rutgers University, and Shari Moskow, Uni­ Andrea R. Nahmod, University of Massachusetts, Amherst, versity of Florida. Bilinear operators in analysis and PDEs. Interpolation Theory and Applications, Michael Cwikel, Edward Odell, University of Texas at Austin, Embeddings Technion, Laura De Carli, Florida International Univer­ in Banach space theory. sity, and Mario Milman, Florida Atlantic University. Karen V. H. Parshall, University of Virginia, The British de­ Invariants of Low-Dimensional Manifolds, Thomas G. velopment of the theory of invariants, 1841-1895. Leness, Florida International University, and Nikolai N. Michael S. Vogelius, Rutgers University, Electromagnetic Saveliev, University of Miami, Coral Gables. imaging-An applied analyst's perspective. Mathematical Models in Image and High-Dimensional Data Analysis, Hanna E. Makaruk and Robert M. Owczarek, Los Special Sessions Alamos National Laboratory, and Nikita Sakhaneko, Uni­ Approximation Theory and Orthogonal Polynomials, Doron versity of New Mexico and Los Alamos National Labora­ S. Lubinsky, Georgia Institute of Technology, and Edward tory. B. Saff, Vanderbilt University. Monomials and Resolutions, Joseph P. Brennan, North Commutative Algebra and Algebraic Geometry, Laura Dakota State University, and Heath M. Martin, University Ghezzi, Florida International University, Huy T<'li Ha, of Central Florida.

APRIL 2006 NOTICES OF THE AMS 509 Meetings & Conferences

Nonlinear Waves, Andrea R. Nahmod, University of Mass­ Christopher M. Skinner, University of Michigan, Modular achusetts, Amherst, and Sijue Wu, University of Michigan forms and special values of L-functions. at Ann Arbor. Partial Differential Equations and Several Complex Vari­ Special Sessions ables, Shiferaw Berhanu, Temple University, and Hamid Algebraic Structures of Exactly Solvable Models, Michael Meziani, Florida International University. Gekhtman, University of Notre Dame, Mikhail Shapiro, Qualitative Analysis of Partial Differential Equations, Con­ Michigan State University, and Alexander Stolin, Univer­ gming Li, University of Colorado, and Wenxiong Chen, sity of Gothenburg. Yeshiva University. Analysis and Geometry of Non-linear Evolution Equations, Recent Developments on Fluid and Geophysical Fluid Dy­ Alexandrou A. Himonas and Gerard K. Misiolek, Univer­ namics, C. Cao and T. Tachim Medjo, Florida Interna­ sity of Notre Dame. tional University, and X. Wang, Florida State University. Combinatorial Algebraic Geometry, juan C. Migliore, Uni­ Singular Integrals, Geometric Analysis, and Free Boundary versity of Notre Dame, and Uwe R. Nagel, University of Ken­ Problems, Laura De Carli, Florida International University, tucky. Marianne Korten and Charles N. Moore, Kansas State Uni­ Commutative Algebra and Algebraic Geometry, Alberto versity. Corso, University of Kentucky, Claudia Polini, University Spectral Geometry ofManifolds with Boundary and Singular of Notre Dame, and Bernd Ulrich, Purdue University. Spaces, Juan B. Gil, Pennsylvania State University, Altoona, and Patrick T. McDonald, New College, University of South Developments and Applications in Differential Geometry, Florida. Jianguo Cao, Xiaobo Liu, and Brian Smyth, University of Notre Dame. Structure of Function Spaces and Applications, Jan Lang, The Ohio State University, and Osvaldo Mendez, Univer­ Dynamical Systems, Francois Ledrappier, University of sity of Texas at El Paso. Notre Dame, and Arnie Wilkinson, Northwestern Univer­ sity. Harmonic Analysis, PDE and Geometric Function Theory, Notre Dame, Indiana John L. Lewis, University of Kentucky, and Steve C. Hof­ mann, University of Missouri. University of Notre Dame Holomorphic Methods and Heat Kernels in Harmonic Analy­ April8-9, 2006 sis and Quantization Theory, Brian Hall and William Kir­ Saturday - Sunday win, University of Notre Dame. Mathematical Biology, Mark Alber and Bei Hu, University Meeting #1 016 of Notre Dame. Central Section Model Theory and Computability, Steven Allen Buechler Associate secretary: Susan]. Friedlander and Julia Knight, University of Notre Dame, Steffen Lempp, Announcement issue of Notices: January 2006 University of Wisconsin, and Sergei Starchenko, Univer­ Program first available on AMS website: February 23, 2006 sity of Notre Dame. Program issue of electronic Notices: April 2006 Issue of Abstracts: Volume 27, Issue 2 New Developments in Optimization, Leonid Faybusovich, University of Notre Dame. Deadlines Nonlinear Waves, Mark S. Alber and Pavel Lushnikov, For organizers: Expired University of Notre Dame, and Ildar Gabotiv and Vladimir For consideration of contributed papers in Special Sessions: E. Zakharov, University of Arizona. Expired Number Theory, Scott T. Parsell and Jonathan P. Soren­ For abstracts: Expired son, Butler University. Invited Addresses Numerical Solution of Polynomial Systems, Christopher S. Peterson, Colorado State University, and Andrew J. Douglas N. Arnold, Institute for Math and Applications, Somrnese, University of Notre Dame. University of Minnesota, Finite element exterior calculus and its applications. PDEs and Geometric Analysis, Matt Gursky and Qing Han, University of Notre Dame. Bela Bolio bas, University of Memphis and Cambridge Uni­ versity, Inhomogeneous random graphs (Erdos Memorial Several Complex Variables, Nancy K. Stanton and Jeffrey Lecture). A. Diller, University of Notre Dame. Steven C. Hofmann, University of Missouri, Local Tb the­ Special Functions and Orthogonal Polynomials, Diego Do­ orems and applications in PDE. minici, State University of New York at New Paltz. Michael Larsen, University of Indiana, Representation zeta Topics in Representation Theory, Sam Evens, University of functions. Notre Dame, and Jiu-Kang Yu, Purdue University.

510 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Meetings & Conferences

Topology and Physics, Stephan A. Stolz and Bruce Williams, Banach Spaces ofAnalytic Functions, Rita A. Hibschweiler, University of Notre Dame. University of New Hampshire, and Thomas H. MacGregor, Undergraduate Mathematical Research, Francis X. Con­ SUNY Albany and Bowdoin College. nolly, University of Notre Dame, and Zsuzsanna Szanis­ Discrete and Convex Geometry, Daniel A. Klain, University zlo, Valparaiso University. of Massachusetts (Lowell), Barry R. Monson, University of Water Waves, David Nicholls, University of Illinois at New Brunswick, and Egon Schulte, Northeastern Univer­ Chicago. sity. Galois Theory in Arithmetic and Geometry, Florian Pop and David Harbater, University of Pennsylvania, and Rachel Durham, New J. Pries, Colorado State University. Geometric Methods in Group Theory and Topology, Kim Hampshire Ruane, Tufts University, Jennifer Taback, Bowdoin Col­ University of New Hampshire lege, and Peter N. Wong, Bates College. Global Perspectives on the Geometry of Rier.zann Surfaces, April 22-23, 2006 Eran Makover and Jeffrey K. McGowan, Central Con­ Saturday - Sunday necticut State University. Hopf Algebras and Galois Module Theory, Timothy Kohl, Meeting #1 01 7 Boston University, and Robert G. Underwood, Auburn Eastern Section University Montgomery. Associate secretary: Lesley M. Sibner Mathematical Challenges in Physical and Engineering Sci­ Announcement issue of Notices: February 2006 ences, Marianna A. Shubov, University of New Hampshire. Program first available on AMS website: March 9, 2006 Program issue of electronic Notices: April 2006 Quantum Invariants of Knots and 3-Manifolds, Charles D. Issue of Abstracts: Volume 27, Issue 2 Frohman, University of Iowa, and Razvan Gelca, Texas Tech University. Deadlines Symplectic and Contact Topology, Weimin Chen, Michael For organizers: Expired G. Sullivan, and Hao Wu, University of Massachusetts, For consideration of contributed papers in Special Sessions: Amherst. Expired Topological Algebras and Applications, Alexander A. Katz, For abstracts: Expired St. John's University, and Genady Y. Grabarnik, IBM T. ]. Invited Addresses Watson Research Center. Ailana M. Fraser, University of British Columbia, Title to be announced. San Francisco, Dmitri Nikshych, University of New Hampshire, Algebraic theory of tensor categories. California Florian Pop, University of Pennsylvania, From topological covers to algebraic numbers. San Francisco State University Konstantina Trivisa, University of Maryland, College Park, Title to be announced. April 29-30, 2006 Saturday - Sunday Special Sessions Meeting #1 018 Algebraic Groups, George J. McNinch, Tufts University, and Western Section Eric Sommers, University of Massachusetts, Amherst. Associate secretary: Michel L. Lapidus Arithmetic Geometry and Modular Forms, Paul E. Gun­ Announcement issue of Notices: February 2006 nells and Farshid Hajir, University of Massachusetts, Program first available on AMS website: March 16, 2006 Amherst. Program issue of electronic Notices: April 2006 Arrangements and Configuration Spaces, Graham C. Den­ Issue of Abstracts: Volume 27, Issue 2 ham, University of Western Ontario, and Alexander I. Suciu, Northeastern University. Deadlines Banach Lattices, Regular Operators, and Applications, A. For organizers: Expired K. Kitover, Community College of Philadelphia, M. Orhon, For consideration of contributed papers in Special Sessions: University of New Hampshire, and A. W. Wickstead, Expired Queen's University of Belfast. For abstracts: Expired

APRIL 2006 NOTICES OF THE AMS 511 Meetings & Conferences

Invited Addresses Institute of Technology & University of California Los An­ geles, and Balint Virag, University of Toronto. Lincoln Chayes, University of California Los Angeles, Title to be announced. Q-series and Partitions, Neville Robbins, San Francisco State University. C. Robin Graham, University of Washington, Ambient met­ rics, jet isomorphism and parabolic invariant theory in conformal geometry. Vadim Kaloshin, California Institute of Technology, Non­ Salt Lake City, Utah local instabilities of the planar three body problem. University of Utah Benoit B. Mandelbrot, Yale University, The nature of rough­ ness in mathematics, science, and art (Einstein Public Lec­ October 7-8,2006 ture in Mathematics). Saturday - Sunday Yuval Peres, University of California Berkeley, Hex, Meeting#l019 random-turn games, and the infinity Laplacian. Western Section Special Sessions Associate secretary: Michel L. Lapidus Announcement issue of Notices: August 2006 Computational Arithmetic Geometry, Kenneth A. Ribet, Uni­ Program first available on AMS website: August 24, 2006 versity of California Berkeley, and Kristin Estrella Lauter, Program issue of electronic Notices: October 2006 Microsoft Corporation. Issue of Abstracts: Volume 27, Issue 3 Elliptic Methods in Geometry, C. Robin Graham, University of Washington, and Rafe Mazzeo, Stanford University. Deadlines Enumerative Aspects of Polytopes, Federico Ardila and For organizers: Expired Matthias Beck, San Francisco State University. For consideration of contributed papers in Special Sessions: Fractal Geometry: Connections to Dynamics, Geometric June 20, 2006 Measure Theory, Mathematical Physics and Number The­ For abstracts: August 15, 2006 ory, MichelL. Lapidus and Erin P. Pearse, University of Invited Addresses California Riverside, and Machiel van Frankenhuijsen, Utah Valley State College. William Arveson, University of California Berkeley, Title Geometric Dynamics and Ergodic Theory, Yitwah Cheung to be announced. and Arek Goetz, San Francisco State University, and Slo­ Alexei Borodin, California Institute of Technology, Title bodan Simic, San Jose State University. to be announced. Geometry of Grabner Bases, Bernd Sturmfels, University Izabella Joanna Laba, University of British Columbia, Title of California Berkeley, and Alexander Yong, University of to be announced. Minnesota and Fields Institute. Darren Long, University of California Santa Barbara, Title Hilbert Functions and Resolutions, Benjamin Richert, Cal­ to be announced. ifornia Polytechnic State University, and Sean Sather­ Wagstaff, California State University, Dominguez Hills. Special Sessions History and Philosophy of Mathematics, Shawnee L. Mc­ Commutative Algebra (Code: SS 3A), Paul Roberts, Anurag Murran, California State University, San Bernardino, and K. Singh, and Oana Veliche, University of Utah. James J. Tattersall, Providence College. Harmonic Analysis: Trends and Perspectives (Code: SS 1A), Homological and K-theoretical Trends in Algebraic Com­ Alex Iosevich, University of Missouri, and Michael T. binatorics, Joseph Gubeladze and Serkan Hosten, San Lacey, Georgia Institute of Technology. Francisco State University. Interface of Stochastic Partial Differential Equations and Liapunov Exponents and Nonuniform Hyperbolicity, Anton Gaussian Analysis (Code: SS 7A), Davar Khoshnevisan, Uni­ Gorodetski and Vadim Kaloshin, California Institute of versity of Utah, and Eulalia Nualart, University of Paris XIII. Technology. Low Dimensional Topology and Geometry (Code: SS 4A), Lie Algebras and Applications, Dimitar Grantcharov, San Mladen Bestvina and Kenneth W. Bromberg, University Jose State University, Vera Serganova, University of Cali­ of Utah. fornia Berkeley, and Arturo Pianzola, University of Al­ Mathematics Motivated by Physics (Code: SS SA), Aaron J. berta. Bertram, Yuan-Pin Lee, and Eric R. Sharpe, University of Partial Differential Equations and Their Applications, Steve Utah. Shkoller, University of California Davis. Nonlinear Differential Equations: Methods and Applica­ Probability and Statistical Physics, Marek Biskup, Univer­ tions (Code: SS 2A), David G. Costa, University of Nevada, sity of California Los Angeles, Noam Berger, California and Zhi-Qiang Wang.

512 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Meetings & Conferences

Theory and Applications of Infinite Dimensional Dynami­ cal Systems (Code: SS 6A), Peter W. Bates, Michigan State Storrs, Connecticut University, and Kening Lu, Brigham Young University. University of Connecticut

October 28-29, 2006 Cincinnati, Ohio Saturday - Sunday University of Cincinnati Meeting #1 021 October 21-22, 2006 Eastern Section Saturday - Sunday Associate secretary: Lesley M. Sibner Announcement issue of Notices: August 2006 Meeting #1 020 Program first available on AMS website: September 14, Central Section 2006 Program issue of electronic Notices: October 2006 Associate secretary: Susan ]. Friedlander Issue of Abstracts: Volume 27, Issue 4 Announcement issue of Notices: August 2006 Program first available on AMS website: September 7, 2006 Deadlines Program issue of electronic Notices: October 2006 For organizers: March 28, 2006 Issue of Abstracts: Volume 27, Issue 3 For consideration of contributed papers in Special Sessions: July 11, 2006 Deadlines For abstracts: September 6, 2006 For organizers: March 21, 2006 For consideration of contributed papers in Special Sessions: Invited Addresses JulyS, 2006 Changfeng Gui, University of Connecticut, Storrs, Title to For abstracts: August 29, 2006 be announced. Invited Addresses Niranjan Ramachandran, University of Maryland, College Park, Title to be announced. Suncica Canic, University of Houston, Title to be an­ nounced. Kannan Soundararajan, University of Michigan, Title to be announced. Bryna R. Kra, Northwestern University, Title to be an­ Katrin Wehrheim, Institute for Advanced Study, Title to nounced. be announced. Ezra N. Miller, University of Minnesota, Title to be an­ nounced. Special Sessions Jon G. Wolfson, Michigan State University, Title to be an­ Analysis and Probability on Fractals (Code: SS 3A), Robert nounced. S. Strichartz, Cornell University, and Alexander Teplyaev, University of Connecticut, Storrs. Special Sessions Combinatorial Methods in Equivariant Topology (Code: SS Analysis and Potential Theory on Metric Spaces (Code: SS 1A), Tara Holm, University of Connecticut, Storrs, and 4A), Thomas Bieske, University of South Florida, and Zair Tom C. Braden, University of Massachusetts, Amherst. Ibragimov and Nageswari Shanmugalingam, University Computability Theory in Honor of Manuel Lerman's Re­ of Cincinnati. tirement (Code: SS 4A), Joseph S. Miller and David Reed Applied Algebraic Geometry and Cryptography (Code: SS Solomon, University of Connecticut, Storrs. 3A), Jintai Ding, Jason Eric Gower, and Timothy J. Hodges, Nonlinear Elliptic and Parabolic Equations (Code: SS SA), University of Cincinnati, Lei Hu, Chinese Academy of Sci­ Yung-Sze Choi, Changfeng Gui, and Joseph McKenna, ences, and Dieter S. Schmidt, University of Cincinnati. University of Connecticut, Storrs. Birational Geometry (Code: SS 2A), Mirel Constantin Caibar Number Theory (Code: SS 2A), Keith Conrad, University and Gary P. Kennedy, Ohio State University. of Connecticut, Storrs, David Pollack, Wesleyan University, and Thomas A. Weston, University of Massachusetts, Ergodic Theory(Code: SS 1A), Nikos Frantzikinakis, Penn­ Amherst. sylvania State University, Bryna R. Kra, Northwestern Uni­ versity, and Mate Wierdl, University of Memphis. Nonlinear Functional Analysis and Applications (Code: SS SA), S. P. Singh and Bruce Watson, Memorial University of Newfoundland.

APRIL 2006 NOTICES OF THE AMS 513 Meetings & Conferences

Meeting #1 023 Fayetteville, ]oint Mathematics Meetings, including the 113th Annual Meeting of the AMS, 90th Annual Meeting of the Mathe­ Arkansas matical Association ofAmerica (MAA), annual meetings of the Association for Women in Mathematics (A HIM) and the University of Arkansas National Association of Mathematicians (NAM), and the winter meeting of the Association for Symbolic Logic (ASL), November 3-4, 2006 with sessions contributed by the Society for Industrial and Friday - Saturday Applied Mathematics (SIAM). Associate secretary: Susan ]. Friedlander Meeting #1 022 Announcement issue of Notices: October 2006 Southeastern Section Program first available on AMS website: November 1, 2006 Associate secretary: Matthew Miller Program issue of electronic Notices: January 2007 Announcement issue of Notices: September 2006 Issue of Abstracts: Volume 28, Issue 1 Program first available on AMS website: September 21, 2006 Deadlines Program issue of electronic Notices: November 2006 For organizers: April1, 2006 Issue of Abstracts: Volume 27, Issue 4 For consideration of contributed papers in Special Sessions: August 1, 2006 Deadlines For abstracts: September 26, 2006 For organizers: April 3, 2006 For consideration of contributed papers in Special Sessions: July 18, 2006 Davidson, North For abstracts: September 12, 2006

Invited Addresses Carolina Richard P. Anstee, University of British Columbia, Title to Davidson College be announced. March 3-4, 2007 Arun Ram, University of Wisconsin, Title to be announced. Saturday - Sunday Donald G. Saari, University of California Irvine, Title to be announced. Meeting #1 024 Andras Vasy, Massachusetts Institute of Technology, Title Southeastern Section to be announced. Associate secretary: Matthew Miller Announcement issue of Notices: To be announced Special Sessions Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Analytic Number Theory and Modular Forms (Code: SS Issue of Abstracts: To be announced 2A), Matthew Boylan, University of Illinois, Urbana­ Champaign, and Gang Yu, University of South Carolina. Deadlines Boundary Operators in Real and Complex Domains (Code: For organizers: August 3, 2006 SS 3A), Loredana Lanzani, University of Arkansas, Fayet­ For consideration of contributed papers in Special Sessions: teville, and David E. Barrett, University of Michigan, Ann To be announced Arbor. For abstracts: To be announced Dirac Operators in Analysis and Geometry (Code: SS 1A), John Ryan, University of Arkansas, Marius Mitrea, Uni­ versity of Missouri, and Mircea Martin, Baker University. Oxford, Ohio Miami University New Orleans, March 16-1 7, 2007 Louisiana Friday - Saturday Meeting #1 025 New Orleans Marriott and Sheraton New Central Section Orleans Hotel Associate secretary: Susan]. Friedlander Announcement issue of Notices: To be announced January 4-7,2007 Program first available on AMS website: To be announced Thursday - Sunday Program issue of electronic Notices: To be announced

514 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Meetings & Conferences

Issue of Abstracts: To be announced For consideration of contributed papers in Special Sessions: To be announced Deadlines For abstracts: To be announced For organizers: To be announced For consideration of contributed papers in Special Sessions: To be announced Warsaw, Poland For abstracts: To be announced University of Warsaw Special Sessions Finite Geometry and Combinatorics (Code: SS 3A), Mark A. July 31 -August 3, 2007 Miller, Marietta College. Tuesday - Friday First ]oint International Meeting between the AMS and the Geometric Topology (Code: SS 2A), Jean-Francois LaFont, Polish Mathematical Society SUNY Binghamton and Ohio State University, and Ivonne Associate secretary: Susan ]. Friedlander J. Ortiz, Miami University. Announcement issue of Notices: To be announced Large Cardinals in Set Theory (Code: SS 1A), Paul B. Lar­ Program first available on AMS website: To be announced son, Miami University, Justin Tatch Moore, Boise State uni­ Program issue of electronic Notices: To be announced versity, and Ernest Schimrnerling, Carnegie Mellon uni­ Issue of Abstracts: To be announced versity. Deadlines For organizers: To be announced Hoboken, New Jersey For consideration of contributed papers in Special Sessions: To be announced Stevens Institute of Technology For abstracts: To be announced April14- 1 5, 2007 Saturday - Sunday Albuquerque, New Meeting # 1 026 Eastern Section Mexico Associate secretary: Lesley M. Sibner Announcement issue of Notices: To be announced University of New Mexico Program first available on AMS website: To be announced October 1 3-14, 2007 Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Saturday - Sunday Western Section Deadlines Associate secretary: Michel L. Lapidus For organizers: September 14, 2006 Announcement issue of Notices: To be announced For consideration of contributed papers in Special Sessions: Program first available on AMS website: To be announced To be announced Program issue of electronic Notices: To be announced For abstracts: To be announced Issue of Abstracts: To be announced Deadlines For organizers: To be announced Tucson, Arizona For consideration of contributed papers in Special Sessions: University of Arizona To be announced For abstracts: To be announced Apri I 21 - 22, 2007 Saturday - Sunday Murfreesboro, Meeting # 1 027 Western Section Tennessee Associate secretary: Michel L. Lapidus Announcement issue of Notices: To be announced Middle Tennessee State University Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced November 3- 4,2007 Issue of Abstracts: To be announced Saturday - Sunday Southeastern Section Deadlines Associate secretary: Matthew Miller For organizers: September 21, 2006 Announcement issue of Notices: To be announced

APRIL 2006 NOTICES OF THE AMS 515 Meetings & Conferences

Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Baton Rouge, Issue of Abstracts: To be announced Deadlines Louisiana For organizers: April4, 2007 Louisiana State University, Baton Rouge For consideration of contributed papers in Special Sessions: To be announced March 28-30, 2008 For abstracts: To be announced Friday- Sunday Southeastern Section Associate secretary: Matthew Miller San Diego, California Announcement issue of Notices: To be announced San Diego Convention Center Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced January 6-9,2008 Issue of Abstracts: To be announced Sunday - Wednesday Deadlines ]oint Mathematics Meetings, including the 114th Annual Meeting of the AMS, 91st Annual Meeting of the Mathe­ For organizers: August 28, 2007 matical Association ofAmerica (MAA), annual meetings of For consideration of contributed papers in Special Sessions: the Association for Women in Mathematics (A V'VM) and the To be announced National Association of Mathematicians (NAM), and the For abstracts: To be announced winter meeting of the Association for Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Applied Mathematics (SIAM). Bloomington, Indiana Associate secretary: Michel L. Lapidus Announcement issue of Notices: October 2007 Indiana University Program first available on AMS website: November 1, 2007 April4-6, 2008 Program issue of electronic Notices: January 2008 Issue of Abstracts: Volume 29, Issue 1 Friday - Sunday Central Section Deadlines Associate secretary: Susan]. Friedlander For organizers: April1, 2007 Announcement issue of Notices: To be announced For consideration of contributed papers in Special Sessions: Program first available on AMS website: To be announced To be announced Program issue of electronic Notices: To be announced For abstracts: To be announced Issue of Abstracts: To be announced

Deadlines New York, New York For organizers: September 4, 2007 For consideration of contributed papers in Special Sessions: Courant Institute of New York University To be announced For abstracts: To be announced March 22-23,2008 Saturday - Sunday Eastern Section Associate secretary: Leslie M. Sibner Claremont, California Announcement issue of Notices: To be announced Claremont McKenna College Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced May 3-4, 2008 Issue of Abstracts: To be announced Saturday - Sunday Southeastern Section Deadlines Associate secretary: Michel L. Lapidus For organizers: August 22, 2007 Announcement issue of Notices: To be announced For consideration of contributed papers in Special Sessions: Program first available on AMS website: To be announced To be announced Program issue of electronic Notices: To be announced For abstracts: To be announced Issue of Abstracts: To be announced

Deadlines For organizers: October 4, 2007

516 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Meetings & Conferences

For consideration of contributed papers in Special Sessions: Deadlines To be announced For organizers: March 24, 2008 For abstracts: To be announced For consideration of contributed papers in Special Sessions: To be announced Rio de Janeiro, Brazil For abstracts: To be announced Instituto Nacional de Matematica Pura e Aplicada (IMPA) Shanghai, People's June 4-7,2008 Republic of China Wednesday - Saturday Fudan University First ]oint International Meeting between the AMS and the Sociedade Brasileira de Matematica. December 17- 21,2008 Associate secretary: Lesley M. Sibner Wednesday - Sunday Announcement issue of Notices: To be announced First ]oint Interntional Meeting Between the AMS and the Program first available on AMS website: To be announced Shanghai Mathematical Society Program issue of electronic Notices: To be announced Associate secretary: Susan ]. Friedlander Issue of Abstracts: To be announced Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Deadlines Program issue of electronic Notices: To be announced For organizers: To be announced Issue of Abstracts: To be announced For consideration of contributed papers in Special Sessions: To be announced Deadlines For abstracts: To be announced For organizers: To be announced For consideration of contributed papers in Special Sessions: To be announced Vancouver, Canada For abstracts: To be announced University of British Columbia October 4- 5, 2008 Washington, District Saturday - Sunday Western Section of Columbia Associate secretary: Michel L. Lapidus Announcement issue of Notices: To be announced Marriott Wardman Park Hotel and Omni Program first available on AMS website: To be announced Shoreham Hotel Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced January 7- 10,2009 Wednesday - Saturday Deadlines ]oint Mathematics Meetings, including the 115th Annual For organizers: March 9, 2008 Meeting of the AMS, 92nd Annual Meeting of the Mathe­ For consideration of contributed papers in Special Sessions: matical Association of America (MAA), annual meetings of To be announced the Associati.on for Women in Mathematics (A VllM) and the For abstracts: To be announced National Association of Mathematicians (NAM), and the winter meeting of the Association for Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Huntsville, Alabama Applied Mathematics (SIAM). Associate secretary: Lesley M. Sibner University of Alabama, Huntsville Announcement issue of Notices: October 2008 Program first available on AMS website: November 1, 2008 October 24- 26, 2008 Program issue of electronic Notices: January 2009 Friday - Sunday Issue of Abstracts: Volume 30, Issue 1 Southeastern Section Associate secretary: Matthew Miller Deadlines Announcement issue of Notices: To be announced For organizers: April1, 2008 Program first available on AMS website: To be announced For consideration of contributed papers in Special Sessions: Program issue of electronic Notices: To be announced To be announced Issue of Abstracts: To be announced For abstracts: To be announced

APRIL 2006 NOTICES OF THE AMS 517 Meetings & Conferences San Francisco, Boston, California Massachusetts Moscone Center West and the San Fran­ john B. Hynes Veterans Memorial Conven­ cisco Marriott tion Center, Boston Marriott Hotel, and January 6-9,2010 Boston Sheraton Hotel Wednesday - Saturday January 4-7, 2012 ]oint Mathematics Meetings, including the 116th Annual Wednesday - Saturday Meeting of the AMS, 93rd Annual Meeting of the Mathe­ ]oint Mathematics Meetings, including the 118th Annual matical Association ofAmerica (MAA), annual meetings of Meeting of the AMS, 95th Annual Meeting of the Mathe­ the Association for Women in Mathematics (A WM) and the matical Association ofAmerica, annual meetings of the As­ National Association of Mathematicians (NAM), and the sociation for Women in Mathematics (A WM) and the National winter meeting of the Association for Symbolic Logic (ASL), Association of Mathematicians (NAM), and the winter meet­ with sessions contributed by the Society of Industrial and ing of the Association for Symbolic Logic (ASL), with sessions Applied Mathematics (SIAM). contributed by the Society for Industrial and Applied Math­ Associate secretary: Matthew Miller ematics (SIAM). Announcement issue of Notices: October 2009 Associate secretary: Michel L. Lapidus Program first available on AMS website: November 1, 2009 Announcement issue of Notices: October 2011 Program issue of electronic Notices: January 2010 Program first available onAMS website: November 1, 2011 Issue of Abstracts: Volume 31, Issue 1 Program issue of electronic Notices: January 2012 Issue of Abstracts: Volume 33, Issue 1 Deadlines For organizers: April1, 2009 Deadlines For consideration of contributed papers in Special Sessions: For organizers: April1, 2011 To be announced For consideration of contributed papers in Special Sessions: For abstracts: To be announced To be announced For abstracts: To be announced New Orleans, San Diego, California Louisiana San Diego Convention Center and San New Orleans Marriott and Sheraton New Diego Marriott Hotel and Marina Orleans Hotel January 9-12,2013 January 5-8,2011 Wednesday - Saturday Wednesday - Saturday ]oint Mathematics Meetings, including the 119th Annual ]oint Mathematics Meetings, including the 117th Annual Meeting of the AMS, 96th Annual meeting of the Mathe­ Meeting of the AMS, 94th Annual Meeting of the Mathe­ matical Association ofAmerica, annual meetings of the As­ matical Association ofAmerica, annual meetings of the As­ sociation for Women in Mathematics (A WM) and the National sociation for Women in Mathematics (A WM) and the National Association of Mathematicians (NAM), and the winter meet­ Association ofMathematicians (NAM), and the winter meet­ ing of the Association for Symbolic Logic (ASL), with sessions ing of the Association for Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Applied Math­ contributed by the Society for Industrial and Applied Math­ ematics (SIAM). ematics (SIAM). Associate secretary: Lesley M. Sibner Associate secretary: Susan J. Friedlander Announcement issue of Notices: To be announced Announcement issue of Notices: October 2010 Program first available on AMS website: To be announced Program first available onAMSwebsite: November 1, 2010 Program issue of electronic Notices: To be announced Program issue of electronic Notices: January 2011 Issue of Abstracts: To be announced Issue of Abstracts: Volume 32, Issue 1 Deadlines Deadlines For organizers: April1, 2012 For organizers: April1, 2010 For consideration of contributed papers in Special Sessions: For consideration of contributed papers in Special Sessions: To be announced To be announced For abstracts: To be announced For abstracts: To be announced

518 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Meetings and Conferences of the AMS

Associate Secretaries ofthe AMS 249), Chicago, IL 60607-7045; e-mail: susan@math. nwu. edu; tele­ Western Section: MichelL. Lapidus, Department of Math­ phone: 312-996-304 1. ematics, University of California, Sproul Hall, Riverside, CA Eastern Section: Lesley M. Sibner, Department of Mathe­ 92521-0135; e-mail: l a pi dus@math. uc r. edu; telephone: 951- matics, Polytechnic University, Brooklyn, NY 11201-2990; 827-5910. e-mail: [email protected] .edu ; telephone: 718-260-3505. Central Section: Susan J. Friedlander, Department of Math­ Southeastern Section: Matthew Miller, Department of Math­ ematics, University of Illinois at Chicago, 851 S. Morgan (MIC ematics, University of South Carolina, Columbia, SC 29208- 0001, e-mail: mill er@math. sc. edu; telephone: 803-777-3690.

The Meetings and Conferences section of the Notices April4-6 Bloomington, Indiana p. 516 gives information on all AMS meetings and conferences May 3-4 Claremont, California p. 516 approved by press time for this issue. Please refer to the page June 4-7 Rio de Janeiro, Brazil p. 517 numbers cited in the table of contents on this page for more October 4-5 Vancouver, Canada p. 517 detailed information on each event. Invited Speakers and October 24-26 Huntsville, Alabama p. 517 Special Sessions are listed as soon as they are approved by December 17-21 Shanghai, People's the cognizant program committee; the codes listed are needed Republic of China p. 517 for electronic abstract submission. For some meetings the list may be incomplete. Information in this issue may be dated. 2009 Up-to-date meeting and conference information can be January 7-10 Washington, DC p. 517 found at www. ams. orglmeeti ngsl. Annual Meeting 2010 Meetings: January 6-9 San Franciso, California p. 518 2006 Annual Meeting April1-2 Miami, Florida p. 509 2011 April8- 9 Notre Dame, Indiana p. 510 January 5-8 New Orleans, Louisiana p. 518 April 22-23 Durham, New Hampshire p. 511 Annual Meeting April29-30 San Francisco, California p. 511 2012 October 7-8 Salt Lake City, Utah p. 512 January 4-7 Boston, Massachusetts p. 518 October 21-22 Cincinnati, Ohio p. 513 Annual Meeting October 28-29 Storrs, Connecticut p. 513 2013 November 3-4 Fayetteville, Arkansas p. 514 January 9-12 San Diego, California p. 518 Annual Meeting 2007 Important Information Regarding AMS Meetings January 4-7 New Orleans, Louisiana p. 514 Potential organizers, speakers, and hosts should refer to Annual Meeting page 296 in the February 2006 issue of the Notices for gen­ March 3-4 Davidson, North Carolina p. 514 eral information regarding participation in AMS meetings and March 16-17 Oxford, Ohio p. 514 conferences. April14-15 Hoboken, New Jersey p. 515 Abstracts April21-22 Tuscan, Arizona p. 515 Speakers should submit abstracts on the easy-to-use interactive July 31- August 3 Warsaw, Poland p. 515 Web form. No knowledge of 0T£X is necessary to submit an October 13-14 Albuquerque, New Mexico p. 515 electronic form, although those who use LAJ"pi: may submit November 3-4 Murfreesboro, Tennessee p. 515 abstracts with such coding, and all math displays and simi­ larily coded material (such as accent marks in text) must 2008 be typeset in 0T£X. Visit http: I jwww. ams. o rg/ cgi -bin/ January 6-9 San Diego, California p. 516 abstracts/abstract. pl. Questions about abstracts andre­ Annual Meeting quests for paper forms may be sent to abs-i nfo@ams. org. Close attention should be paid to specified deadlines in this March 22-23 New York, NY p. 516 issue. Unfortunately, late abstracts cannot be accommodated. March 28-30 Baton Rouge, Louisiana p. 516 Conferences: (see http: 1lwww. ams. orglmeeti ngsl for the most up-to-date information on these conferences.) June 4-June 29, 2006: Joint Summer Research Conferences, Snowbird, Utah (see November 2005 Notices, page 1296). Co-sponsored conferences: 22nd Annual Workshop on Mathematical Problems in Industry, June 12-16, 2006, Olin College, Need­ ham, MA. For details see http: I l proj ects. olin. edulmpi 20061. Recent advances in nonlinear partial differential equations and applications: A conference in honor of Peter D. Lax and Louis Niren­ berg, June 7-10, 2006, Toledo, Spain. For more details see http: I jwww. mat. ucm. esl-l n06/. Poisson 2006: Poisson Geometry in Mathematics and Physics, June 5-9, 2006, Tokyo, Japan. For more details see http:llwww.ams.orglmathcallinfol2006_jun5-9_tokyo . html .

APRIL 2006 NOTICES OF THE AMS 519 CAMBRIDGE I

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