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CALL: 1-800-777-4643 • FAX: (201) 348-4505 • E-MAIL: [email protected] • www.birkhauser.com Please mention promotion #013382 when ordering. Prices are valid in the Americas only and are subject to change without notice. For price and ordering information outside the Americas, please contact Birkhäuser Verlag AG by E-mail: [email protected] 013382x Notices of the American Mathematical Society December 2007 d b a 1 Communications c 1494 WHAT IS...a Cluster 1 Algebra? Andrei Zelevinsky
1500 Emma Lehmer (1906–2007) John Brillhart 1502 Remembering Philip 1477 σ = −1, σ˘ = 0 1458 Rabinowitz Philip J. Davis and Aviezri S. Fraenkel
1507 How to Write Your First Paper Steven G. Krantz Features 1512 Sobolev Institute of Mathematics Celebrates Its 1458 Starting with the Group SL2(R) Fiftieth Anniversary Victor Alexandrov Vladimir V. Kisil The group of two by two real matrices of Commentary determinant one is a well studied elementary math- ematical object. The author considers some of its less 1454 Letters to the Editor well known invariants and the geometries they imply. 1468 The Triumph of Numbers and Karl Pearson—Book Reviews 1477 Irving Kaplansky (1917–2006) Reviewed by Brian Blank Hyman Bass and T. Y. Lam 1497 How Mathematicians Colleagues, students, and family recall the life and work Think—A Book Review of the late American mathematician. Reviewed by Reuben Hersh Notices Departments of the American Mathematical Society About the Cover...... 1516 Mathematics People ...... 1517 EDITOR: Andy Magid Kreps Awarded 2007 CME/MSRI Prize, Singapore National Science and ASSOCIATE EDITORS: Technology Awards Given, NDSEG Fellowships Awarded, Daniel Biss, Susanne C. Brenner, Bill Casselman (Graphics Editor), Robert J. Daverman, Susan B. H. Neumann Award Given, Pi Mu Epsilon Student Paper Presentation Friedlander, Robion Kirby, Steven G. Krantz, Awards. Lisette de Pillis, Peter Sarnak, Mark Saul, John Swallow, Lisa Traynor Mathematics Opportunities...... 1519 SENIOR WRITER and DEPUTY EDITOR: NSF Computing Equipment and Instrumentation Programs, NRC- Allyn Jackson Ford Foundation Diversity Fellowships, NDSEG Fellowships, National MANAGING EDITOR: Sandra Frost Academies Graduate Fellowship Program, Call for Nominations for CONTRIBUTING WRITER: Elaine Kehoe Waterman Award, MfA Fellowship Program, CMI Liftoff Program for PRODUCTION ASSISTANT: Muriel Toupin Summer 2008, Call for Applications for Kovalevskaja Award, Virtual PRODUCTION: Kyle Antonevich, Stephen Moye, Erin Exhibit at the Smithsonian, ETS Visiting Scholars Program. Murphy, Lori Nero, Karen Ouellette, Donna Salter, Deborah Smith, Peter Sykes Inside the AMS...... 1522 ADVERTISING SALES: Anne Newcomb Trjitzinsky Memorial Awards Presented, Epsilon Scholarships Awarded for 2007, Deaths of AMS Members. SUBSCRIPTION INFORMATION: Subscription prices for Volume 54 (2007) are US$447 list; US$358 insti- Reference and Book List...... 1524 tutional member; US$268 individual member. (The subscription price for members is included in the Mathematics Calendar ...... 1530 annual dues.) A late charge of 10% of the subscrip- tion price will be imposed upon orders received from nonmembers after January 1 of the subscription year. New Publications Offered by the AMS...... 1537 Add for postage: Surface delivery outside the United States and India—US$27; in India—US$40; expedited Classified Advertising...... 1544 delivery to destinations in North America—US$35; elsewhere—US$88. Subscriptions and orders for AMS publications should be addressed to the American Meetings and Conferences of the AMS...... 1557 Mathematical Society, P.O. Box 845904, Boston, MA 02284-5904 USA. All orders must be prepaid. Notices 2007 Index ...... 1570 ADVERTISING: Notices publishes situations wanted and classified advertising, and display advertising for San Diego Meeting Registration Forms...... 1589 publishers and academic or scientific organizations. Advertising material or questions may be sent to [email protected] (classified ads) or notices-ads@ Meetings and Conferences Table of Contents...... 1591 ams.org (display ads). SUBMISSIONS: Articles and letters may be sent to the editor by email at [email protected], 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 [email protected]. For more information, see the section “Reference and Book List”. NOTICES ON THE AMS WEBSITE: Most of this publica- tion is available electronically through the AMS website, the Society’s resource for delivering electronic prod- ucts and services. Use the URL http://www.ams. org/notices/ to access the Notices on the website.
[Notices of the American Mathematical Society (ISSN 0002- 9920) is published monthly except bimonthly in June/July by the American Mathematical Society at 201 Charles Street, Providence, RI 02904-2294 USA, GST No. 12189 2046 RT****. Periodicals postage paid at Providence, RI, and additional mailing offices. POSTMASTER: Send address change notices to Notices of the American Mathematical Society, P.O. Box 6248, Providence, RI 02940-6248 USA.] Publication here of the Society’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: [email protected]. © Copyright 2007 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. The AMS Epsilon Fund for Young Scholars
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Whether they become Please give generously. scientists, engineers, or Learn about giving opportunities and estate planning entrepreneurs, young people www.ams.org/giving-to-ams with mathematical talent need to be nurtured. Income from this Contact the AMS Development Office fund supports the Young Scholars 1.800.321.4267 (U.S. and Canada) or Program, which provides grants to 1.401.455.4000 summer programs for talented high (worldwide) school students. email: [email protected] 09/04 Letters to the Editor
Koblitz Article Misleading the paper due to my strong disagree- reality, the precise statement of a I found Koblitz’s essay “The uneasy ment with its contents, but rather due theorem is crucial to its practical relationship between mathematics to the nature of this paper which, in meaning. Indeed, while we all know and cryptography” (Notices, Vol. 54, my opinion, is not a novel technical that the impossibility of angle trisec- No. 8) misleading in several ways. contribution of the type sought by tion depends on the precise defini- Most importantly, I believe that the journal. My opinion was that the tion of allowed operations, none of us Koblitz’s views regarding the subject paper may only be published as a relies on this theorem to protect our are based on several fundamental “position paper”. Since the authors credit card information. Here indeed misconceptions. For example, he refused to revise the title of their cryptographers have sometimes mis- seems to view the unfortunate (and paper accordingly, the editor-in-chief stepped and inadequately modeled rare) cases in which flaws were found was forced to write a special preface the scenarios in which systems could in published claimed “proofs” (of se- that explains that their paper is a be attacked, leading to systems that curity) as indication that proofs are position paper. regardless of their formal analysis useless (w.r.t. security). In my opin- were insecure in practice. But the ion, these incidences merely reinforce —Oded Goldreich problem is not inherently with proofs the importance of careful verification Weizmann Institute of Science of security but rather with cryptogra- of proofs, which constitute our only [email protected] phy itself, a notoriously difficult sub- way of distinguishing facts from con- ject which over its long history has jectures. Furthermore, Koblitz often (Received August 27, 2007) seen many great minds miss subtle confuses proofs with what is being points and design systems that were proved, and consequently does not eventually broken. distinguish between the inadequacy Koblitz Misrepresents In fact, the only way to systemati- of the claim (e.g., an unsatisfactory Cryptography cally improve practical security is to definition of security) and the in- In the famous joke, a mathematician insist on precise modeling, and study correctness of its proof. Finally, he would not infer the color of a sheep’s these models using mathematical often uses unsound reasoning (e.g., right side from its left side. But Neal proofs, on the way refining the mod- inferring that last-minute conference Koblitz, in his article on “The uneasy els and identifying and correcting submissions indicate a rush to pub- relationship between mathematics subtle weaknesses in protocols. In- lish minor results). and cryptography” makes quite a few deed, Koblitz’s anecdote on the MQV The foregoing flaws dominate broad generalizations from a handful and HMQV protocols demonstrates the series of papers by Koblitz and of anecdotes. precisely how careful definitions and Menezes (see references in Koblitz’s Koblitz’s disparagement of secu- insistence on proofs can direct an essay). For a discussion of the main rity proofs is particularly misleading. incremental process towards more flaws, the interested reader is re- Proofs of security of cryptographic secure protocols. ferred to my essay http://eprint. protocols are standard mathemati- iacr.org/2006/461. Let me just cal proofs and in that sense are no —Boaz Barak stress that, in contrary to Koblitz’s more “over-hyped” (to use Koblitz’s Princeton University belief, the fact that this essay does term) than proofs in calculus. Ko- [email protected] not criticize the papers of Koblitz blitz gives examples of mistakes in and Menezes for inadequate refer- security proofs, but as we know such (Received August 30, 2007) ences to prior work does not mean examples can be found in any area of that such cases are not numerous. mathematics. He also criticizes these On the contrary. Koblitz’s essay suf- proofs for relying on unproven con- Publication of Koblitz’s Article fers from the same problems, and jectures. This is indeed most often Questioned in addition it provides a distorted the case, as is not surprising in such I was shocked and dismayed that the account of my own essay (e.g., the a young and vibrant field. Eventually AMS Notices published Neal Koblitz’s (legitimate) controversy regarding we might prove these conjectures article [“The uneasy relationship be- the “Random Oracle Model” is far (although some seem as hard as the tween mathematics and cryptog- from being the focus of my essay and hardest open problems in mathemat- raphy”, September 2007] without, was certainly not the source of my ics) but regardless, it’s much better to apparently, any editorial oversight. concerns regarding the Koblitz and use a protocol proven secure under As one who works in the field of Menezes papers). a well-defined and widely believed “provable security”, I vehemently I also wish to correct Koblitz’s conjecture than a protocol with no disagree with Koblitz’s main argu- account of the events related to the analysis at all. ment—more on this below—but this publication of his paper with Mene- Koblitz points out the obvious is not my primary complaint. Instead, zes in the Journal of Cryptography. I truth that in cryptography, as in what I found abhorrent is that the did not object to the publication of any mathematical field that models article crosses the line from academic
1454 Notices of the AMS Volume 54, Number 11 Letters to the Editor argument to personal screed, from of publishing “contributions” of this driven design” methodology is a tes- constructive criticism to belliger- nature. tament to the fundamental role of the ent name-calling. I cannot imagine theory of cryptography in bringing the Notices publishing a similarly —Jonathan Katz more secure systems to practice. disparaging article about any other University of Maryland There is no better way to as- academic discipline. [email protected] sess the value of the HMQV pro- By another fault of the editors, tocol than reading the paper itself readers were not given the oppor- (Received August 30, 2007 posted under http://eprint.iacr. tunity to read a companion article org/2005/176. In particular, the in- containing a countervailing point of Koblitz’s Arguments troduction and concluding remarks view. Without dissecting Koblitz’s Disingenuous section in the paper, unchanged since arguments point-by-point, let me Addressing Neal Koblitz’s disingenu- the original publication, already con- assure readers that proofs in mod- ous arguments against theoretical ern cryptography are as meaningful cryptography in his recent article in tain answers to many of the points as proofs in any other field. Can a the Notices requires far more elabora- raised by Koblitz against our meth- scheme that has been proven secure tion than allowed by the space allo- odology. Also note the preface where still succumb to a real-world attack? cated for this letter (see http://www. I comment on a correction pointed Yes, but this does not invalidate the ee.technion.ac.il/~hugo/ams- out by Alfred Menezes that, contrary proof. (A proof is given with respect letter). Let me thus focus only on to Koblitz’s misleading account, did to a particular definition; any single some of Koblitz’s unfounded claims not change in any essential way the definition is not appropriate for all against my work on the HMQV proto- results and value of the work, neither possible environments in which a col that he uses as a way to discredit with respect to its provability nor scheme may be deployed.) Are most the entire field of complexity-based the substantial practical benefits of results in cryptography conditional? cryptography (what he refers to as HMQV. Yes, but this has been shown to be “provable security”) and to deny the Let me end by stressing a very inherent until the P vs. NP question significant achievements of this field, important point in understanding the is settled, and should not hold back in particular its important contribu- role of theory when designing and research. Do mistakes happen? Oc- tions to the practice of cryptogra- analyzing real-world cryptographic casionally, though rarely. But this phy. systems: By its very nature, there surely does not diminish the impor- Contrary to what Koblitz claims, is no (and cannot be) empirical evi- tance of proofs in the first place. the HMQV work represents a prime dence for the security of a design. Frankly, I cannot understand why example of the success of theoreti- any mathematician would discour- cal cryptography, not only in laying Indeed, no concrete measurements age the use of definitions, proofs, rigorous mathematical foundations or simulations can show that attacks and formal reasoning in any field. for cryptography at large, but also in against a cryptographic scheme are (Indeed, these elements have helped its ability to guide us in the design not feasible. The only way to do so cryptography progress from an art of truly practical solutions to real- is to develop a formal mathemati- to a science.) Koblitz’s article clari- world problems. Indeed, the HMQV cal model and language in which fies his motivation: sheer elitism. key-agreement protocol that resulted to reason about such schemes. The According to Koblitz, cryptographers from this work not only improved area of theoretical cryptography and publish papers of “little originality” significantly on its predecessor, the its applications has been remarkably and containing “tiny improvements”; MQV protocol, in terms of analysis successful in developing such mod- when we do publish something of and security guarantees, but the pro- els. They are certainly not perfect potential interest, it is likely to be tocol itself became more practical, and will be further improved over wrong. According to Koblitz, cryp- improving performance and lowering time, but the foundations laid so tographers are simply incapable of the dependency on external mecha- far are outstanding. Whoever finds writing correct proofs, hence his nisms such as trust in certification them insufficient should be encour- admonition that anyone other than authorities and key derivation func- aged to improve upon them or come “trained mathematicians” simply tions. up with alternatives. Emotional and give up on the goal. This is snobbery This double improvement, in both unfounded attacks against a whole at its purest. security and performance, is no coin- research area and its individuals, as Publication of Koblitz’s article cidence. It is the very understanding has the potential to cause serious that one obtains through the process carried by Koblitz, are of no use. damage: not to the field of cryptog- of formally proving (or disproving) a raphy—which will continue to do cryptographic protocol that allows —Hugo Krawczyk fine with or without Koblitz’s sup- us to eliminate safety margins that IBM T. J. Watson Research Center port—but to the future involvement are often added to cryptographic [email protected] of mathematicians in this field. In schemes when there is not enough the future, the editors should more confidence in the strength of the (Received September 3, 2007) carefully weight the pros and cons design. The success of this “proof-
December 2007 Notices of the AMS 1455 Letters to the Editor ASSISTANT, ASSOCIATE OR FULL PROFESSOR (2 POSITIONS) CENTER FOR ALGORITHMS AND Reply to Katz, Goldreich, and HMQV protocol had been deployed INTERACTIVE SCIENTIFIC SOFTWARE (CAISS) Krawczyk in its original form as published, not only would the advertised “prov- FLSA Status: Exempt Jonathan Katz misstates what I wrote in my article and attributes to me able security” guarantee have been Compensation: Assistant false, but in certain settings HMQV Professor: $38,801 - $67,092, things I never said, all to justify accus- Associate Professor: $50,321 - ing me of “sheer elitism” and “snob- could have been breached by a mali- $80,020, Professor: $62,268 - bery at its purest”. I never objected cious adversary. That’s not a minor $95,997 (Salaries effective matter. (See http://eprint.iacr. 9/19/2007) to cryptographers making a carefully reasoned, rigorous argument in sup- org/2005/205 for detailed explana- College Web Site: www.ccny.cuny.edu tions of the security flaws that have Notice Number: FY13784 port of a claim. Indeed, in my papers with Menezes on “provable security” been found in HMQV.) Indeed, if Closing Date: Open until filled. Krawczyk believes that fallacies in POSITION DESCRIPTION AND DUTIES we give detailed explanations of the need for precision in definitions and proofs are so unimportant, then why The City College of the City University of New York as bother to give proofs at all? part of its continuing efforts to achieve the highest security analysis, and we describe
quality in teaching and research is making available some of the best examples of early —Neal Koblitz two new tenure-track positions in one or other of and more recent research along these the Department of Mathematics or Computer University of Washington lines. In my article what I took issue Science, or the School of Education. These [email protected] positions will be closely associated with CAISS, the with was all the hype, misleading ter-
Center for Algorithms and Interactive Scientific minology, and easily misunderstood (Received September 14, 2007) Software. CAISS has embarked on a number of and misinterpreted “theorems” that projects involving symbolic computation (see our website: www.caissny.org and the associated site one finds in much of the “provable se- www.grouptheory.org). The objective is to build curity” literature. It is hard to escape on past efforts culminating in an ongoing, the impression that mathematical innovative, high-risk, creative, research group with jargon and the theorem-proof para- wide-ranging interests. It is expected that successful applicants will have digm are often used to kick dust in almost limitless creative license with the the eyes of outsiders. expectation that funding from a number of sources, It is Oded Goldreich, not me, who both government and commercial, will flow gives a misleading version of the from these efforts. These positions promise to be extremely exciting opportunities for the events surrounding his last-minute right candidates. effort to prevent publication of my Collaboration with the Computer Science and article with Menezes in the Jour- Mathematics Departments and the School of nal of Cryptology. Our paper had Education, and working on some research with one gone through the refereeing process or two undergraduates will be expected. almost two years before, and had Teaching responsibilities will be reduced for the first two years (and possibly the third year) to one been judged to be of sufficient tech- course per semester. nical novelty to merit acceptance. QUALIFICATION REQUIREMENTS Goldreich’s essay “On post-modern A PhD in Mathematics, Computer Science or a cryptography” finds fault with our closely related field is required. Rank and salary to article not on technical, but rather be determined depending on the experience of on philosophical grounds. Calling the applicant. Menezes and me “post-modern [and] Energy, a high degree of programming skills, interest in open source and zero learning curve reactionary”, he is incensed by some software, a deep knowledge of the Linux and of our conclusions—notably, that Microsoft operating systems, evidence of an ability “our confidence in the random oracle to build and design software, new learning tools assumption is unshaken” and that and/or academic accomplishments and a willingness to go out on a limb and work very hard cryptography “is as much an art as are a must. a science”. Whatever Goldreich’s rea- TO APPLY sons might have been for attempting Applicants should send a curriculum vitae and at to block our article on the eve of its least 3 letters of recommendation to: Mary Tramel, publication, in the scientific world Mathematics Dept, CAISS NAC 8/133, City College of New York, 160 Convent Ave, New York, such conduct by an editorial board NY 10031, E-mail: [email protected]. member is irregular and improper. Hugo Krawczyk’s letter itself is The City University of New York is an Equal Employment Opportunity/Affirmative an illustration of what I find so ex- Action/Immigration Reform and Control Act/ Americans with asperating in the “provable security” Disabilities Act Employer field. In order to advertise his work as “a prime example of the success of theoretical cryptography,” Kraw- czyk minimizes the fact that his published proof was fallacious. If the
1456 Notices of the AMS Volume 54, Number 11 The American Mathematical Mathematics Society introduces Research Communities
The AMS invites mathematicians just beginning their research careers to become part of Mathematics Research Communities, a new program to develop and sustain long-lasting cohorts for collaborative research projects in many areas of mathematics. Qualified women and underrepresented minorities are especially encouraged to participate. The AMS will provide a structured program to engage and guide all participants as they start their careers. The program will include:
• One-week summer conferences for each topic • Special Sessions at the national meeting • Discussion networks by research topic • Ongoing mentoring • Longitudinal study of early career mathematicians
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2008 Topics:
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• Computational Algebra & Convexity Organizers: Henry Schenck (UIUC), Michael Stillman (Cornell), Jan Verschelde (UIC)
www.ams.org/amsmtgs/mrc.html Starting with the Group SL2(R) Vladimir V. Kisil
he simplest objects with noncommuta- only due to the big numerator but also due to the tive multiplication may be 2×2 matrices undeservedly small denominator. As we shall see with real entries. Such matrices of de- below Klein’s approach provides some surprising terminant one form a closed set under conclusions even for such over-studied objects as multiplication (since det(AB) = det A · circles. Tdet B), the identity matrix is among them, and any such matrix has an inverse (since det A ≠ 0). In other words those matrices form a group, the Make a Guess in Three Attempts It is easy to see that the SL (R) action (1) makes SL2(R) group [8]—one of the two most important 2 Lie groups in analysis. The other group is the sensealso as a map of complex numbers z = x+iy, Heisenberg group [3]. By contrast the “ax + b”- i2 = −1. Moreover, if y > 0 then g · z has a positive group, which is often used to build wavelets, is a imaginary part as well, i.e., (1) defines a map from subgroup of SL2(R), see the numerator in (1). the upper half-plane to itself. The simplest nonlinear transformations of the However there is no need to be restricted to real line—the linear-fractional or Möbius maps— the traditional route of complex numbers only. may also be associated with 2 × 2 matrices [1, Less-known double and dual numbers [9, Suppl. C] Ch. 13]: = + (1) also have the form z x iy but different as- 2 = 2 = ax + b a b sumptions on the imaginary unit i: i 0ori 1 g : x ֏ g · x = , where g = , x ∈ R. correspondingly. Although the arithmetic of dual cx + d c d! and double numbers is different from the complex An easy calculation shows that the composition of ones, e.g., they have divisors of zero, we are still two transforms (1) with different matrices g and 1 able to define their transforms by (1) in most cases. g2 is again a Möbius transform with matrix the Three possible values −1, 0, and 1 of σ := product g1g2. In other words (1) is a (left) action i2 will be referred to here as elliptic, parabolic, of SL2(R). According to F. Klein’s Erlangen program and hyperbolic cases respectively. We repeated- (which was influenced by S. Lie) any geometry is ly meet such a division of various mathematical dealing with invariant properties under a certain objects into three classes. They are named by group action. For example, we may ask: What kinds the historically first example—the classification of geometry are related to the SL2(R) action (1)? of conic sections—however the pattern persis- The Erlangen program has probably the highest tently reproduces itself in many different areas: rate praised among mathematical theories, not actually used equations, quadratic forms, metrics, manifolds, operators, etc. We will abbreviate this separation Vladimir V. Kisil is Reader of Analysis at the University of as EPH-classification. The common origin of this Leeds. He is on leave from Odessa University. His email address is [email protected]. fundamental division can be seen from the simple Dedicated to the memory of Serge Lang. picture of a coordinate line split by zero into
1458 Notices of the AMS Volume 54, Number 11 negative and positive half-axes: Subgroups A and N act in (1) irrespective of the value of σ : A makes a dilation by α2, i.e., z ֏ α2z, and N shifts points to left by ν, i.e. z ֏ z + ν. (2)
Connections between different objects admit- ting the EPH-classification are not limited to this common source. There are many deep results link- ing, for example, ellipticity of quadratic forms, metrics and operators.
(a)
(b) Figure 2. K-orbits as conic sections: circles are sections by the plane EE′; parabolas are sections by PP ′; hyperbolas are sections by HH′. Points on the same generator of the cone correspond to the same value of φ.
By contrast, the action of the third matrix from the subgroup K sharply depends on σ , see Figure 1. In the elliptic, parabolic and hy- perbolic cases K-orbits are circles, parabolas and (equilateral) hyperbolas correspondingly. Thin tra- versal lines in Figure 1 join points of orbits for the same values of φ and grey arrows repre- sent “local velocities”—vector fields of derived representations. Figure 1. Action of the K subgroup. The Definition 1. The common name cycle [9] is corresponding orbits are circles, parabolas, used to denote circles, parabolas, and hyperbolas and hyperbolas. (as well as straight lines as their limits) in the respective EPH case. To understand the action (1) in all EPH cases we It is well known that any cycle is a conic section = use the Iwasawa decomposition [8] of SL2(R) and an interesting observation is that correspond- ANK into three one-dimensional subgroups ing K-orbits are in fact sections of the same A, N, K: two-sided right-angle cone, see Figure 2. More- (3) over, each straight line generating the cone, see a b α 0 1 ν cos φ − sin φ Figure 2(b), crosses corresponding EPH K-orbits = −1 . c d! 0 α ! 0 1! sin φ cos φ ! at points with the same value of the parameter φ
December 2007 Notices of the AMS 1459 from (3). In other words, all three types of orbits A generic cycle is the set of points (u, v) ∈ R2 are generated by the rotations of this generator defined for all values of σ by the equation along the cone. (4) k(u2 − σv2) − 2lu − 2nv + m = 0. K-orbits are K-invariant in a trivial way. More- over since actions of both A and N for any σ This equation (and the corresponding cycle) is are extremely “shape-preserving” we find natural defined by a point (k, l, n, m) from a projective invariant objects of the Möbius map: space P3, since for a scaling factor λ ≠ 0 the point (λk,λl,λn,λm) defines the same equation (4). We Theorem 2. Cycles from Definition 1 are invariant call P3 the cycle space and refer to the initial R2 under the action (1). as the point space. ∈ In order to get a connection with the Möbius Proof. We will show that for a given g SL2(R) and a cycle C its image gC is again a cycle. Fig- action (1) we arrange the numbers (k, l, n, m) into ure 3 gives an illustration with C as a circle, but the matrix our reasoning works in all EPH cases. + − s = l ˘ısn m (5) Cσ˘ − + , k l ˘ısn! with a new imaginary unit˘ıand an additional pa- rameter s usually equal to ±1. The values of σ˘ := ˘ı2 are −1, 0, or 1 independently of the value of σ . The matrix (5) is the cornerstone of the (extended) Fillmore–Springer–Cnops construction (FSCc) [2] and closely related to the technique recently used by A. A. Kirillov to study the Apollonian gasket [4]. The significance of FSCc in the Erlangen frame- work is provided by the following result: ˜s s Figure 3. Decomposition of an arbitrary Theorem 3. The image Cσ˘ of a cycle Cσ˘ under ∈ Möbius transformation g into a product transformation (1) with g SL2(R) is given by = ′ ′ g gagngkgagn. similarity of the matrix (5): ˜s = s −1 (6) Cσ˘ gCσ˘ g . For a fixed C there is always a unique pair of In other words FSCc (5) intertwines Möbius ac- ′ ′ ∈ transformations gn from the subgroup N and ga tion (1) on cycles with a linear map (6). ′ ′ A such that the cycle gagnC is exactly a K-orbit. We ′ ′ − There are several ways to prove (6): either by a make a decomposition of g(g g ) 1 into a product a n brute-force calculation (fortunately performed by as in (3): a CAS) [7] or through the related orthogonality of g(g′ g′ )−1 = g g g . a n a n k cycles [2]; see the end of the next section. ′ ′ ′ ′ = Since gagnC is a K-orbit we have gk(gagnC) The important observation here is that FSCc (5) ′ ′ gagnC, then: uses an imaginary unit ˘ıwhich is not related to = ′ ′ −1 ′ ′ = ′ ′ i defining the appearance of cycles on plane. In gC g(gagn) gagnC gagngkgagnC ′ ′ ′ ′ other words any EPH type of geometry in the cycle = g g g (g g C) = g g g g C. a n k a n a n a n space P3 allows one to draw cycles in the point Since the subgroups A and N obviously preserve space R2 as circles, parabolas, or hyperbolas. We the shape of any cycle this finishes our proof. may think of points of P3 as ideal cycles while their depictions on R2 are only their shadows on According to Erlangen ideology we should now the wall of Plato’s cave. study invariant properties of cycles. Figure 4(a) shows the same cycles drawn in = l − n different EPH styles. Points ce,p,h ( k , σ k ) are Invariance of FSCc their respective e/p/h-centers. They are related to Figure 2 suggests that we may get a unified each other through several identities: treatment of cycles in each EPH case by consider- 1 ation of higher-dimensional spaces. The standard (7) ce = c¯h, cp = (ce + ch). mathematical method is to declare objects un- 2 der investigation (cycles in our case, functions in Figure 4(b) presents two cycles drawn as parabo- n functional analysis, etc.) to be simply points of las; they have the same focal length 2k and thus some bigger space. This space should be equipped their e-centers are on the same level. In other with an appropriate structure to hold externally words concentric parabolas are obtained by a ver- information that was previously inner properties tical shift, not scaling as an analogy with circles of our objects. or hyperbolas may suggest.
1460 Notices of the AMS Volume 54, Number 11 the absolute values of trace or determinant are irrelevant, unless they are zero. Alternatively we may have a special arrange- ment for normalization of quadruples (k, l, n, m). For example, if k ≠ 0 we may normalize the l n m quadruple to (1, k , k , k ) with the cycle’s center s highlighted. Moreover in this case det Cσ˘ is equal to the square of cycle’s radius, cf. the next to last s = section below. Another normalization det Cσ˘ 1 is used in [4] to get a nice condition for touching circles. (a) We still get important characterizations even with non-normalized cycles, e.g., invariant classes (for different σ˘ ) of cycles are defined by the condi- s = tion det Cσ˘ 0. Such a class is parameterized by two real numbers and as such is easily attached to R2 s a certainpointof . For example, the cycle Cσ˘ with s = = − det Cσ˘ 0, σ˘ 1 drawn elliptically represents a l n point ( k , k ), i.e., an (elliptic) zero-radius circle. The same condition with σ˘ = 1 in hyperbolic drawing l n produces a null-cone originating at a point ( k , k ): l n (u − )2 − (v − )2 = 0, k k i.e., a zero-radius cycle in the hyperbolic metric. In general for every notion there are nine pos- (b) sibilities: three EPH cases in the cycle space times three EPH realizations in point space. The nine Figure 4. (a) Different EPH implementations of cases for “zero radius” cycles are shown in Fig- the same cycles defined by quadruples of ure 5. For example, p-zero-radius cycles in any numbers. implementation touch the real axis. (b) Centers and foci of two parabolas with the same focal length.
Figure 4(b) also presents points, called e/p/h- foci: s l det Cσ˘ (8) fe,p,h = , − , k 2nk which are independent of the sign of s. If a cycle is depicted as a parabola then h-focus, p-focus, e-focus are correspondingly geometrical focus of the parabola, its vertex, and the point on the directrix nearest to the vertex. As we will see, cf. Theorems 5 and 7, all three centers and three foci are useful attributes of a Figure 5. Different i-implementations of the cycle even if it is drawn as a circle. same σ˘ -zero-radius cycles and corresponding foci. Invariants: Algebraic and Geometric We use known algebraic invariants of matrices to build appropriate geometric invariants of cycles. This “touching” property is a manifestation of It is yet another demonstration that any division the boundary effect in the upper-half plane geom- of mathematics into subjects is only illusive. etry [7, Rem. 3.4]. The famous question on hearing For 2 × 2 matrices (and thus cycles) there are the shape of a drum has a sister: only two essentially different invariants under Can we see/feel the boundary similarity (6) (and thus under Möbius action (1)): from inside a domain? the trace and the determinant. The latter was al- Both orthogonality relations described below are ready used in (8) to define a cycle’s foci. However “boundary aware” as well. It is not surprising after P3 due to the projective nature of the cycle space all since the SL2(R) action on the upper-half plane
December 2007 Notices of the AMS 1461 was obtained as an extension of its action (1) on According to the categorical viewpoint internal the boundary. properties of objects are of minor importance in comparison to their relations with other objects from the same class. As an illustration we may put the proof of Theorem 3 sketched at the end of the next section. Thus from now on we will look for invariant relations between two or more cycles.
Joint Invariants: Orthogonality The mostexpectedrelationbetweencycles is based on the following Möbius invariant “inner product” built from the trace of the product of two cycles as matrices: s ˜s = s ˜s (9) Cσ˘ , Cσ˘ tr(Cσ˘ Cσ˘ ). D E Bytheway,aninnerproductofthistypeisused,for example, in the GNS construction to make a Hilbert space out of C∗-algebras. The next standard move is given by the following definition.
Definition 4. Two cycles are called σ˘ -orthogonal s ˜s = if Cσ˘ , Cσ˘ 0. D E For the case of σσ˘ = 1, i.e., when geometries of the cycle and point spaces are both either elliptic or hyperbolic, such an orthogonality is the standard one, defined in terms of angles between tangent lines in the intersection points of two cycles. How- ever in the remaining seven (= 9 − 2) cases the innocent-looking Definition 4 brings unexpected relations. Elliptic (in the point space) realizations of Def- inition 4, i.e., σ = −1 are shown in Figure 6. The first picture corresponds to the elliptic cycle space, e.g., σ˘ = −1. The orthogonality between the red circle and any circle from the blue or green families is given in the usual Euclidean sense. The central (parabolic in the cycle space) and the last (hyperbolic) pictures show the non-local nature of the orthogonality. There are analogous pictures in parabolic and hyperbolic point spaces as well [7]. This orthogonality may still be expressed in the Figure 6. Orthogonality of the first kind in the traditional sense if we associate to the red circle elliptic point space. the corresponding “ghost” circle, which is shown Each picture presents two groups (green and by the dashed line in Figure 6. To describe the blue) of cycles that are orthogonal to the red ghost cycle we need the Heaviside function χ(σ ): s s cycle Cσ˘ . Point b belongs to Cσ˘ and the family ≥ of blue cycles passing through b is orthogonal = 1, t 0; s (10) χ(t) − to Cσ˘ . They all also intersect in the point d, ( 1, t < 0. s which is the inverse of b in Cσ˘ . Any orthogonality is reduced to the usual Theorem 5. A cycle is σ˘ -orthogonal to the cycle s orthogonality with a new (“ghost”) cycle Cσ˘ if it is orthogonal in the usual sense to the σ - ˆs (shown by the dashed line), which may or may realization of the “ghost” cycle Cσ˘ , which is defined s not coincide with Cσ˘ . For any point a on the by the following two conditions: “ghost” cycle the orthogonality is reduced to ˆs (i) The χ(σ )-center of Cσ˘ coincides with the the local notion in the terms of tangent lines σ˘ -center of Cs . at the intersection point. Consequently such a σ˘ (ii) The cycles Cˆs and Cs have the same roots, point a is always the inverse of itself. σ˘ σ˘ ˆ1 = χ(σ)˘ moreover det Cσ det Cσ .
1462 Notices of the AMS Volume 54, Number 11 The above connection between various centers of cycles illustrates their meaningfulness within our approach. One can easily check the following orthogonal- ity properties of the zero-radius cycles defined in the previous section: s s = s (i) Since Cσ˘ , Cσ˘ det Cσ˘ zero-radius cycles are self-orthogonal (isotropic) ones. s (ii) A cycle Cσ˘ is σ -orthogonal to a zero- s s radius cycle Zσ˘ if and only if Cσ˘ passes s through the σ -center of Zσ˘ . Sketch of proof of Theorem 3. The validity of The- orem 3 for a zero-radius cycle − ¯ 1 −¯ s = z zz = z z 1 z Zσ˘ − − 1 z¯ ! 2 1 1! 1 z¯! with center z = x + iy is straightforward. This im- plies the result for a generic cycle with the help of Möbius invariance of the product (9) (and thus the orthogonality) and the above relation (ii) be- tween the orthogonality and the incidence. See [2] for details.
Higher Order Joint Invariants: s-Orthogonality With appetite already whetted one may wish to build more joint invariants. Indeed for any ho- mogeneous polynomial p(x1, x2,...,xn) of several non-commuting variables one may define an in- j s variant joint disposition of n cycles Cσ˘ by the condition: 1 s 2 s n s = tr p( Cσ˘ , Cσ˘ ,..., Cσ˘ ) 0. However it is preferable to keep some geometrical meaning for constructed notions. An interesting observation is that in the ma- trix similarity of cycles (6) one may replace the ∈ element g SL2(R) by an arbitrary matrix cor- responding to another cycle. More precisely the Figure 7. Orthogonality of the second kind for s ˜s s product Cσ˘ Cσ˘ Cσ˘ is again the matrix of the form (5) circles. To highlight both similarities and and thus may be associated to a cycle. This cycle distinctions with the ordinary orthogonality ˜s s may be considered as the reflection of Cσ˘ in Cσ˘ . we use the same notation as that in Figure 6. s Definition 6. A cycle Cσ˘ is s-orthogonal to a cycle ˜s ˜s s Cσ˘ if the reflection of Cσ˘ in Cσ˘ is orthogonal (in the sense of Definition 4) to the real line. Analytically s Theorem 7. A cycle is s-orthogonal to a cycle Cσ˘ this is defined by: if it is orthogonal in the traditional sense to its s- s ˜s s s = ˜σ˘ = χ(σ)Rσ˘ χ(σ) (11) tr(Cσ˘ Cσ˘ Cσ˘ Rσ˘ ) 0. ghost cycle Cσ˘ Cσ˘ σ˘ Cσ˘ , which is the reflec- tion of the real line in Cχ(σ) and χ is the Heaviside Due to invariance of all components in the σ˘ above definition s-orthogonality is a Möbius in- function (10). Moreover ˜σ˘ variant condition. Clearly this is not a symmetric (i) The χ(σ )-Center of Cσ˘ coincides with s ˜s ˜s s relation: if Cσ˘ is s-orthogonal to Cσ˘ then Cσ˘ is not the σ˘ -focus of Cσ˘ , consequently all lines necessarily s-orthogonal to Cs . s σ˘ s-orthogonal to Cσ˘ are passing through Figure 7 illustrates s-orthogonality in the ellip- the respective focus. tic point space. By contrast with Figure 6 it is not a s ˜σ˘ (ii) The cycles Cσ˘ and Cσ˘ have the same roots. local notion at the intersection points of cycles for all σ˘ . However it may be again clarified in terms Note the above intriguing interplay between a of the appropriate s-ghost cycle, cf. Theorem 5. cycle’s centers and foci. Although s-orthogonality
December 2007 Notices of the AMS 1463 may look exotic it will naturally appear again at the end of the next section. Of course, it is possible to define other inter- esting higher-order joint invariants of two or even more cycles.
Distance, Length, and Perpendicularity Geometry in the plain meaning of this word deals with distances and lengths. Can we obtain them from cycles? We mentioned already that for circles normal- = s = ized by the condition k 1 the value det Cσ˘ (a) s s Cσ˘ , Cσ˘ produces the square of the traditional circle radius. Thus we may keep it as the definition of the radius for any cycle. But then we need to accept that in the parabolic case the radius is the (Euclidean) distance between (real) roots of the parabola, see Figure 8(a). Having radii of circles already defined we may use them for other measurements in several dif- ferent ways. For example, the following variational definition may be used:
Definition 8. The distance between two points is the extremum of diameters of all cycles passing through both points, see Figure 8(b). (b) If σ˘ = σ this definition gives in all EPH cases the following expression, see Figure 8(b):
2 2 2 (12) de,p,h(u, v) = (u + iv)(u − iv) = u − σv . 2 = 2 The parabolic distance dp u algebraically sits between de and dh according to the general prin- ciple (2) and is widely accepted [9]. However one may be unsatisfied by its degeneracy. An alternative measurement is motivated by the fact that a circle is the set of equidistant points from its center. However the choice of “center” is now rich: it may be any point from among three centers (7) or three foci (8). ------→ Definition 9. The length of a directed interval AB is the radius of the cycle with its center (denoted (c) ------→ ------→ by lc (AB)) or focus (denoted by lf (AB)) at the point Figure 8. (a) The square of the parabolic A that passes through B. diameter is the square of the distance between roots if they are real (z1 and z2), This definition is less common and has------→ some otherwise the negative square of the distance unusual properties like non-symmetry: l (AB) ≠ ------→ f between the adjoint roots (z3 and z4). lf (BA). However it comfortably fits the Erlangen (b) Distance as extremum of diameters in
program due to its SL2(R)-conformal invariance: elliptic (z1 and z2) and parabolic (z3 and z4) cases. Theorem 10 ([7]). Let l denote either the EPH dis- (c) Perpendicular as the shortest route to a tances (12) or any length from Definition 9. Then line. for fixed y, y ′ ∈ Rσ the limit: · · + ′ l(g y, g (y ty )) ∈ lim ′ , where g SL2(R), t→0 l(y, y + ty ) exists and its value depends only on y and g and is independent of y ′.
1464 Notices of the AMS Volume 54, Number 11 We return from distances to angles recalling Of course, there is no reason to limit the Er- that in Euclidean space a perpendicular provides langen program to SL2(R) only, other groups may the shortest length from a point to a line, see be more suitable in different situations. However
Figure 8(c). SL2(R) still possesses a big unexplored potential and is a good object to start with. Definition 11. Let l be a length or distance. We ------→ say that a vector AB is l-perpendicular to a vector ------→ ------→ ------→ Note: Graphics for this article were created by CD if the function l(AB + εCD) of ε has a local the author with the help of Open Source Software: extremum at ε = 0. MetaPost (http://www.tug.org/metapost.html),
A pleasant surprise is that lf -perpendicularity Asymptote (http://asymptote.sourceforge.net/), obtained through the length from focus (Defi- and GiNaC (http://www.ginac.de/). nition 9) coincides with s-orthogonality already defined in the preceding section, as follows from References Theorem 7(i). [1] Alan F. Beardon, Algebra and Geometry, Cam- All these notions are waiting to be general- bridge University Press, Cambridge, 2005. MR 2153234 (2006a:00001) ized to higher dimensions, and Clifford algebras [2] Jan Cnops, An Introduction to Dirac Operators on provide a suitable language for this [7]. Manifolds, Progress in Mathematical Physics, vol. 24, Birkhäuser Boston Inc., Boston, MA, 2002. MR Erlangen Program at Large 1917405 As we already mentioned the division of mathe- [3] Roger Howe, On the role of the Heisenberg group matics into areas is only apparent. Therefore it is in harmonic analysis, Bull. Amer. Math. Soc. (N.S.) 3 unnatural to limit the Erlangen program only to (1980), no. 2, 821–843. MR 81h:22010 [4] A. A. Kirillov, A tale on two fractals, 2007. (To “geometry”. We may continue to look for SL (R) in- 2 appear). variant objects in other related fields. For example, [5] Vladimir V. Kisil, Meeting Descartes and Klein transform (1) generates unitary representations on somewhere in a noncommutative space, High- certain L2 spaces, cf. (1): lights of Mathematical Physics, 2002. 165–189, American Mathematical Society. http://arXiv. 1 ax + b org/abs/math-ph/0112059. MR 2005b:43015 (13) g : f (x) ֏ f . [6] , Spectrum as the support of function- + m + (cx d) cx d al calculus, Functional Analysis and Its Appli- = For m 1, 2, …the invariant subspaces of L2 are cations, 2004, pp. 133–141. http://arXiv.org/ Hardy and (weighted) Bergman spaces of complex abs/math.FA/0208249. MR MR2098877 analytic functions. All the main objects of complex [7] , Erlangen program at large–1: Geometry of analysis (Cauchy and Bergman integrals, Cauchy- invariants (2005). http://arXiv.org/abs/math. CV/0512416. preprint LEEDS–MATH–PURE–2005–28. Riemann and Laplace equations, Taylor series, [8] Serge Lang, SL2(R), Graduate Texts in Mathematics, etc.) may be obtained in terms of invariants of the vol. 105, Springer-Verlag, New York, 1985. Reprint of discrete series representations of SL2(R) [5, § 3]. the 1975 edition. MR 803508 (86j:22018) Moreover two other series (principal and compli- [9] I. M. Yaglom, A Simple Non-Euclidean Geometry and mentary [8]) play the similar roles for hyperbolic Its Physical Basis, Springer-Verlag, New York, 1979. and parabolic cases [5,7]. An elementary account of Galilean geometry and the Moving further we may observe that trans- Galilean principle of relativity, Heidelberg Science Li- form (1) is defined also for an element x in any brary, translated from the Russian by Abe Shenitzer, algebra A with a unit 1 as soon as (cx+d1) ∈ A has with the editorial assistance of Basil Gordon. MR 520230 (80c:51007) an inverse. If A is equipped with a topology, e.g., is a Banach algebra, then we may study a functional calculus for the element x [6] in this way. It is defined as an intertwining operator between the representation (13) in a space of analytic functions and a similar representation in a left A-module. In the spirit of the Erlangen program such a functional calculus is still a geometry, since it is dealing with invariant properties under a group action. However even for a simplest non-normal operator, e.g., a Jordan block of the length k, the space obtained is not like a space of points but is rather a space of k-th jets [6]. Such non-point behavior is often attributed to noncommutative geometry, and the Erlangen program provides an important input on this fashionable topic [5].
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The Triumph of Numbers: How Counting Shaped Modern Life and Karl Pearson: The Scientific Life in a Statistical Age Reviewed by Brian Blank
The Triumph of Numbers: How Counting of commercial transactions that took place more Shaped Modern Life than 5,000 years ago. It is not surprising that I. Bernard Cohen numbers are found so frequently in prehistoric W. W. Norton, 2006 artifacts: these numerical symbols predate writ- US$14.95, 224 pages, ISBN: 978-0393328707 ing precisely because they played a crucial part in the development of writing. As the historian Will Karl Pearson: The Scientific Life in a Statistical Durant once remarked, “The berated bourgeoisie Age may take consolation in the thought that literature Theodore M. Porter originated in bills of lading.” Princeton University Press, 2005 Cohen begins his study of numbers with the US$23.95, 352 pages, ISBN: 978-0691126357 Narmer Macehead, which dates from the thirty-first century B.C.E. and is now housed in Oxford’s Ash- “We live in a world of numbers.” So begins The molean Museum. It has much to excite different Triumph of Numbers, the posthumously published sorts of historians. The writing it bears is among book of the eminent historian of science, I. Bernard the earliest yet found. The king it depicts is one of Cohen. The numbers he refers to are wages, taxes, the earliest named historical persons, the pharaoh consumer prices, financial records, economic Narmer. The event it commemorates is thought to reports, demographic data, sports statistics, and have taken place shortly after the unification of the like. When did this numerical deluge begin? Upper and Lower Egypt and the establishment of Too long ago for us to know! Even before the for- the first Egyptian dynasty. All very important to some, but to a historian like Cohen the relic con- mation of the first civilizations, several Neolithic tains another lesson that should not be overlooked: communities scattered about the Mediterranean the census figures that are carved in relief on the left behind stones and ostraca bearing numerical macehead—120,000 (adult men), 400,000 (cattle), archives of trade, personal services, interest rates, and 1,422,000 (small animals)—show that the an- and gambling debts. As civilizations emerged, cient Egyptians were able to write very large num- societies and governments increasingly relied on bers. Such capabilities were essential for the con- numbers to manage the ever greater complexi- struction of the Great Pyramid at Giza, which was ties of daily life. Archeologists have unearthed erected according to a meticulous design requiring thousands of Sumerian tablets bearing records some 2,300,000 stone blocks. After discussing the Brian Blank is professor of mathematics at Washington Great Pyramid in thorough numerical detail, Cohen University in St. Louis. His email address is brian@math. concludes his first chapter by outlining those as- wustl.edu. pects of the world of numbers that are the focus
1468 Notices of the AMS Volume 54, Number 11 of his investigation: “Our mission from here on is regularities, such as the near con- to explicate some of the important and interesting stant percentage of deaths attribut- steps whereby the analysis of society, the conduct able to suicide, Graunt published of government, the regulation of daily life, and the credible estimates of London’s pop- understanding of nature came to be.” ulation and its growth rate. Allow- The second chapter provides a good illustra- ing for plagues, he found that “in tion of Cohen’s plan and execution, his successes eight times eight years the whole and failures. After a capsule treatment of Kepler’s People of the City shall double third law, Cohen turns to Galileo’s study of con- without the access of Foreigners.” stantly accelerated motion. Here, as elsewhere, Graunt, it should be noted, was well the discussion is aimed at a general reader. Thus, aware that a constant doubling time when Galileo is said to have confirmed his formula is not indefinitely sustainable. As for free fall by experimenting with a ball rolling he quaintly explained, “One couple, down an inclined plane, Cohen assumes that the viz. Adam and Eve, doubling them- reader will not notice that the second dynamical selves every 64 years of the 5610 problem involves rotational kinetic energy, which years, which is the age of the World is not present in the first. Cohen’s survey of the according to the Scriptures, shall seventeenth century, an era with which he was produce far more People, than are now in it.” particularly associated, continues with William Kepler, Galileo, Harvey, Leeuwenhoek, Halley, Harvey’s investigation of the circulation system, Graunt, and Petty—all these essays are packed into Anton Van Leeuwenhoek’s estimate of Earth’s a mere twenty-one pages of text. It is inevitable carrying capacity, Edmond Halley’s contributions that such brevity results in missed opportunities to insurance annuities, John Graunt’s analysis of and incompleteness. Petty’s political arithmetic, a London’s Bills of Mortality, and William Petty’s statistically based statecraft using algebra for its advocacy of “Political Arithmetick”. It is a tribute analysis, is explained in theory, but no application to Cohen’s narrative skill that he is able to string is mentioned. It might have interested the reader to together such a hodgepodge of case studies so learn that Petty advised the government to combat coherently. Moreover, he has enlivened each topic the plague on economic rather than humanitarian with fascinating, unsuspected details. Thus, even grounds. Funds devoted to fighting the plague, the reader who is well acquainted with elementary Petty argued, would be profitably invested: the mechanics may be surprised to learn Galileo’s money used to save adult lives would preserve the own description of his discovery: “The distances far greater sums that had already been expended traversed during successive equal intervals of time in rearing those individuals to maturity. We are by a body falling from rest stand to one another in familiar with similar public policy reasoning in the same ratio as the odd numbers beginning with modern life, but other suggested uses of political unity.” Leeuwenhoek’s pioneering contributions arithmetic now seem startling. As Theodore Porter to microscopy are well known, but how did they relates in his study of social statistics [9], “Petty lead to an interest in demography? The answer, proposed that all Irishmen, save a few cowherds, we discover, is that Leeuwenhoek found it dif- should be forceably transported to England, for ficult to convey the minuteness of spermatozoa since the value of an English life far surpassed that at a time when microscopes were in the hands of of an Irish one, the wealth of the kingdom would only a few scientists. He solved his problem by es- thereby be greatly augmented.” timating that the Earth could support a maximum The last six of Cohen’s nine chapters are largely of 13,385,000,000 human beings, a population concerned with social and medical statistics. One dwarfed by the number of “little animals in the chapter is devoted to the Belgian sociologist and milt of a cod”. statistician Adolphe Quetelet (1796–1874), who The discourse on John Graunt touches upon has been receiving his due in recent years [9], two of Cohen’s themes: the analysis of society [12], [13]. Less well known is the Parisian lawyer and the conduct of government. Following an and amateur statistician André-Michel Guerry episode of plague that struck London in 1603, (1802–1866), who is the subject of Cohen’s short- the government published broadsides called Bills est chapter. Simultaneously but independently, of Mortality at regular intervals. By listing the both men analyzed the records of criminal activity number of deaths per week according to cause, that the French government began to publish in these advisories alerted rich Londoners to any 1827. Both were astonished by the regularity with increase of disease that might suggest a precau- which lawbreaking occurred. Guerry found it dif- tionary retreat to the shelter of the countryside. ficult to reconcile the constant crime rate with “the Graunt’s study of the Bills, originally published in infinite number of circumstances that can cause 1661 or 1662, is considered to be the first statis- the commission of a crime.” Quetelet, believing tical analysis ever undertaken. In addition to call- that the puzzling data could be explained only by ing attention to numerous unsuspected societal a societal component of criminality, concluded that
December 2007 Notices of the AMS 1469 “Society prepares the crime and the guilty person case with the goal of survival and future immunity. is merely the instrument by which it is executed.” This procedure, known as variolation, originated There were many other unsuspected patterns for in seventeenth-century China and reached colonial Guerry and Quetelet to pry from the data. Contrary America by the early 1700s. Despite some fervent to expectation, regions with the most educated advocates, the Puritan minister Cotton Mather inhabitants had the highest incidences of crime. included, variolation was distrusted by the public. One curious correlation noticed by Guerry when As a result, waves of smallpox epidemics ravaged he scrutinized suicide records was paraphrased the colonies; one outbreak in 1736 took Benjamin by an astounded English contemporary in this way: Franklin’s four-year-old son. In 1756 and 1759, “The method by which a person destroys himself Franklin, who bitterly regretted not having inocu- is almost as accurately and invariably defined by lated his child, published pamphlets in support of his age as the seasons are by the sun.” variolation. To “Parents who omit that Operation By contrasting nineteenth-century France with on the Supposition that they should never forgive seventeenth-century London, we recognize the themselves if a child died under it,” Franklin cau- dawning of a new age of numerical information, tioned, “my Example [shows] that the Regret may an era characterized not only by the assiduousness be the same either way, and that therefore the safer with which nineteenth-century governments col- [option] should be chosen.” lected and disseminated demographic, social, and Franklin’s strategy for convincing parents of the medical data but also by a heightened awareness wiser choice was to analyze the relevant mortality that useful information could be gleaned from a statistics. Of course, his tactic now seems obvious, careful study of the tables that were pouring forth. so far has our acceptance of statistical argument Whereas the Bills of Mortality waited nearly sixty come, but it was inventive for its time. At one years for a John Graunt, the Compte Général…de smallpox hospital Franklin found the death rate la Justice Criminelle was mined almost instantly by from inoculation to be 6 out of 1,601. At the same Guerry and Quetelet. Nevertheless, as the bandy- hospital 1,002 patients out of 3,856 who contracted ing of statistics grew, so did the number of critics. smallpox “in the common way” died. Cohen does In The Triumph of Numbers, Cohen uses Charles not directly say whether Franklin published or Dickens, depicted as a well-meaning reaction- even determined the incidence of smallpox in the ary who “abhorred the introduction of numbers population, a statistic that is necessary for decid- into discussions of human affairs,” to represent ing the more prudent course. However, Cohen does the opposition to statistics. Other historians are quote Franklin’s claim that the chance in favor of more receptive to the contention of Dickens that inoculation was as high as thirty to one (Franklin’s statistics were being manipulated to discount the emphasis). Given that epidemics in Boston in 1752 plight of the working poor. According to Michael and Charleston in 1738 infected, respectively, Cullen [3, pp. 136, 144], to cite one example, “the 37.5 percent and 50 percent of the inhabitants of statisticians were uniformly committed to policies those cities, we see that Franklin’s analysis was of economic laissez-faire” and, in order to block on the mark [1], [4]. The question then becomes, labor reform, “disguised propaganda as facts.” It Did Franklin’s advocacy of variolation have any was in the nineteenth-century, after all, that Saint measurable effect? Cohen is silent on the subject, Thomas Aquinas’s long-standing classification of but the evidence suggests that Franklin’s efforts lies into three kinds—namely the jocose, the of- did not result in any triumph of numbers. Smallpox ficious, and the malicious—evolved into the jibe continued to decimate America for decades. Four “There are liars, there are outrageous liars, and years after Franklin’s first pamphlet, Charleston there are scientific experts,” which in turn mutated suffered an epidemic that afflicted 75 percent of its into the surviving barb “There are three kinds of citizens. In 1776, during Benedict Arnold’s siege of lies: lies, damned lies, and statistics.” (Cohen’s as- Quebec in the American Revolutionary War, 1,200 sertion that “scholars now assume that this saying of his 3,200 troops suffered from smallpox—the was invented by Mark Twain” is misleading: some same incidence rate found in Boston a quarter of scholars cite earlier uses [5], [6].) a century earlier [2]. Statistical applications in healthcare also had to The Triumph of Numbers concludes with a overcome entrenched resistance. Physicians, striv- chapter that is entirely devoted to Florence Night- ing for the ideal of certitude, often regarded proba- ingale. Her interest in statistics, although fairly bilistic reasoning as unscientific. Furthermore, well known, is usually covered so perfunctorily in when the first attempts to introduce statistics into the mathematical literature (and often only for her medical practice were made, the general public was commentary on Quetelet) that it is good to have not ready to place its trust in numbers. The deadly Cohen’s lengthier discussion. In 1854 Nightingale scourge of smallpox in the American colonies is was dispatched to the Crimea in response to reports a case in point. Before Edward Jenner discovered that appalling conditions were prevalent in British vaccination in 1796, the only preventive measure military hospitals. To convince the authorities that against smallpox was deliberate exposure to a mild lives were being needlessly lost on a tragic scale,
1470 Notices of the AMS Volume 54, Number 11 Nightingale carefully collected hospital records loss of vision, Cohen was able, for the most part, and analyzed the causes of death. In one finding, to bring to his last book the care and scholarship she determined that the annual mortality rate in with which he was long associated. The Numbers in British hospitals in Turkey and the Crimea was History section of the first chapter does, however, 1,174 deaths per 10,000 patients, of which 1,023 require some caution. Here Gaspard Monge is said per 10,000 were the result not of battle wounds to have invented projective geometry rather than but of disease. Her recommendations called for descriptive geometry. Several factoids, apparently clean water, improved ventilation, proper sewage repeated from inaccurate secondary sources, are disposal, and hot water for laundering, measures misleading or wrong. Dates are sometimes incor- that sharply reduced hospital mortality within one rect (the Narmer Macehead) or inconsistent (the month of their implementation. This lesson should Great Pyramid). Few miscues appear after this have been learned once and for all, but Nightingale section, but several of Cohen’s opinions may strike had to repeat her algorithm of mortality analysis, the reader as questionable. Christiaan Huygens, nagging, and cajolery in order to win sanitary for example, is described as “really comparable reform in other locales. Realizing that “none but with Isaac Newton.” Fourier, according to Cohen, scientific men ever look into the appendices of a is “particularly remembered for his contributions Report,” she took pains to augment her tabulated to the mathematical theory of probability.” The data with graphical presentations. To that end she assertion that Laplace was “the most important introduced a type of pie chart, which she called a mathematician since Isaac Newton” will be consid- coxcomb, in which the sectors have equal angles ered heretical by readers who see it as a demotion but variable radii. of Euler in the pantheon. With the story of Florence Nightingale drawing The Triumph of Numbers is neither the capstone to a close, the reader may sense that a triumph of of a long, distinguished career nor a technical his- numbers is near at last. Up until this point, Cohen’s tory of statistical methods. Cohen has written a case studies have represented the significant steps modest book aimed at the nonspecialist who has that are promised in the first chapter but not the never noticed the prevalence of numerical data in triumph that is announced in the title. Thus, when modern society or who, having taken such notice, Cohen relates Nightingale’s reduction of mortality has wondered how society came to be so quanti- in India from 69 British troops per 1,000 to 18 per fied. Because The Triumph of Numbers demands 1,000, it appears that he is finally ready to drive nothing more of its readers than a desire to be home George Sarton’s maxim “Humanity must be entertained and informed, it will reward a wide protected by the watchful Statistician.” Instead, audience. For the mathematically sophisticated, its Cohen concludes with a less decisive message sent greatest value will lie in stimulating an interest in by Nightingale to Francis Galton in 1891, a letter the rise of statistics. To those who wish to delve that thirty years later prompted Karl Pearson to deeper, the books by Porter [9], [10] and Stigler despair, “We are only just beginning to study social [12], [13] are especially recommended. problems—medical, educational, commercial—by For readers of the Notices, Theodore Porter’s adequate statistical methods, and that study has newest book, a study of Karl Pearson, also comes at present done very little in influencing legisla- to mind as a natural follow-up to The Triumph of tion.” It is a dispiriting note on which to end and an Numbers. Indeed, the continuation of the timeline enigmatic one too: why did Cohen bring his story is perfect: Pearson began his statistical work the to an abrupt halt a decade short of the twentieth year after the Nightingale letter with which Cohen’s century? A two-page epilogue mentions the punch history concludes. From that time until the arrival card that Herman Hollerith invented to process the of Ronald Aylmer Fisher decades later, Pearson American census data of 1890, but this brief refer- dominated (in more than one sense) the statisti- ence is not a convincing bridge between Florence cal scene. During its first ten years his biometric Nightingale and the digital age. Perhaps a clue to school contributed about half of Britain’s statisti- Cohen’s seemingly arbitrary close may be found in cal work. Before these efforts only about 2 percent his New York Times obituary, which states that The of papers presented to the Royal Statistical Society Triumph of Numbers was originally envisioned as dealt with statistical methods [7]. It was largely The Fate of Mankind in a World of Numbers [11]. Pearson who transformed statistics into a branch The Times writer goes on to say that the manu- of mathematical analysis. script was mailed the week before its 89-year-old The pertinent facts of Karl Pearson’s life can author died. We are left to wonder if Cohen had be outlined as follows. He was born in 1857; was planned a somewhat different book but wrapped educated at Cambridge, where he took third wran- it up when time grew short. gler in 1879; was appointed Goldsmid Professor of Whatever the case may be, what we do have is Applied Mathematics and Mechanics at University a well-written, engaging work that requires few al- College London in 1884; and was selected for the lowances for the adverse conditions under which Gresham Lectureship of Geometry in 1891. After it was completed. Despite declining health and being influenced by the economist Francis Ysidro
December 2007 Notices of the AMS 1471 Edgeworth in 1891 and the biolo- addition to more obvious ones such as statistics gist Walter Frank Raphael Weldon and the history of science). For some time now, in 1892, Pearson took up statistics an expert synthesis has been overdue. Although as his vocation in 1892, published Egon’s memoir [8], written in the year of Karl’s his first paper in statistics and death when memories were fresh and documents coined the term standard deviation handy, remains a valuable resource, it is more of in 1893, introduced the product- a guide to Karl’s career in statistics than a proper moment correlation formula in biography. Furthermore, Egon lacked three im- 1896, developed the χ2 test for portant advantages that are available to a modern goodness of fit in 1900, cofounded scholar: the passage of enough time to assess the the journal Biometrika in 1901, lasting value of Karl’s contributions to science, founded the Biometric Laboratory the objectivity that so polemical a subject as Karl in 1903, became the first director requires in a biographer, and the extensive histori- of Galton’s Eugenics Laboratory in cal and sociological research that has appeared in 1907, became the first Galton Pro- the seventy years since Karl’s death. fessor of Eugenics in 1911, founded For the connoisseur of biography, the first six the Annals of Eugenics in 1925, chapters of Porter’s Karl Pearson will prove impres- retired in 1933, and died in 1936. His son, Egon sive. Such a reader is bound to marvel at the extent Sharpe Pearson (1895–1980), also a leading stat- to which Porter immersed himself in Pearson’s istician, became Karl’s first biographer in 1936. world, studying the cultural and intellectual cur- (For the remainder of this review, “Pearson” will rents that Pearson navigated, absorbing the books refer to Karl.) that influenced Pearson, and travelling down the As Stigler has remarked, throughout his adult many dead ends of Pearson’s wander years. This years Pearson did the work of three men, but be- part of Karl Pearson can be recommended as a self- fore his conversion to statistics it was always in contained Bildungsroman to both the historian of three different fields [12]. In fact, German history, the Victorian era and the aficionado of biography. literature, folklore, philosophy, intellectual poli- The last of these chapters, “Intellectual Love and tics, applied mathematics, physics, and engineer- the Woman Question”, presents a particularly ing all occupied Pearson to some extent before he intriguing account of Pearson’s courtships, all of turned to biometry, heredity, and other topics that which were pursued with characteristic eccentric- share a statistical theme. Prior to his first paper on ity. In the wooing of Maria Sharpe, for example, statistics, Pearson’s publication list numbered one Pearson followed his marriage proposal with hundred items, nine of which were books (includ- written confessions of his personality defects. ing a fictionalized autobiography and a passion The tactic of admitting hypersensitiveness, self- play). Porter describes the Pearson of these years consciousness, selfish tendencies, and “an almost as a “thoroughly restless intellectual.” Toward the equally vicious tendency to periods of depression end of this formative period, Pearson expressed & moroseness” did not disarm Maria immediately, misgivings that he was regarded as a “second-rate but after nearly a year of anxious deliberation she mathematician who indulges in extreme views & did accept his offer. dabbles in journalism.” He confessed to his fian- Pearson’s marriage in 1890 is the last event in cée, “I look back & round on all the odds & ends the biographical thread of Porter’s book. Excluding of careless & superficial work, which mark my a brief epilogue, the final quarter of Porter’s study life & make me shudder sometimes at the energy concerns the “statistical impulse” that seized Pear- & time frittered away attempting what was not son soon after he wed. How Pearson discovered his within my powers.” Nearly forty years later, a few vocation after so many false starts is an interesting years after he had been offered the Order of the question. In 1934 Pearson himself asserted that “It British Empire, which he refused, and a few years was Galton who first freed me from the prejudice before he was offered a knighthood, which he that sound mathematics could only be applied to also refused, Pearson continued to express doubts natural phenomena under the category of causa- about his research: “Twenty years hence a curve or tion.” As Porter has previously observed [9, p. 299], a symbol will be called ‘Pearson’s’ & nothing more Pearson was either forgetful or disingenuous in remembered of the toil of the years.” this reference to Galton’s Natural Inheritance. In Pearson was prescient in that his name is com- fact, reporting on Galton’s book in 1889, Pearson memorated primarily by the Pearson correlation warned of the “considerable danger in applying the coefficient—it is Egon who is remembered when method of exact science to problems in descriptive Neyman-Pearson is cited. However, the toil of science.” Stigler has suggested that Edgeworth was Karl’s years, far from being forgotten, has inspired the pivotal figure in Pearson’s turn to statistics [12, a vast literature scattered among the journals of p. 305], whereas Porter in his first book concluded diverse fields such as sociology, economics, psy- that it was only through Weldon that Pearson chology, genetics, medicine, and epidemiology (in came to accept Galton’s statistical approach [9,
1472 Notices of the AMS Volume 54, Number 11 p. 299]. In his new book, Porter attributes Pearson’s be benevolent motivations when Porter refers to conversion to “a new vision, one that he acquired them, but Pearson’s own words reveal a more sin- not through a single Eureka experience but episodi- ister intent: “It is cruel to the individual, it serves cally, over about three years.” Porter continues, no social purpose, to drag a man of only moderate “Initially his new interest in statistics grew out of intellectual power from the hand-working to the an ideal of education, the cultivation of a more brain-working group,” “Intelligence can be aided effective citizenry.” This judgment is reinforced and trained, but no training or education can create pages later when Porter endorses the widely held it. You must breed it,” and “From a bad stock can view that eugenics motivated and even shaped come only bad offspring, and if a member of such Pearson’s statistical work. a stock is, owing to special training and education, The two chapters that Porter allots to Pearson an exception to his family, his offspring will still the statistician concentrate on Pearson’s most be born with the old taint.” creative period, the last decade of the nineteenth- Though fairness may be in danger when a century. Because of this narrow focus, very little is historical figure is judged by the ethos of a later said about Pearson’s final thirty years, a period of era, the declaration of Pearson-trained statistician rapid progress in statistics during which Pearson George Udny Yule that “Votes for women is for me continued to play a prominent though diminishing nearly as loathworthy as eugenics” demonstrates role. In contrast to the attention lavished on the the abhorrence of eugenics among even those cultural and social milieux that molded Pearson, of Pearson’s contemporaries who do not seem the statistical enterprise that Pearson so greatly enlightened by present-day standards. If nothing influenced is largely neglected. William Sealy Gos- else, Pearson’s biometric interests remind us that set, better known by the pseudonym “Student”, is anthropometry and craniometry were flourishing mentioned only once, even though he published at the end of the nineteeth-century. The cephalic almost exclusively in Pearson’s Biometrika. Flor- index would soon be joined by other contentious ence Nightingale David, an assistant of Pearson’s numerical measures such as IQ, the general intel- who went on to have a distinguished independent ligence factor g, and Pearson’s own coefficient of career, is not mentioned at all. Pearson’s profes- racial likeness. Cohen did not have to confront sional disputes were too numerous to receive full such argumentative quantities in his book because coverage, but it is regrettable that Porter does not of his choice of time frame. Will future historians include a detailed account of Pearson’s especially who update his story be able to speak of a triumph acrimonious and protracted feud with Fisher. Their of numbers? squabble, which started during World War I, flared Because Porter’s Karl Pearson is neither a up in 1922, and occupied Pearson on multiple conventional biography nor a thorough account fronts in the last few months of his life, is of great of Pearson’s career in statistics, it will seem to fall between two stools. If a classifying label is interest not only for the many important statisti- needed, then microhistory, the term Porter uses cal issues that were raised but also for the nature in his epilogue, is apt. Many readers of the Notices of the personal combat. “Wasting your time fitting will regret that Porter did not write a different curves by moments, eh!” was the opening salvo sort of book, but no reader of Karl Pearson will in one of Pearson’s last papers, a public reply to have cause to wish for a more successful book of Fisher’s attacks on his method of moments and its type. Of course, there is still much work for a system of frequency curves. Pearson scholar to do. Now that nineteenth-century Porter, in a reflexive epilogue concerning meth- Pearson has received so outstanding a treatment, odology and conclusions, confesses that contrary we may hope for the arrival of another biographer to the general view of scholars he has come to to chronicle twentieth-century Pearson with equal regard Pearson as a figure that is more tragic mastery. than malign. Porter’s readers may be disinclined to share this assessment, particularly because References Porter does not conceal Pearson’s offensive be- [1] Nicolau Barquet and Pere Domingo, Smallpox: liefs and behaviors. Thus, Porter characterizes The triumph over the most terrible of the ministers the personal insults that Pearson liberally inserted of death, Annals of Internal Medicine 127 (1997), into Biometrika as “pointless and petulant” and 635–642. describes Pearson’s writings as “uncourtly”, “un- [2] Ann M. Becker, Smallpox in Washington’s army: Stra- attractive”, and “disturbing”. Porter also exposes tegic implications of the disease during the American Pearson’s smug racism and deeply held eugenic Revolutionary War, The Journal of Military History 68 sentiments. Here, however, his analysis would (2004), 381–430. [3] Michael John Cullen, The Statistical Movement in have had even greater effect had he quoted Pear- Early Victorian Britain: The Foundations of Empirical son more extensively. Consider, for example, the Social Research, The Harvester Press Limited, John ideal of education and the cultivation of a more Spiers, 1975. effective citizenry, the key factors Porter cites in [4] John Duffy, Epidemics in Colonial America, Louisiana Pearson’s conversion to statistics. These seem to State University Press, 1953.
December 2007 Notices of the AMS 1473 [5] Stephen Goranson, Lies, statistics: Talleyrand? (or dead end?), American Dialect Society list server, Janu- ary 26, 2007, http://listserv.linguistlist.org/ cgi-bin/wa?A2=ind0701d&L=ads-l&D=0&P=12466. IMA [6] I. D. Hill, Review of “Quotes, Damned Quotes, and…”, J. Roy. Statist. Soc. Series A 146 (1983), 431. INSTITUTE FOR MATHEMATICS [7] Donald A. Mackenzie, Statistics in Britain 1865– AND ITS APPLICATIONS 1930: The Social Construction of Scientific Knowledge, University of Minnesota, Minneapolis, Minnesota Edinburgh University Press, 1981. [8] Egon Pearson, Karl Pearson: An Appreciation of Some Aspects of His Life and Work, Cambridge University !SSOCIATE $IRECTOR Press, 1938. [9] Theodore M. Porter, The Rise of Statistical Thinking 1820–1900, Princeton University Press, 1986. The Institute for Mathematics and its Applications (IMA) [10] , Trust in Numbers: The Pursuit of Objectivity seeks a person to serve as Associate Director for two years ——— in Science and Public Life, Princeton University Press, beginning in August 2008. 1995. The IMA is located on the Minneapolis campus of the [11] Wolfgang Saxon, I. Bernard Cohen, 89, Dies; Pio- University of Minnesota. Since its founding in 1982 as the neer in History of Science, New York Times, June 28, result of a national competition, the IMA has been a major 2003. national institute with the mission of fostering [12] Stephen M. Stigler, The History of Statistics: The interdisciplinary research which links high caliber Measurement of Uncertainty before 1900, Harvard mathematics with important problems from other University Press, 1986. disciplines, technology, industry, and society, and of [13] ——— , Statistics on the Table: The History of Statisti- strengthening the talent base engaged in mathematical cal Concepts and Methods, Harvard University Press, research applied to or relevant to such problems. The IMA 1999. runs visitor programs involving around 1,200 scientists per year, with an annual budget of about $6M. The major source of funding of the IMA is the National Science Foundation which has funded the IMA through 2010. The IMA's many programs and activities are documented at its web site, www.ima.umn.edu.
The Associate Director will work with the Director and Deputy Director, the IMA staff, and program organizers on a variety of IMA activities, especially program implementation, participant applications, publicity and special projects. A successful candidate should have a PhD in mathematics or a related field and a successful record in and broad view of mathematical research, a commitment to the IMA mission, and administrative or organizational experience. The Associate Director will work in the exceptional scientific environment provided by the IMA, and will contribute to shaping the international research agenda. Research activity is compatible with the position, and a visiting faculty position at the University of Minnesota may be arranged if appropriate.
Applicants should send a CV, a statement of their interest in the position and other relevant information not contained in the CV, and have at least three letters of reference sent. All materials should be sent by email to [email protected] and inquiries can be directed to the same address. Review of applications will begin on January 14, 2008. Applications will be accepted until the position is filled.
The University of Minnesota is an equal opportunity educator and employer.
The IMA is an NSF funded Institute
1474 Notices of the AMS Volume 54, Number 11 INSTITUTE FOR PURE AND APPLIED MATHEMATICS Los Angeles, California IPAM/CCB Summer School: MATHEMATICS IN BRAIN IMAGING July 14 – 25, 2008
ORGANIZING COMMITTEE: Paul Thompson (UCLA School of Medicine), Michael Miller (Johns Hopkins University), Russ Poldrack (UCLA Dept. Psychology), Tom Nichols (Oxford University/GlaxoSmithKline), Keith Worsley (McGill University), Jonathan Taylor (Stanford)
Introduction This two-week intensive workshop will focus on mathematical techniques applied to brain images to measure, map and model brain structure and function. Topics will range from modeling anatomical structures in MRI scans, and mapping connectivity in diffusion tensor images, to statistical analysis of functional brain images from fMRI, EEG, and MEG. Current applications in radiology and neuroscience will be highlighted, as will new directions in the mathematics of structural and functional image analysis. In the second week on Functional Brain Mapping, a series of lectures on diffusion tensor imaging will discuss mathematics and tools for registration, segmentation, fiber tracking and connectivity modeling in tensor and “beyond-tensor” (high-angular resolution) diffusion images, using metrics on Riemannian manifolds. Software implementing a wide range of algorithms will be demonstrated; tutorial notes will be provided. Talks will interest newcomers as well as experts in the field. Morning lectures on the principles behind the methods; afternoon lectures will go in-depth into applications.
Program Schedule • Week 1: Computational Anatomy, July 14 – 18, 2008 • Week 2: Functional Brain Mapping, July 21 – 25, 2008
Participation Additional information about this program including links to register and to apply for funding, can be found on the webpage listed below. Encouraging the careers of women and minority mathematicians and scientists is an important component of IPAM's mission, and we welcome their applications. www.ipam.ucla.edu/programs/mbi2008
IPAM is an NSF funded institute A m e r i c a n M a t h e m a t i c a l S o c i e t y
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* The AMS will pay your California sales tax on all on-site AMS purchases. For many more publications of interest, visit the AMS Bookstore www.ams.org/bookstore Irving Kaplansky 1917–2006 Hyman Bass and T. Y. Lam
I liked the algebraic way of looking at things. Hyman Bass I’m additionally fascinated when the algebraic method is applied to infinite objects. Some Biographical Vignettes of Kap —Irving Kaplansky Mathematicians are conventionally measured by the depth and creativity of their contributions to Introduction research. On these grounds alone Kaplansky is a towering figure. But another, perhaps comparably On June 25, 2006, mathematics lost one of its important, way to contribute to the advancement leading algebraists, Irving Kaplansky. He passed of mathematics lies in the building of human away at age eighty-nine after a long illness, at the capacity, in the formation of productive young home of his son, Steven. Eight months earlier he researchers, through teaching, mentoring, and was still doing mathematics (Diophantine equa- written exposition. In this regard, Kaplansky, with tions). Steven’s question, “What are you working an astonishing fifty-five doctoral students (among whom I count myself), and 627 mathematical on, Dad?” brought only, “It would take too long to descendants, has had a singular impact on our explain.” From a generous teacher and elegant ex- field. positor who inspired generations of students and Kap was born March 22, 1917, in Toronto, young researchers, this utterance offers perhaps a the youngest of four children, shortly after his hint of the weary burdens of his final illness. parents had emigrated to Canada from Poland. His father, having studied to be a rabbi in Poland, “Kap”, as Kaplansky became universally known worked in Toronto as a tailor. His mother, with among friends and colleagues, was not only a little schooling, was enterprising and built up a brilliant research mathematician and teacher, but business, “Health Bread Bakeries”, that supported also an accomplished musician, a distinguished (and employed) the whole family. institutional leader, and a devoted husband and father. The remembrances and tributes that follow are from some of the many colleagues, students, friends, and family who Kap influenced and in- spired. We hope that they adequately convey the awesome breadth of Kap’s life and work—as a mathematician, teacher, writer, administrator, musician, and father—that we celebrate here.
Hyman Bass is professor of mathematics and mathematics education at the University of Michigan. His email address is [email protected]. T. Y. Lam is professor of mathematics at the University of California, Berkeley. His email address is [email protected]. Family photo, approximately 1921. Irving All photos in this article, except where otherwise noted, Kaplansky (second from left, see red arrow) and are courtesy of Alex Kaplansky. his parents and siblings.
December 2007 Notices of the AMS 1477 Kap showed an early and evolving talent for gigantic appointments—Saunders Mac Lane, An- music, as he himself recounts [1]: toni Zygmund, André Weil, and Shiing-Shen Chern —followed by waves of talented young faculty and At age four, I was taken to a Yiddish graduate students. Among the younger colleagues musical, Die Goldene Kala (The Golden who greatly influenced Kap were Irving Segal, Paul Bride). It was a revelation to me that Halmos, and Ed Spanier. there could be this kind of entertain- Kap’s life style, outside his family and music, ment with music. When I came home was rigorous and austere. He scheduled classes I sat down and played the show’s hit and meetings at (defiantly) early hours of the song. So I was rushed off to piano les- morning. Daily swimming was a lifelong practice; sons. After 11 years I realized there was he loved the Lake Michigan shoreline. Lunch was no point in continuing; I was not going lean in time as well as substance. With students to be a pianist of any distinction.…I he was generous and indulgent in mathematical enjoy playing piano to this day.…God conversation, but entertained little else. intended me to be the perfect accompa- After Chicago, Kap, succeeding Shiing-Shen nist—or better, the perfect rehearsal pi- Chern, became the second director of the Math- anist. I play loud, ematical Sciences Research Institute (MSRI) in I play in tune, but Berkeley, 1984–1992. Also in 1984, Kap was elected I don’t play very president of the AMS. So we see here a career tra- well. jectory from a precocious college student to a Indeed, Kap became dedicated, well established and prolific researcher, a popular accompa- to a leader of some of the premier institutions of nist and performer the profession. Along the way, Kap was honored by through much of his election to the National Academy of Sciences and career. At one point, to the American Academy of Arts and Sciences, and to demonstrate the vir- he was named an honorary member of the London tues of a structure he Mathematical Society. In 1989 the AMS awarded discovered common to him the Steele Prize, Career Award. First Putnam Fellow, his favorite songs, he To understand Kap’s mathematical accomplish- Harvard, around 1940. says, “I tried to show ments, it is important to speak of his students as that you could [use it well as his publications, to distinguish and com- to] make a passable pare what these two records tell us. Kap’s mathe- song out of such an unpromising source of the- matical work is distributed across several different matic material as the first 14 digits of π.” The areas of mathematics. For purposes of surveying resulting “Song about π” was later given lyrics them, I have somewhat arbitrarily grouped them by Enid Rieser and is often performed by Kap’s as follows, the major areas in bold font: daughter, Lucy, herself a popular folk singer- TA: Topological algebra, including operator songwriter [3]. Some more personal family vi- algebras, *-algebras, locally compact gnettes of Kap can be found below in Lucy’s remi- rings, etc. niscences of her father. Q: Quadratic and higher forms, both As a senior at the University of Toronto in 1938, abstract and arithmetic aspects Kap won the very first Putnam Competition, as did C: Commutative and homological algebra the Toronto team. This won him a fellowship to R: Ring theory (noncommutative) Harvard, where he earned his Ph.D. in 1941, under Lie: Lie theory—groups and algebras, the direction of Saunders Mac Lane. He stayed on including infinite dimensional and as a Benjamin Peirce Instructor till 1944, when characteristic p Mac Lane brought him to the Applied Mathematics #: Combinatorics and number theory Group at Columbia University in 1944–45, which M: Module theory, including abelian groups was doing work to support the war effort. Kap re- L: Linear algebra counts, “So that year was spent largely on ordinary G Miscellaneous, including general differential equations. I had a taste of real life and topology, group theory, game theory found that mathematics could actually be used for PS: Probability and statistics something.” In the following chronological chart I have From there Kap moved to the University of color-coded Kap’s journal articles, books, and Chicago in the fall of 1945, where he remained till monographs according to which of these areas his retirement in 1984, having chaired the depart- they belong. The data are taken from MathSciNet. ment during 1962–67. A year after Kap’s arrival, Not included are the numerous contributions to Marshall Stone came to Chicago to direct and build the Problem sections of the American Mathemati- up the mathematics department, ushering in what cal Monthly; Kap remained throughout a virtuoso some have called “the Stone Age”. Stone made four problem solver and contributor.
1478 Notices of the AMS Volume 54, Number 11 GRAPH OF KAPLANSKY’S PUBLICATIONS
‘99 U Toronto WWII work U Chicago ‘45 - ’84 Pres AMS ‘85 - ’86 Retired Putnam ‘38 ‘44-’45 Harvard @ Columbia U Dept Chair ‘62 - ’67 Director MSRI ‘84 - ’92 In Berkeley PhD ’41 30 PS 29 PS TA Topological algebra, operator algebras, etc. 28 PS Q Quadratic and other forms, both arithmetic and algebraic theory 27 M M (B) C Commutative and homological algebra 26 M M R Ring theory, including differential algebra 25 R M Lie Lie theory, including infinite dimensional 24 R R # Combinatorics, and some number theory 23 R R M Module theory, including abelian groups 22 R Q L Linear algebra 21 G Q G General, including general topology, group theory, and game theory 20 C Q PS Probability and statistics 19 # Q B = book or monograph 18 # TA 17 # TA 16 # TA 15 # TA 14 # TA 13 TA TA 12 TA TA L 11 TA TA GTop (B) Lie 10 TA TA Q Lie 9 TA TA TA (B) TA (B) Lie L 8 TA TA L (B) Lie (B) Lie G Q 7 TA TA DAlg (B) R (B) Lie (B) Lie G G Q 6 TA TA R M (B) R R G TA TA Q 5 PS TA TA Lie Lie M R (B) R Lie Lie Lie Q 4 PS TA TA M C C C (B) R Lie Lie Lie Q 3 # TA TA C C Q C (B) R L R L Q G 2 R TA TA TA C Q C (B) M R R R Q Q 1 R TA TA TA (B) C Q C C (B) R R Q Q Q #
December 2007 Notices of the AMS 1479
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super-symmetry, and related classification prob- field. In commutative and homological algebra, in lems. Peter Freund writes vividly below about their contrast, the field was already in rapid motion, collaboration. into which Kap boldly ventured as more of an ap- •Quadratic (and higher) forms: This subject, prentice, guiding a flock of similarly uninitiated from the beginning to the end of Kap’s career, graduate students and postdocs with him. was dear to his heart. This interest was first in- Homological algebra was spawned from alge- spired by his attending L. E. Dickson’s lectures in braic topology. In the hands of Eilenberg, Mac Lane, number theory and quadratic forms at Chicago Grothendieck, and others it evolved into a new in the 1940s. It was rekindled during the years of branch of algebra, embracing category theory and his retirement, when he turned to the arithmetic other new constructs. Meanwhile, the Grothen- theory of such forms, partly in collaboration with dieck-Serre reformulation of algebraic geometry William Jagy. A charming account of a significant demanded that its foundations in commutative piece of this work can be found in the contribution algebra be deepened and expanded. of Manjul Bhargava below. A basic new concept of homological algebra •In the eyes of many mathematicians today, was that of global homological dimension, a new commutative and homological algebra is the field ring-theoretic invariant. This turned out to be with which they now most associate Kaplansky’s uninteresting for the most investigated rings, finite name. Yet we see that its (yellow) color occupies dimensional (noncommutative) algebras. On the remarkably little of the chart of publications. How other hand, a landmark discovery (of Auslander- can we explain this paradox? Well, for one thing, Buchsbaum and Serre) was that, for a commutative Kap’s publications in this area include several noetherian local ring A, the global dimension of A books and monographs (lecture notes), and these is finite if and only if A is regular (the algebraic contain a number of new results and methods that expression of the geometric notion of nonsingu- were not elsewhere published. This also reflects larity). This equivalence, and Serre’s homological the fact that Kap was generating mathematics in formulation of intersection multiplicities, firmly this rapidly evolving field more through instruc- established homological algebra as a fundamental tion than through papers written in solitude. And tool of commutative algebra. so what he was producing mathematically was However, these developments were known significantly embodied in the work of the students mainly on a Cambridge (MA)–Paris–axis. It was in this context that Kap offered a Chicago graduate who were learning from him. course introducing these new ideas, methods, and •We can see this phenomenon in the preceding results, then still very much in motion. Use of these chart of Kap’s Ph.D. students, again color-coded by methods led to the general proof (by Auslander- the areas of their dissertations. Buchsbaum) of unique factorization for all regular The first thing to notice in comparing these two local rings. Kap’s course, and its sequels, lifted a charts is that the “relative masses” of topological whole generation of young researchers (myself algebra and commutative algebra have been ap- included) into this field. This played out for Kap proximately reversed, of course with a time shift. In over the next two decades, with many students and topological algebra, Kap was a pioneer and a major, several books to show for it. intensely productive, conceptual developer of the
1480 Notices of the AMS Volume 54, Number 11 In mathematical style, Kap was a problem solver …the thing that bedevils the mathemati- of great virtuosity. For course goals he sought cal profession—the difficulty we have problems, and theorems of great pedigree, and in telling the world outside mathemat- probed them deeply. His main focus was on proofs ics what it is that mathematicians do. (pathways), more than on theorems (destinations). And for shame, for shame, right within He sought geodesics, and the most economic (high mathematics itself, we don’t tell each mileage) means to get there. Proof analysis led to other properly. double-edged kinds of generalization/axiomatiza- And here is a sampling of how Kap was seen by tion: others, including some of his students: – A given proof yields more than claimed. The given hypotheses deliver more than the stated “He was not only a fantastic mathematician but a theorem promises. marvelous lecturer, and he had a remarkable talent – The hypotheses can be weakened. We can for getting the best out of students.” get the same results more cheaply, and so more —Richard G. Swan generally. The strength of this disposition was per- “I knew Kaplansky in his later years, and also haps sometimes over-zealous, pushing toward through some of his books. Cheerful, gracious, and “premature maturation” of the mathematics. But elegant are some of the words that come to mind it was an effective mode of instruction, yielding powerful conceptual command of the territory when I think of him.” covered. —Roger Howe As the record above indicates, and the testimo- nials below will affirm, Kap was a gifted teacher, “The mathematical community in India is shocked mentor, and writer. Here are a few of the things he to have news of the demise of Professor Irving Ka- himself has said in reflection on this. plansky. We all feel very sad at this irreparable loss. Professor Kaplansky was a source of inspiration I like the challenge of organizing my for mathematicians around the world. He will no thoughts and trying to present them in doubt live for all time through his mathematical a clear and useful and interesting way. contributions. We will miss his personal wit, charm On the other hand, to see the faces light and warm personality.” up, as they occasionally do, to even get —I. B. S. Passi, President, them excited so that maybe they can do Indian Mathematical Society a little mathematical experimentation themselves—that’s possible, on a limited “I did know about the work of Emmy Noether and scale, even in a calculus class. it may have influenced my choice of area, algebra, although I think the teaching of Irving Kaplansky Advice to students: “Look at the first was what really inspired me.” case, the easiest case that you don’t —Vera Pless understand completely. Do examples, a million examples, ‘well chosen’ ex- Kaplansky’s books “have one feature in common. amples, or ‘lucky’ ones. If the problem is The content is refreshing and the style of exposi- worthwhile, give it a good try—months, tion is friendly, informal (but at the same time maybe years if necessary. Aim for the mathematically rigorous) and lucid. The author less obvious, things that others have not gets to the main points quickly and directly, and likely proved already.” selects excellent examples to illustrate on the way.” And: “Spend some time every day learn- —Man Keung Siu ing something new that is disjoint from the problem on which you are currently “I learnt from his books in my youth, and would working (remember that the disjoint- not have survived without them. Even today, I ask ness may be temporary). And read the my students to read them, to learn the ‘tricks’ of masters.” the trade.” —Ravi Rao When a great mathematician has mas- tered a subject to his satisfaction and “Kaplansky was one of my personal heroes: during is presenting it, that mastery comes my student years, I discovered his little volume on through unmistakably, so you have abelian groups and noticed that algebra too has an excellent chance of understanding stories to tell…” quickly the main ideas. [He cites as ex- —Birge Huisgen-Zimmermann amples, Weil, Serre, Milnor, Atiyah.]
December 2007 Notices of the AMS 1481 Kap as a Thesis Advisor: “I was very young and very Kap offered me my first job as an instructor at immature when I was Kap’s student. I’m deeply Chicago were perhaps not statistically independent indebted to Kap for putting up with me and help- events. The time was forty years ago, when Kap was ing me to develop in my own eccentric way. I asked finishing his five-year term as Chicago chair. The Kap for a thesis problem that didn’t require any offer was consummated by a Western Union tele- background and, surprisingly, he found one with gram—the 1960s equivalent of email. Kap didn’t enough meat in it to allow me to get a feeling for ask for my C.V. (I wouldn’t have known what that doing research. was); nor did he want to know my “teaching phi- “It wasn’t until I had my own thesis students losophy” (I had none). For my annual salary, Kap that I realized how hard it must have been to ac- offered me US$8,000—a princely sum compared commodate my special needs and help me develop to my then T.A. stipend of US$2,000 at Columbia in my way, not in his way.” University. I have joked to my colleagues that I’ll —Donald Ornstein (Kap Ph.D., 1957) always remember Kap as the only person through my whole career to have ever quadrupled my For me Kap’s transition from course instructor salary. But in truth, a ticket to Chicago’s famed to thesis advisor was almost imperceptible, since Eckhart Hall for postdoctoral studies was more I had become deeply engrossed than anything a fledgeling algebraist could have in his courses on commuta- dreamed. For this wonderful postdoctoral experi- tive and homological algebra ence Kap afforded me through his unconditional and questions about projective confidence in a mathematical grandson, I have modules, an exciting territory always remained grateful. wide open for exploration, and for which Kap had laid a solid I met Kap for the first time in the fall of 1967 groundwork. He did float a few when I reported to work in Hyde Park. By that time, other problems to me, such as Kap had already taught for twenty-two years at the the structure of certain infinite University of Chicago. Although he was Canadian dimensional Lie algebras, whose by birth, Chicago had long been his adopted home significance I only later came to and workplace: it is, appropriately, the city where appreciate. But I didn’t take that our “tale” begins. bait then. He was a generously For students interested in abstract algebra, Ka- available and stimulating advi- plansky is virtually a household name. In graduate sor, often sharing promising school, I first learned with great delight Kap’s mar- ideas that he had not yet had velous theorem on the decomposition of projective Chair, University of Chicago time to pursue. What I remem- modules, and his surprisingly efficient treatment Math Department, ber most of that time is the of homological dimensions, regular local rings, and 1962–1967. brilliance of his courses, and the UFDs. It was to take me forty more years, however, richness and excitement of the to get a fuller glimpse of the breadth and depth of mathematical milieu that he had created among Kap’s total mathematical output. In these days of his many students then. This milieu powerfully increasing specializations in mathematics, we can amplified the many mathematical resources that only look back in awe to Kap’s trail-blazing work Kap had to offer. I think that it is fair to say that through an amazingly diverse array of research Kap’s students are an important part of his oeuvre. topics, ranging from valuation theory, topological One could hardly have asked for a better teacher algebra, continuous geometry, operator algebras and advisor. and functional analysis, to modules and abelian References groups, commutative and homological algebra, P.I. rings and general noncommutative rings, infinite- [1] Donald J. Albers, Interview with Irving Kaplansky, The College Mathematics Journal 22 (1991), no. 2, pp. dimensional Lie algebras, Lie superalgebras (su- 99–117. persymmetries), as well as the theory of quadratic [2] Richard V. Kadison, Irving Kaplansky’s role in mid- forms in both its algebraic and arithmetic flavors. twentieth century functional analysis, Notices of the Kap was master of them all. In between the “big- AMS (to appear). ger” works, Kap’s publications also sparkled with [3] Ivars Peterson, A song about π, MathTrek, MAA an assortment of shorter but very elegant notes, Online, July 12, 1999. in number theory, linear algebra, combinatorics, statistics, and game theory. All of this, still, did not include the many other works recorded in T. Y. Lam “fourteen loose-leaf notebooks” (referred to in the preface of [1]) that Kap had kept for himself over Kap: A Tale of Two Cities the years. One cannot help but wonder how many Through Professor Hyman Bass, Kap was my more mathematical gems have remained hidden mathematical grandfather. This and the fact that in those unpublished notebooks!
1482 Notices of the AMS Volume 54, Number 11 For me, reading one of Kap’s papers has always proved to be a richly rewarding experience. There Books of Irving Kaplansky are no messy formulas or long-winded proofs; Infinite Abelian Groups, 1954, 1969 instead, the reader is treated to a smooth flow of An Introduction to Differential Algebra, 1957, 1976 novel mathematical ideas carefully crafted to per- Introduction to Galois Theory (in Portuguese), 1958 fection by an artisan’s hand. Some authors dazzled Rings of Operators, 1968 us with their technical brilliance; Kap won you Fields and Rings, 1969, 1972 over by the pure soundness of his mathematical Linear Algebra and Geometry. A Second Course, 1969, thought. In his publications, Kap was much more 1974 given to building new conceptual and structural Commutative Rings, 1970, 1974 frameworks, than going down single-mindedly Lie Algebras and Locally Compact Groups, 1971, 1974 into a path of topical specialization. This style Set Theory and Metric Spaces, 1972, 1977 of doing mathematics made him a direct intel- Matters Mathematical (with I. Herstein), 1978 lectual descendant of Emmy Noether and John Selected Papers and Other Writings, 1995 von Neumann. As a consequence, many of Kap’s mathematical discoveries are of a fundamental worked. Many of Kap’s fifty-five Ph.D. students nature and a broad appeal. The famous Kaplansky from Chicago are now on the senior faculty at Density Theorem for unit balls and his important major universities in the U.S. Currently, the Math- inaugural finiteness result in the theory of rings ematics Genealogy Project listed Kap as having with polynomial identities are only two of the most 627 descendants—and counting. This is the second outstanding examples. highest number of progeny produced by mathema- Those of us who have had the privilege of lis- ticians in the U.S. who had their own Ph.D. degrees tening to Kap all knew that he was extremely well awarded after 1940. We leave it as an exercise for spoken and had indeed a very special way with the reader to figure out who took the top honor in words. However, this gift did not always manifest that category, with the not-too-useful hint that this itself when Kap was in social company with Chellie. mathematician was born a year after Kap. It was quite clear to all his colleagues who Kap While Kap had clearly exerted a tremendous in- thought was the better orator in the family. Din- fluence on mathematics through his own research ner parties the Kaps attended were often replete work and that of his many Ph.D. students, the with Chellie’s amusing stories about the Chicago books written by him were a class by themselves. department and its many colorful mathematical The eleven books listed in the sidebar on this page personalities, from an austere André Weil down traversed the whole spectrum of mathematical to the more transient, sometimes bungling gradu- exposition, from the advanced to the elementary, ate students over the years. As Chellie recounted reaching down to the introduction of mathemat- such funny stories with her characteristic zest and ics to non-majors in the college. Differential Al- candor, Kap would listen admiringly on the side gebra typified Kap’s broad-mindedness in book —without interruption. Only at the end of a story writing, as its subject matter was not in one of would he sometimes add a clarifying comment, Kap’s specialty fields. On the other hand, Infinite perhaps prompted by his innate sense of math- Abelian Groups introduced countless readers to ematical precision, such as “Oh, that was 1957 the simplicity and beauty of a subject dear to summer, not fall.” Kap’s heart, while Rings of Operators served as a Kap’s extraordinary gift in oral (and written) capstone for his pioneering work on the use of al- expression was to find its perfect outlet in his gebraic methods in operator algebras. Lie Algebras, teaching, in which it became Chellie’s turn to play Commutative Rings, as well as Fields and Rings, all a supporting role. In the many lecture courses originating from Kap’s graduate courses, extended Kap gave at the University of Chicago in a span his classroom teaching to the mathematical com- of thirty-nine years, he introduced generation munity at large, and provided a staple for the edu- after generation of students to higher algebra and cation of many a graduate student worldwide, at a analysis. In those courses he taught that were of time when few books covering the same materials an experimental nature, Kap often directly inspired at the introductory research level were available. his students to new avenues of investigation, and In these books, Kap sometimes experimented with even to original mathematical discoveries at an rather audacious approaches to his subject mat- early stage. (Schanuel’s Lemma on projective reso- ters. For instance, Commutative Rings will probably lutions, proved by Stephen Schanuel in Kap’s fall go down on record as the only text in commutative 1958 Chicago course in homological algebra, was algebra that totally dispensed with any discussion perhaps the best known example.) It was thus no of primary ideals or artinian rings. accident that Chicago graduate students flocked As much as his books are appreciated for their to Kap for theses supervision. Over the years, Kap valuable and innovative contents, Kap’s great fame directed doctoral dissertations in almost every one as an author derived perhaps even more from his of the mathematical fields in which he himself had very distinctive writing style. There is one com-
December 2007 Notices of the AMS 1483 mon characteristic of Kap’s books: they were all cally with Kap—discussing with him issues in short—something like 200 pages was the norm. quadratic forms and ring theory. Kap seemed to (Even Selected Papers [1] had only 257 pages, by favor the written mode of communication (over his own choice.) Kap wrote mostly in short and the oral), but his letters were just as concise as simple sentences, but very clearly and with great his books. I still have in my prized possession an precision. He never belabored technical issues, and almost comical sample of Kap’s terseness, in the always kept the central ideas in the forefront with form of a covering letter for some math notes he an unerring didactic sense. The polished economy sent me. Written out on a standard-size 8½ by of Kap’s writing makes it all at once fresh, crisp, 11 MSRI letterhead, the letter consisted of twelve and engaging for his readers, while his mastery words: “Dear Lam: I just did a strange piece of ring and insight shone on every page. The occasional theory. Kap.” It was briefly—but of course unam- witty comments and asides in his books—a famous biguously—dated: “Apr. 11 /97”. Kaplansky trademark—are especially a constant Another interaction with Kap in 1998 led to source of pleasure for all. In retrospect, Kap was some mathematical output. In preparation for a not just a first-rate author; he was truly a superb special volume in honor of Bass’s sixty-fifth birth- expositor and a foremost mathematical stylist of day, I was very much hoping to commission an his time. article from Kap. In his usual self-effacing fashion, After I moved from Chicago to Berkeley, my Kap protested that he had really nothing to write contacts with Kap became sadly rather infrequent. about. However, after much persuasion on my part So imagine my great surprise and delight, sixteen (stressing that he must write for Bass), he gave in years later, when word first came out that Kap and wrote up in his impeccable hand a short note was to retire from the University of Chicago, in in number theory [2]. Glad that my tactics had paid order to succeed Chern as the director of MSRI! off, I worked all night to set Kap’s written note in In the spring of 1984, the Kaplanskys arrived and TEX, and delivered a finished printout to him early established their new abode a few blocks north of the next morning. Kap was surprised; he thanked the university campus—in Berkeley, California, the me profusely, but said that maybe he shouldn’t second city of our tale. have written his article. It was too late. The math departments at Chicago and Berkeley One of Kap’s best known pieces of advice to share much more than the “U.C.” designation of young mathematicians was to “spend some time the universities to which they belong. There has every day learning something new that is disjoint been a long (though never cantankerous) history from the problem on which you are currently of the Berkeley department recruiting its faculty working,…and read the masters” [3]. Amazingly, from the Chicago community, starting many years even after reaching his seventies, Kap still took ago with Kelley, Spanier, and Chern. Indeed, when his own advice personally and literally. In all the Kap himself joined the U.C.B. faculty in 1984, there years he was in Berkeley, Kap made it his habit to were at least as many as sixteen mathematicians go to every Monday’s Evans-MSRI talk and every there who had previously been, in one way or an- Thursday’s Math Department Colloquium talk. He other, associated with the University of Chicago. even had a favorite seat on the left side of the front It must have given Kap a tinge of “nostalgia” to row in the colloquium room, which, in deference to be reunited, in such an unexpected way, with so him, no local Berkeley folks would try to occupy. many former graduate students, postdocs, and In the years 1995–97 when I worked at MSRI, I saw colleagues from his beloved Chicago department. Kap quite frequently at the periodicals table in the But if anyone had speculated that, by coming West, library, poring over the Mathematical Reviews to Kap was to spend his golden years resting on his keep himself abreast with the latest developments laurels, he or she could not have been more wrong. in mathematics. And he read the masters too, In fact, as soon as Kap arrived at Berkeley in 1984, e.g., in connection with his work on the integral he was to take on, unprecedentedly, two simultane- theory of quadratic forms. Members of MSRI have ous tasks of herculean proportions: (a) to head a reported sightings of Kap using a small step- major mathematics research institute in the U.S., ladder in the library to reach a certain big book on and (b) to preside over the largest mathematical a high shelf, and putting the book back in the same 1 society in the world—the AMS . fashion after using it (instead of leaving it stray Other contributors to this memorial article are on a table). That tome was an English translation in a much better position than I to comment on of Disquisitiones Arithmeticae: the fact that even Kap’s accomplishments in (a) and (b) above, so I a six-foot-tall Kap needed a step-ladder to access defer to them. In the following, my reminiscences it was perhaps still symbolic of the lofty position on Kap’s Berkeley years are more of a personal of the work of the twenty-year-old Carl Friedrich nature. From 1984 on, I certainly had more oc- Gauss. casions than ever before to interact mathemati- My two-years’ stay at MSRI was rich with other 1Kaplansky served as AMS president-elect in 1984, and remembrances about Kap. Undoubtedly, a high- president for the 2-year period 1985–86. light was Kap’s eightieth birthday fest in March
1484 Notices of the AMS Volume 54, Number 11 1997, which was attended by three MSRI directors Richard Kadison and six MSRI deputy directors, as well as visiting dignitaries such as Saunders Mac Lane, Tom Lehrer, Letter from Richard Kadison to Section 11 and Constance Reid. Another most memorable gathering was the holiday party in December 1996, (Mathematics) of the National Academy of where a relaxed and jovial Kap sang some of his Sciences signature songs for us all, accompanying himself (Addressed to the Chair, Paul Rabinovich, July 1, on the piano in the MSRI atrium. His energetic, 2006) sometimes foot-stomping performance really brought down the house! It saddens me so much Dear Paul, to think that, now that Kap is no longer with us, these heart-warming events will never be repeated Just about ten minutes ago, I sat down to my again. email; I had looked at it at about 9:30 a.m.—before Twenty years may have been only about a third the sad news about Kap arrived. So, I saw the mes- of Kap’s professional life, but I hope that Kap cher- sage appended below (you have it as well) only a ished his twenty years in Berkeley with as much few minutes ago. I was shocked by the news. “Sad” fondness as he had cherished his thirty-nine years really doesn’t begin to describe my feelings; Kap in Chicago. Those were the two cities (and universi- was almost as close, where I’m concerned, as a ties) of his choice, for a long and very distinguished beloved parent. Of all my graduate school teachers career in mathematics. In Chicago, Kap was a re- (Stone, Zygmund, Chern, Spanier, Halmos, Segal, searcher, a chairman, a teacher, a mentor, and an Weil, Graves, Hestenes, Mac Lane, Albert, etc.), and author. In Berkeley, while remaining a steadfast I revered each and every one of them, Kap was my researcher, Kap also became a scientific leader, favorite. A half-hour-to-hour conversation with a senior statesman, and a universal role model. him about mathematics generated so much excite- In each of these roles, Kap served his profession ment that I spent the rest of the day walking on a with devotion, vigor, wisdom, and unsurpassed cloud. Irv was immensely popular with the gradu- insight. His lifetime work has profoundly impacted ate students; he was always ready to talk math twentieth century mathematics, and constituted with us and make good and useful suggestions for for us an amazingly rich legacy. our work, but he was also somewhat “scary” for many of the students. His “social” behavior was On a personal level, Kap—mathematical grandpa even more peculiar than the “standard” behavior and algebraist par excellence—will continue to oc- of dedicated mathematicians. Most of us have an cupy a special place in my heart. I shall miss his exaggerated sense of the “futility” of small talk; great generosity and easy grace, but thinking of Irv’s view of that had to be described as “exces- Kap and his towering achievements will always sive”. For example, if you met him in the hallway enable me to approach the subject of mathematics and stopped for a conversation with him, when with hope and joy. the conversation was clearly over, he just walked on, turned and walked away, whatever—absolutely References no decompression stage (or phrases, e.g., the cur- [1] I. Kaplansky, Selected Papers and Other Writings, rently popular, and almost always, fatuous “have Springer-Verlag, Berlin-Heidelberg-New York, 1995. a nice day”—recently inflated to “have a great [2] , A salute to Euler and Dickson on the occasion ——— day”). Handshakes? Forget it! As fast and smart of Hy’s 65th birthday, in Algebra, K-theory, groups, and education (T. Y. Lam, A. Magid, eds.), Contemp. and creative as he was, and all that (genuine, not Math. 243 (1999), pp. 79–81, Amer. Math. Soc., Provi- affected) no-nonsense behavior of his, we loved dence, R.I. (“worshipped” might be more accurate) him. Chatting with him in his office, after a few years, [3] ——— , Response to Steele Prize citation (Career Award), Notices of the Amer. Math. Soc. 36 (1989), p. he asked me a nice question that had occurred 836. to him, a fine blend of algebra and analysis (nil- potents of index 2 and approximation). I thought about it for fifteen minutes or so that evening and didn’t see how to get started. Being busy with other things I dropped it and didn’t get back to it until I met and talked to him a day or two later. He asked me if I had thought about the problem. I said that I had, but hadn’t been able to get started on it, and then asked him if he really thought it
Richard Kadison is the Gustave C. Kuemmerle Professor of Mathematics at the University of Pennsylvania. His email address is [email protected].
December 2007 Notices of the AMS 1485 student days, we took too long with it. I remember a bunch of us walking down the stairs of Eckhart Hall on our way over to lunch at the Commons. Irv came bouncing past us, evidently on the same mission. The Commons is a few hundred meters from Eckhart. We sauntered over arriving in time to see Kap emerging from the Commons, lunch over. Chellie probably slowed him down over the years. In my first years at Chicago, Irv had no discernible social life. He liked swimming in Lake Michigan during the summer and did so early each morning. Then, in 1949 he had a few quarters off and went out for a stay at UCLA. The rumors flew back to Chicago, Kap had bought himself a convertible, now drank liquor, socially, and smoked. One day it was said that the “new” Kaplansky had returned. A day later, I happened on my dear pal and fellow graduate student, Arnold Shapiro. He told me that Kap and Chellie Kaplansky on their wedding day, 1951. he had just talked with Irv a few hours ago. I asked, “The new Kaplansky?” Arnold’s reply was, “What was true. His response was, “When God whispers new Kaplansky? It’s just the old Kaplansky—with a theorem in your ear, you should listen.” Now, of a smile on his face.” Shortly after that, a few of us course I understood his “cute” way of giving me finished our Ph.D. requirements. As tradition had some valuable mathematical advice, but I chose to it, we invited one and all to a party. Walking around misinterpret it. When I reported this to the other with a tray and two drinks (“highballs”) on it, one graduate students, I told them the story and added primarily scotch and the other bourbon, I offered that Kaplansky had finally revealed himself, and one of those drinks to Irv. His question for me was, as many of us had suspected, he was God. In those “Which is the perfume and which is the hair tonic?” very early years (end of the 1940s), Irv lived an aus- That, apparently, was the “new” Kaplansky. Of tere life. He rented a single room in a house near course, I could tell you so many more stories about the U. of Chicago campus, paid $5 a week for it, if Irv, many of them that have some mathematical I remember correctly, and saved almost all the rest significance. They are all memories I treasure. Kap of his salary. The word was that he was (relatively) is one of the very few people I’ve known well most wealthy in those days. (Of course, that could mean of my working life of whom I can say that I have anything from someone with a bank balance of nothing but enjoyable memories. $100 and up and no debts, in those days and our —Kindest, society.) Chellie (Rochelle), Irv’s wife of fifty-five Dick years was tremendous fun, great sense of humor. I had the impression that she could wrap him around her little finger, he knew it, and he enjoyed Peter G. O. Freund it. She entered the picture in 1951. Some years later 2 (about five), George Mackey was having dinner Irving Kaplansky and Supersymmetry with Karen and me at our apartment in Cambridge I arrived in Chicago some two decades after Irving (we were visiting MIT that year). The conversation Kaplansky, and I met Kap, as we all called him, turned to Irv. (Kap and George were great friends.) shortly after my arrival here. We became friends When the subject of Kap’s purported wealth came later, in 1975, while collaborating on a paper on up, George told of a conversation he and Chellie supersymmetry. Lie superalgebras, graded coun- had had some little time back. He said that he had terparts of ordinary Lie algebras, play a central asked Chellie if she didn’t feel that she was lucky role in string theory and other unified theories. A to have married a wealthy man—to which she re- classification of the simple ones was of essence. I plied, with a (feigned) surprised smile, “Oh, that—it took some initial steps, but the real work started was only about $30,000 and I went thru that in no when Yitz Herstein put me in touch with Kap. At time!” George paused after reporting that, assumed first, communication was not easy. We couldn’t a troubled, somber look and said, “Fair, sent a chill quite make out each other’s reasoning, much as down my spine!” It probably helps to know that, in we agreed on results. It didn’t take long however, those days, George was still a bachelor, and lived a frugal, austere existence—completely by choice. Peter G. O. Freund is professor emeritus in the department Both Kap and Mackey were perfectly willing to of physics at the University of Chicago. His email address spend their money when the occasion warranted is [email protected]. 2 it. I’ve had many fine meals with each of them. Based on remarks at the Irving Kaplansky Memorial at Kap did not eat lunch with us during our graduate MSRI, Berkeley, CA, February 23, 2007.
1486 Notices of the AMS Volume 54, Number 11 to get used to the other’s way of looking at things. early work, Killing had found almost all of them, Mathematicians and physicists think in similar but he “somehow overlooked one,” namely the ways after all, all that was needed was a dictionary. exceptional 52-dimensional simple Lie algebra This was during the early phase of the rapproche- F4, which remained to be discovered later by Élie ment between mathematics and theoretical phys- Cartan. Apparently, G(3) is the exceptional Lie ics. After the glorious first half of the twentieth superalgebra which carries on that curse of the century—when the likes of Poincaré, Hilbert, Weyl, ordinary exceptional Lie algebra F4. von Neumann, Élie Cartan, Emmy Noether, and I mentioned the almost total lack of contact others made major contributions to the then-new between theoretical physicists and mathemati- physical theories of general relativity and quantum cians, when this work got going. It went so deep mechanics, while physicists like Jordan, Dirac, that in 1975 most physicists, if asked to name Casimir, and Feynman made major contributions a great modern mathematician, would come up to mathematics—physics entered a period best with Hermann Weyl, or John von Neumann, both described as phenomenological. During this pe- long dead. Mathematicians had it a bit easier, for riod, some advanced complex function theory if they read the newspapers, they could at least aside, very little modern mathematics was drawn keep track of the Nobel Prizes, whereas newspaper on. To give you an idea, when in his celebrated editors rarely treated Fields Medal awards as “news “Eightfold Way” paper, Murray Gell-Mann wrote fit to print.” down a basis of the three-dimensional representa- I recall that while standing by the state-of-the- tion of the su(3) Lie algebra, this was heralded by art Xerox machine to produce some ten copies of physicists as a great mathematical feat. “Imagine, our paper in about…half an hour’s time, I asked he found a 3 × 3 generalization of the famous 2 Kap, “Who would you say, is the greatest mathema- × 2 Pauli matrices,” is what most people said. To tician alive?” He immediately took me to task: my get there, Murray had consulted with Block and question was ill-defined, did I mean algebraist, or Serre! topologist, or number-theorist, or geometer, or dif- It was in the fields of supersymmetry and gauge ferential geometer, or algebraic geometer, etc.…I theory that the initial steps in modern mathemati- replied that I did not ask for a rigorous answer, cal physics were taken. This convergence of the but just a “gut-feeling” kind of answer. “Oh, in that paths of mathematics and of theoretical physics case the answer is simple: André Weil,” he replied, is typical of times when major new physical theo- without the slightest hesitation, a reply that should ries—gauge theory and string theory in this case not surprise anyone, who has heard today’s talks. —are being born. The earliest example of such “You see,” Kap went on, “We all taught courses a convergence is the creation of calculus at the on Lie algebras or Jordan algebras, or whatever birth of Newton’s mechanics and of his theory we were working on at the time. By contrast, Weil of gravitation. Weyl’s spectacular work on group called all the courses he ever taught simply ‘math- theory under the impact of the newborn quantum ematics’ and he lived up to this title.” mechanics is another such example. Kap went on to tell me about Weil’s legendary A few words about our joint paper [1] are in first colloquium talk in Chicago. This was the first order here. In it we found all the infinite families of time I heard that very funny story. Weil had been simple Lie superalgebras, as well as 17-, 31- and 40- recruited for the Chicago mathematics department dimensional exceptional ones. We also discussed by its chairman, Marshall Stone. With Stone sitting real forms and explained why supersymmetry can in the first row, Weil began his first Chicago col- act on 4-dimensional anti-de Sitter but not on de loquium talk with the observation, “There are three Sitter space, a result essential for understanding types of department chairmen. A bad chairman will why the remarkable duality discovered by Malda- only recruit faculty worse than himself, thus lead- cena [2] in the 1990s, is of the AdS/CFT and not of ing to the gradual degeneration of his department. the dS/CFT type. We were convinced that we had A better chairman will settle for faculty roughly of found all simple Lie superalgebras (as we actually the same caliber as himself, leading to a preserva- had), but we lacked a proof of this fact. The proof tion of the quality of the department. Finally, a came from the powerful independent work of good chairman will only hire people better than Victor Kac [3]. Amusingly, in his beautiful proof, himself, leading to a constant improvement of his Kac somehow overlooked one of the exceptional department. I am very pleased to be at Chicago, superalgebras, namely the 31-dimensional super- which has a very good chairman.” Stone laughed algebra G(3), whose Bose (even) sector consists of it off; he did not take offense. the ordinary Lie algebra g2 + sl (2), the only simple The lack of communication between mathema- Lie superalgebra to have an exceptional ordinary ticians and physicists was to end soon. By 1977, Lie algebra as one of the two constituents of its we all knew about Atiyah and Singer, and then Bose sector. I said “amusingly” above because, as I the floodgates came down fast, to the point that learned from Kap, in the classification of ordinary an extremely close collaboration between math- simple Lie algebras, in his extremely important ematicians and physicists got started and, under
December 2007 Notices of the AMS 1487 the leadership of Ed Witten and others, is ongoing skills would be essential ingredients in making this and bearing beautiful fruit to this day. By the way, program the great success that it was. on Kap’s desk I noticed some work of his on Hopf Almost at the same time, the Board of Trustees algebras. I asked him about Hopf algebras, and got of MSRI selected Kap to succeed Shiing-Shen Chern the reply, “They are of no relevance whatsoever for as director of MSRI in 1984. We served together at physics.” I took his word on this, was I ever gullible. MSRI, he as director and I continuing as deputy In the wake of our joint work, Kap and I became director for a year before I left MSRI for an ad- good friends. This friendship was fueled also by ministrative post in the University of California. It our shared love of music; he was a fine pianist, and was a wonderful learning and teaching experience I used to sing. For me, the most marvelous part of for both of us. I learned much from Kap’s wisdom my collaboration and friendship with Kap was that and experience, and I in turn tried to convey to for the first time I got to see up-close how a great him what I knew about MSRI operations. I subse- mathematician thinks. quently watched more from a distance, and it was clear that MSRI grew and prospered under his eight References years of excellent leadership as director. He also [1] P. G. O. Freund and I. Kaplansky, Simple supersym- maintained a lively research program while serv- metries, J. Math. Phys. 17 (1976), 228–31; I. Kaplan- ing as director and for many years after stepping sky, Superalgebras, Pacific J. Math. 86 (1980) 93–8. down. We all miss this generous and wise man of [2] J. M. Maldacena, The large N limit of superconformal many talents. theories and supergravity, Adv. Theor. Math. Phys. 2 (1988), 231–52; archive hepth/9711200. Susanna S. Epp and E. Graham [3] V. G. Kac, Classification of simple superalgebras, Funct. Anal. (English) 9 (1975), 263-5; ibid. 10,163 Evans Jr. (1976); A sketch of Lie superalgebra theory, Comm. Math. Phys. 53 (1977) 31–64. Kap as Advisor We are two of the fifty-five students who com- Calvin C. Moore pleted a doctorate with Kaplansky between 1950 and 1978. This is an astonishing number. Indeed Kap Encounters in Chicago and Berkeley during the years 1964–1969, when we were at the University of Chicago, Kap oversaw an average I first “met” Kap mathematically when I was a of three completed dissertations a year despite graduate student at Harvard working in functional serving as department chair from 1962–1967. His analysis and read and studied his striking 1951 secret, we think, was an extraordinary instinct for papers on C*-algebras in which he defined and productive avenues of research coupled with a explored the properties of what he called CCR generous willingness to spend time working with and GCR algebras. His algebraic insight into these his students. He also often encouraged students objects arising in analysis turned out to be of to run a seminar, with beginning students present- seminal importance and indeed were years ahead ing background material and advanced students of their time. I also read about what was by then presenting parts of their theses. called the Kaplansky density theorem for von When Evans worked with him, Kap was teach- Neumann algebras dating from 1952 and studied ing the commutative algebra course that was his wonderful 1948 paper on groups with repre- published soon afterward by Allyn and Bacon. sentations of bounded degree and its connection As with each course he taught, he filled it with with polynomial identities. new thoughts about the subject. For instance, I was eager to meet this algebraist whose work at one memorable point he experimented to see had been so influential in my own studies in a very how much he could deduce if he knew only that different field, and I had that opportunity when Ext1(A,B) was zero. He managed to get pretty I had a postdoctoral appointment at Chicago in far, but eventually the proofs became unpleas- 1960–61. However, I soon left Chicago for Berkeley, antly convoluted. So he abruptly announced that and it was many years before our paths crossed henceforth, he would assume the full structure of again. When we were planning a full year program Extj(A,B), and the next day he resumed lecturing in at the Mathematical Sciences Research Institute his usual polished fashion. This episode was atypi- (MSRI) in 1983-84 on the topic of infinite dimen- cal in that he first developed and then cut off a line sional Lie algebras, Kap’s broad and deep insight of inquiry. More frequently, after commenting on and understanding in algebra led us to select him Susanna S. Epp is Vincent de Paul Professor of Mathemat- as the chair of the program committee. We also ics at DePaul University. Her email address is sepp@ recognized that his subtle and effective diplomatic condor.depaul.edu. Calvin C. Moore is emeritus professor of mathematics at E. Graham Evans Jr. is emeritus professor of mathemat- the University of California, Berkeley. His email address ics at the University of Illinois at Urbana-Champaign. His is [email protected]. email address is [email protected].
1488 Notices of the AMS Volume 54, Number 11 new insights of his own, he would interject ques- on the weather, Kap responded with a smile, “Cut tions for students to explore and develop. In his the crap. Let’s talk mathematics.” They did, and he lectures he made the role of non-zero divisors, and became one of the many students Kap mentored hence regular sequences, central in the study of long after he retired. commutative rings. At one point he gave an elegant proof, avoiding the usual filtration argument, that Joseph Rotman the zero divisors are a finite union of prime ide- als in the case of finitely generated modules over Student Memories of Kap a Noetherian ring. Then he asked Evans to try to determine what kinds of non-Noetherian rings As a graduate student at the University of Chi- would have the property that the zero divisors cago, I attended many of Kaplansky’s elementary of finitely generated modules would always be a courses: complex variables, group theory, set finite union of primes. One of the ideas in Kap’s theory, point-set topology; later, I attended more proof was just what Evans needed to get the work advanced courses: commutative algebra, Hilbert’s on his thesis started. fifth problem, abelian groups, homological algebra. The year that Epp worked with Kap, he was not Every course, indeed, every lecture, was a delight. teaching a course but had gone back to a previ- Courses were very well organized, as was each lec- ous and recurring interest in quadratic forms. A ture. Results were put in perspective, their applica- quintessential algebraist, he was interested in ex- tions and importance made explicit. Humor and ploring and expanding classical results into more droll asides were frequent. Technical details were abstract settings. Just as in his courses he tossed usually prepared in advance as lemmas so as not to out questions for further investigation, in private cloud the main ideas in a proof. Hypotheses were sessions with his students he suggested various stated clearly, with examples showing why they lines of inquiry beyond his own work. In Epp’s case were necessary. The exposition was so smooth and this meant exploring the results Kap had obtained exciting that I usually left the classroom feeling in generalizing and extending H. Brandt’s work on that I really understood everything. To deal with composition of quaternary quadratic forms and such arrogance, Kap always assigned challenging trying to determine how many of these results problems, which made us feel a bit more humble, could be extended to general Cayley algebras. but which also added to our understanding. He was Kap typically scheduled an early morning a wonderful teacher, both in the short term and for weekly meeting with each student under his di- the rest of my mathematical career. His taste was rection. For some it was much earlier than they impeccable, his enthusiasm was contagious, and would have preferred, but for him it followed a he was the model of the mathematician I would daily swim. He led our efforts mostly by express- have been happy to be. ing lively interest in what we had discovered since Kap was my thesis advisor. I worked in abe- the week before and following up with question lian groups (at the same time, he had five other after question. Can you prove a simpler case? Or a advisees: two in homological algebra and three in more general one? Can you find a counterexample? functional analysis). He set weekly appointments When one of us arrived disappointed one day, hav- for me. When I entered his office, he was usually ing discovered that a hoped-for conjecture was sitting comfortably at his desk, often with his feet false, Kap said not to be discouraged, that in the up on the desk. He’d greet me with “What’s new?” search for truth negative results are as important I would then talk and scribble on the blackboard as positive ones. He also counseled persistence in as he listened and asked questions. Once I had other ways, commenting that he himself had had axiomatized a proof of his and Mackey’s, enabling papers rejected—a memorable statement because me to generalize their result. “How did you think it seemed so improbable. Having made contribu- of that?” he asked. I replied that that was the way tions in so many fields and having experienced he had taught me to think; he smiled. the benefits of cross-fertilization, he advised Both of us spent a sabbatical year in London being open to exploring new areas. Some of his at Queen Mary College. Of course, I continued to students may have taken this advice further than enjoy his mathematics, but I saw another side of he perhaps intended, ultimately working far from him as well. N. Divinsky was another sabbatical their original topic areas at the National Security visitor (as was H. Flanders), and I was dubbed Rot- Agency, at the Jet Propulsion Laboratory, and in mansky to go along with Kaplansky and Divinsky. K–12 mathematics education, for example. Kap discovered cricket, and often went to Lord’s Cricket Grounds. But Kap really loved Gilbert and Kap derived a great deal of pleasure from hav- Sullivan. He arranged an evening in which we per- ing generated 627 mathematical descendants, formed Iolanthe. Kap was at the piano, Divinsky perhaps especially from meeting his mathemati- cal grandchildren and great-grandchildren. When Joseph Rotman is emeritus professor of mathematics at one encountered him at the MSRI bus stop one the University of Illinois at Urbana-Champaign. His email day and, not knowing what to say, commented address is [email protected].
December 2007 Notices of the AMS 1489 did the patter songs, Flanders was on the recorder, and I got into a “discussion” on how Kap (of Ca- and I was Strephon. nadian origin like Yitz) pronounced “schedule”. I There are few giants in the world, and now there maintained that Kap would pronounce it with an is one less. “sk” as Americans do and Yitz, of course, said that Kap would say “shedule”, as Canadians and Britons Lance Small do. So, Yitz and I bet a quarter. When Kap arrived at tea, Yitz and I bounded up to him and told him Kap as Teacher and Mentor of our bet. Kap thought for an instant and, then, carefully pronounced “skedule” remarking that I was not a student of Kaplansky—at least, not faculty shouldn’t take money from students and in the sense we usually mean in mathematics. He that Yitz “should pay up.” However, I only got 15 was, however, my teacher in a number of courses, cents. undergraduate and graduate, and was chairman Kap’s rhetorical flourishes are well known; but, of the University of Chicago math department sometimes they had unintended consequences. when I was a graduate For my first job, I needed official certification student. Kap’s “style”, that I had completed the Ph.D. A letter from the mathematical as well as chairman would suffice. Kap wrote such a letter personal, shone through concluding “…and, barring catastrophe, he will everywhere. receive the degree on June 11….” This was deemed Nowadays, most math insufficient by a departmental administrator at departments offer a Berkeley who quoted the “barring catastrophe” “bridge” course for their remark. Kap washed his hands of it and sent me off majors. This course is de- to the Dean of Students in the Division of Physical signed to ease the transi- Sciences for a “really” official letter. tion to real, upper-divi- Even, at the last moment, during my final oral sion mathematics from exam, Kap’s style was apparent. He asked me (increasingly) less rig- where would you find a commutative ring with orous calculus courses. some property or other. I started to construct the Chicago has had such a course for years. In my ring when he interrupted: “No, no, in what book day, it was Math 261; at would you look for it?” I replied, “Nagata” and was present it has the fash- off the hook! ionably inflated number Kap’s lessons and advice remain fresh to this 26100. Currently, just day. His books and his expositions are as attrac- as it did several decades tive to the current generation of students as they Photograph courtesy of Julius Zelmanowitz, MSRI. Zelmanowitz, Julius of courtesy Photograph ago, the course covers were to mine. Kaplansky at the keyboard. “sets, relations, and func- tions; partially ordered Manjul Bhargava sets; cardinal numbers; Zorn’s lemma, well-ordering, and the axiom of Kap Across Generations choice; metric spaces; and completeness, compact- I was a graduate student at Princeton in the year ness, and separability.” When I took the course, 1999. And being a student of algebra, I obviously Kap used notes of Ed Spanier on “Set Theory and knew of Professor Kaplansky, though I knew of Metric Spaces”. Spanier never got around to writing him more as a “legend” than as a person. His these notes up as a book. Kap, however, did! Set name was one that was attached to a number of Theory and Metric Spaces appeared in 1972 and great theorems, some going back to the 1940s. continues in the AMS Chelsea series. Kaplansky’s At the time I suspect it never occurred to me that style is as appealing to current students as it was to he might be an actual person who was still doing us decades ago. I have used the book in my classes great mathematics. for many years. One of my recent students enjoyed While working on my dissertation, I became in- the book so much that she bought it as a birthday terested in a classical problem from number theory present for her engineer father! relating to quadratic forms. (It was not really a As chairman, Kap maintained a keen interest problem in the “Kaplansky style”, or so I thought!) in graduate students and the graduate program. The question was: When does a positive-definite His sensitivity to grad student-advisor dynamics can be illustrated by the following anecdote. One integral quadratic form represent all positive afternoon at math tea, my advisor, Yitz Herstein, integers? (For example, Lagrange’s Four Squares Form a2 b2 c2 d2 gives such an expression—i.e., + + + Lance Small is professor of mathematics at the University Manjul Bhargava is professor of mathematics at Prince- of California, San Diego. His email address is lwsmall@ ton University. His email address is bhargava@math. ucsd.edu. princeton.edu.
1490 Notices of the AMS Volume 54, Number 11 every positive integer can be written as a sum of renewing our sessions!,” and there were always four square numbers.) This was a beautiful ques- new and exciting things to discuss; every year I tion of Ramanujan that Professor Conway taught looked forward to it.) Until the very last summer, me about and got me hooked on. when I heard the sad and devastating news. I’ve After working on the question for some time, I since always felt that it was unfair that I got to realized that some good headway could be made know him only toward the later years of his life. provided that one could understand the clas- Of course, deep down I know I should be grateful sification of what are known as “regular ternary that I got to meet him at all, and to have been one forms”. In particular, I needed to know: How many of the lucky ones in my generation to have had the such regular ternary forms are there? I did some privilege of knowing him. He was so encouraging searches on MathSciNet, and soon enough found a to me always, as a person, as a musician, and most 1997 (!) paper by W. Jagy and I. Kaplansky entitled: of all, as a mathematician. I will always cherish the “There are 913 regular ternary forms”. memories of his enthusiasm, brilliance, generosity, Here was the exact answer to my question in the and friendship. I will miss him very much. very title of a paper written only two years ago! It was quite exciting, and I thought to myself “Surely David Eisenbud this is not the same Kaplansky!,” but after some research I soon discovered that it was. I emailed Jagy and Kaplansky later that week, Kap at MSRI and heard back from both almost immediately. Kap was enormously influential in many fields Kap and Will (Jagy) were also both very excited that of mathematics, through his papers, through his their recent work had found applications so soon. I books, and perhaps most of all through his Ph.D. mentioned to them that I would be in Berkeley for students and the many many additional students a few weeks that summer to learn tabla with my who, like myself, listened raptly to his courses. I re- teacher, and Kap kindly invited me to visit MSRI member well his highly entertaining and beautifully while I was there. polished lectures from my student days in Chicago Kap asked David Eisenbud, the director of MSRI, —whatever he taught, I signed up for the course, it to give me an office for the summer, and David being such a pleasure to listen to him. From being generously agreed. That summer turned out to be on the first winning team of the Putnam competi- one of my most productive summers ever. I worked tion to being president of the AMS and National on mathematics during the day and played tabla Academy member, his career was truly remarkable by night. Rather than working in my private office, —you can find more information starting from I found myself mostly working in Kap’s office! We the AMS website, http://www.ams.org/ams/48- didn’t really work together, but rather we worked kaplansky.html. independently and then shared what we had dis- As the second director of MSRI, Kap served the covered or learned at various intervals throughout Institute directly from 1984 through 1992. He the day. Kap, Will, and I discussed and learned greatly developed the reputation and influence various mathematical topics together in what were of MSRI, building on the start provided by the some extremely enjoyable sessions. Kap’s love, founders, Chern, Moore, and Singer. My own first enthusiasm for, and unique view of mathematics experiences at MSRI were under Kaplansky’s direc- were constantly evident and always inspiring! torship. As with everything he did, he paid atten- tion to every detail of the operation—he boasted In addition, I talked to Kap a lot about other to me once that he personally read and signed things; we shared common interests not only in every single letter of invitation that the Institute mathematics but also in music, making it a rather frequent topic of conversation. In the process, I sent out during his eight years in office. He and his also learned a great deal about Kap’s amazingly wife, Chellie, were also very present and available regular life and his other associated charming to the members—literally thousands will remem- idiosyncrasies. He brushed his teeth more often ber Kap’s musical performances at the Christmas than anyone I’ve ever known. And no matter how parties. Among the many marks Kap left on MSRI exciting a particular conversation or work session was the start of fundraising activity. For example was, if it was time for his daily noon swim, then Kap formed the “International Board of Friends there was no stopping him from running off to of MSRI”, and the connections made through this the pool! (The same occurred when it was time group are still of the utmost importance to us. for his chosen 5:14 p.m. end-of-the-day bus from Kap’s first paper appeared in 1939. After step- MSRI.) I found myself changing my own schedule to ping down as MSRI director, at seventy-five, Kap match his work schedule better (including waking went back to full time research mathematics, and up rather early!). returned to number theory, one of his first loves. The same schedule was adhered to the follow- David Eisenbud is professor of mathematics at the Univer- ing few summers, as he always generously invited sity of California, Berkeley. He is former director of MSRI. me back (He would write, “Looking forward to His email address is [email protected].
December 2007 Notices of the AMS 1491 Some of his most recent He was elected to the Council in 1951 as a young work, on integral quadratic faculty member, and later was elected to the Board forms, was published in of Trustees (as an older one). He was elected vice 2003, when he was eighty- president in 1974, putting him back on the Council, six. and finally in 1985–86 was elected president of the Mathematically, Kap AMS. All together, he served a total of ten years on was my brother: he, the the Council and seven years on the Board—a great first student of Mac Lane, many meetings for anyone! I, nearly the last. But he The four years from 1984–87, which included was much more an uncle his time as president elect and past president, to me who had been down were particularly eventful for the Society. Kap most of the avenues that I played a key role in every one of those events. later began to explore. He The AMS hosted the 1986 International Congress, Photograph courtesy of Julius Zelmanowitz, MSRI. Zelmanowitz, Julius of courtesy Photograph AMS president, was always generous in which took place in Berkeley; Kap was on the local 1985–86. advice, counsel, and in giv- organizing committee and oversaw many aspects ing credit. I saw him nearly of the Congress. The Society was undergoing some every day in my student days at Chicago, and again, radical changes during this time, including its re- nearly every day, over the first eight years I was covery from a disastrous financial situation earlier MSRI director. Interacting with Kap was always a in the decade and a restructuring of Mathematical pleasure, crisp, clear, and somehow uplifting. It is Reviews administration; again, Kap played a key one that I shall deeply miss. role in reshaping the AMS. And it was during this period that the AMS decided to create a premier John Ewing journal—the Journal of the American Mathematical Society. Kap was the one who championed this idea Kap and the AMS (which came from the Committee on Publications) For more than forty years, Irving Kaplansky was and helped bring the journal to life by carefully active in the American Mathematical Society, and choosing the first editorial board. for much of that time he was a driving force. He The most remarkable feature about Kap’s ser- began as associate editor of the Bulletin at the age vice to the Society was his style. In every job he of twenty-eight in 1945—the same year that he undertook—in everything he did—he was forceful joined the faculty of the University of Chicago. Two and yet graceful, eloquent and yet thoughtful, en- years later, he became an editor for the Transac- ergetic and yet polite. When he received the AMS tions, and ten years after that he was an editor for Steele Prize, Career Award in 1989, the citation the Proceedings. He thus served as editor for the acknowledged that style by honoring him for “his entire complement of AMS journals at the time. energetic example, his enthusiastic exposition, In addition to his role with journals, Kap was and his overall generosity.” It went on to point active in the Society’s governance for many years. out that he “has made striking changes in math- ematics and has inspired generations of younger John Ewing is executive director of the AMS. His email address is [email protected]. mathematicians.” Kap left his mark on many parts of mathemat- ics, but he especially left his mark on the Society. Lucy Kaplansky Kap Was My Father My dad, Irving Kaplansky, was a mathematician, but he was also a teacher, and he taught me many things. When I was a little girl he taught me to play checkers. In our games together he would start with half his checkers, and he’d beat me anyway. But whenever I played checkers with other kids, I demolished them. He got a huge kick out of that. He taught me math. I would come home from school when I was in grade school and high school and he would re-teach me that day’s math lesson. He was always patient and clear, and he made it all make sense. I’d go back to school the next day, and 1972 family photo. Top row, left to right: Steven, Alex, and Singer-songwriter Lucy Kaplansky lives in New York City. Lucy. Front: Irving and Chellie Kaplansky. Her website is www.lucykaplansky.com
1492 Notices of the AMS Volume 54, Number 11 often I was the only one who would understand what was going on in math class. I ran into a couple of my math teachers from grade school recently and they told me when they found out I was in their classes they were petrified because they knew exactly who my dad was! My dad taught me to be organized in everything, reliable, and punctual. I think I’m the only musician I know who always shows up on time and actually does what I say I’m going to do. He taught me that I should love what I do for a living. Throughout my childhood he would sit in his study, classical music always on the radio, doing math. Sometimes he’d look like he was doing nothing, maybe even sleeping, but he’d always say he was “thinking mathematics”. He instilled in me one of the central ideas that has informed my life, that making money for money’s sake was not im- portant, that doing work you love is everything. I asked him once why he loved math. He re- sponded simply “it’s beautiful.” He taught me that learning was fun. He espe- cially loved learning about history and he was forever reading about and discussing history, all kinds. Because of him, I, too, love to learn about history; because of him I love to learn, period. And perhaps most of all he taught me to love music. He was a gifted pianist, and there’s a story I’ve heard my whole life that when he was three years old he and his family attended a Yiddish musical in Toronto, and when they got home he sat down at the piano and played the show’s main song perfectly, note for note. From as early as I can remember I would sing while he played the piano. He taught me dozens of songs from the 1930s and 1940s, as well as from Gilbert and Sullivan operettas. I still remember most of these songs. When I was older and pursued a career as a singer-songwriter, I started performing songs that he had written; one of the most popular was “A Song About Pi”. To this day it’s one of my most requested songs. When my dad was already in his eighties, my parents often went on the road with me when I was doing concerts. We’d all get in the car and stay in hotels, and he would sell my CDs for me after the show, sometimes he was even asked for autographs. And if there was a piano on stage he would accompany me on a couple of his songs. He always brought down the house. I’m so grateful we were able to share this. The last time he sat in with me onstage he was 88 years old. I’ve heard from so many of my dad’s students over the years what a wonderful teacher he was. I know that. He was my teacher.
December 2007 Notices of the AMS 1493 WHATIS... ? a Cluster Algebra? Andrei Zelevinsky
Cluster algebras, first introduced in [2], are con- Before discussing the general definition, let us structively defined commutative rings equipped look at cluster algebras of rank two. One associates with a distinguished set of generators (cluster such an algebra A(b, c) with any pair (b, c) of variables) grouped into overlapping subsets (clus- positive integers. The cluster variables in A(b, c) ters) of the same finite cardinality (the rank of an are the elements xm, for m ∈ Z, defined recursively algebra in question). Among these algebras one by the exchange relations finds coordinate rings of many algebraic varieties ( xb + 1 if m is odd; that play a prominent role in representation theo- = m xm−1xm+1 c ry, invariant theory, the study of total positivity, xm + 1 if m is even. etc. For instance, homogeneous coordinate rings Iterating these relations, we can express each xm of Grassmannians, Schubert varieties, and other as a rational function of x1 and x2. Thus, A(b, c) related varieties carry a cluster algebra structure is the subring generated by all the xm inside the (after a minor adjustment). Potential applications field of rational functions Q(x1, x2). The clusters of this structure include explicit constructions of are the pairs {xm, xm+1} for m ∈ Z. Starting with the (dual) canonical basis and toric degenerations the initial cluster {x1, x2}, we can reach any other for these varieties. cluster by a series of exchanges Since its inception, the theory of cluster algebras has found a number of exciting connections and · · · ↔ {x0, x1} ↔ {x1, x2} ↔ {x2, x3} ↔ .... applications: quiver representations, preprojec- For an arbitrary rank n, the construction is tive algebras, Calabi-Yau algebras and categories, similar. Each cluster x = {x1, . . . , xn} is a collection Teichmüller theory, discrete integrable systems, of algebraically independent elements of some Poisson geometry... The current state of these ambient field, and each cluster variable xk can developments, including links to papers, work- be exchanged from x by forming a new cluster ing seminars, conferences, etc., is represented at 0 0 0 x = x−{xk}∪{xk}. Here xk and xk are related by an the online Cluster Algebras Portal created and exchange relation of the following form: the product maintained by S. Fomin [1]. 0 xkxk is equal to the sum of two disjoint monomi- Although some of the above connections are 0 als in the variables from x ∩ x = x − {xk}. (For rather technical, cluster algebras themselves are simplicity, we restrict ourselves to the coefficient- defined in an elementary manner not requiring any free case, where both monomials appear with tools beyond high-school algebra. On the other the coefficient 1.) The exponents in these two hand, they have an unusual feature that both monomials are encoded by an n × n integer matrix generators and algebraic relations among them are B = (bij ) called the exchange matrix; it is usu- not given from the outset but are produced by an ally assumed to be skew-symmetrizable, that is, iterative process of seed mutations. di bij = −dj bji for some positive integers d1, . . . , dn. The corresponding exchange relations take the Andrei Zelevinsky is professor of mathematics at North- eastern University. His email address is [email protected]. form 0 Y [bik]+ Y [−bik]+ He thanks Sergey Fomin for helpful suggestions. xkxk = xi + xi , The author was partially supported by NSF grant DMS i i 0500534 and a Humboldt Research Award. where we use the notation [b]+ = max(b, 0).
1494 Notices of the AMS Volume 54, Number 11 A pair (x,B) as above is called a seed. To begin invited to check that if bc = 1 (respectively 2; 3) the iterative process, we extend, for each index k, then the sequence of cluster variables is periodic the transformation x , x0 of clusters to the trans- with period 5 (respectively 6; 8). These three cases 0 0 formation (x,B) , (x ,B ) of seeds called the seed are naturally associated with the root systems A2, mutation in direction k. Its key ingredient is the B2, and G2. 0 0 matrix mutation µk : B , B = (bij ) given by the The study of cluster algebras of finite type rule brings to light new combinatorial and geometric structures associated to root systems. For example, −bij if i = k or j = k; 0 by a result of F. Chapoton, S. Fomin, and the author, b = b + [b ] [b ] ij ij ik + kj + the cluster complex (the simplicial complex whose −[−b ] [−b ] otherwise. ik + kj + vertices are cluster variables and whose maximal The corresponding cluster algebra is then defined simplices are clusters) can be identified with the as the subring of Q(x1, . . . , xn) generated by all dual face complex of a simple convex polytope, cluster variables, that is, by the union of all clusters the generalized associahedron. These polytopes obtained from the initial cluster x by iterating seed include as special cases the Stasheff associahedron mutations in all directions. (in type An), and the Bott-Taubes cyclohedron (in The definition of matrix mutations may look type Bn). strange at first. (Remarkably, the same rule came We conclude this brief tour of cluster alge- up in recent work on Seiberg dualities in string bras with the following informal question: how theory.) One of its main consequences—and one of can one detect a cluster algebra structure in a the main reasons for introducing it—is the Laurent commutative ring of interest? A possible strategy phenomenon: every cluster variable, which a priori is to look for three-term relations satisfied by is just a rational function in the elements of a some naturally arising elements of the ring and given cluster, is in fact a Laurent polynomial with try to interpret them as exchange relations. The integer coefficients. For instance, in each rank 2 mutation mechanism will take care of the rest. algebra A(b, c), every cluster variable xm is a One recent example (due to S-W. Yang and the Laurent polynomial in x1 and x2. As a corollary, if author): consider the variety Xn of tridiagonal we specialize all elements of some cluster to 1 then (n + 1) × (n + 1) unimodular complex matrices all cluster variables become integers. This is rather U = (uij ) with ui,i+1 = ui+1,i = 1 for all i. The top- unexpected since, in the process of seed mutations, left-corner principal minors D1,...,Dn of U satisfy every cluster variable eventually appears as the the recurrence relations Dk+1 = uk+1,k+1Dk − Dk−1, denominator of the expression used for producing which feature prominently in the classical theory a new one. The cluster algebra machinery pro- of orthogonal polynomials. Rewritten in the form vides a unified explanation of several previously Dkuk+1,k+1 = Dk+1 + Dk−1, these relations acquire known phenomena of this kind. One example is distinct “cluster flavor”. Indeed, the coordinate ring the Somos-5 sequence discovered some years ago of Xn can be made into a cluster algebra (of finite by M. Somos: its first five terms are equal to 1, type An) with the initial cluster x = {D1,...,Dn}, and the rest are given by the recurrence relation so that the above relations are exactly the exchange xmxm−5 = xm−1xm−4 + xm−2xm−3. The fact that all relations from x. Other examples of three-term terms of this sequence are integers can be deduced relations leading to cluster algebras include short from the Laurent phenomenon for cluster algebras. Plücker relations between Plücker coordinates on The Laurent polynomial expressions for cluster Grassmannians, and Ptolemy relations between variables are not yet well understood. S. Fomin Penner coordinates on the decorated Teichmüller and the author conjectured that all coefficients space of a bordered Riemann surface. in these Laurent polynomials are positive; this is still open in general. The strongest known result Further Reading in this direction (by P. Caldero and M. Reineke), 1. S. Fomin, http://www.math.lsa.umich.edu/ which establishes the conjecture in many special efomin/cluster.html. cases, uses some heavy machinery: it is based 2. S. Fomin and A. Zelevinsky, Cluster algebras I: on a beautiful geometric interpretation (due to Foundations, J. Amer. Math. Soc. 15 (2002), 497–529. P. Caldero and F. Chapoton) of the coefficients 3. , Cluster algebras II: Finite type classification, in question as Euler-Poincaré characteristics of Invent. Math. 154 (2003), 63–121. certain quiver Grassmannians. A cluster algebra is of finite type if it has finitely many seeds. As shown in [3], these algebras are classified by the same Cartan-Killing types (or Dynkin diagrams) as semisimple Lie algebras, finite root systems, and many other important structures. In particular, the rank-two cluster algebra A(b, c) is of finite type if and only if bc ≤ 3; the reader is
December 2007 Notices of the AMS 1495 Book Review
How Mathematicians Think Reviewed by Reuben Hersh
cognitive scientists, sociologists, a computer sci- How Mathematicians Think: Using Ambiguity, entist, and an anthropologist. Contradiction, and Paradox to Create Mathematics There’s not much consensus, but at least one William Byers thing has been pretty generally taken for granted: Princeton University Press, 2007 mathematical thinking and discourse is supposed 425 pages, US$35.00, ISBN 978-0691127385 to be precise—that is to say, unambiguous. A mathematical statement is supposed to have a single definite meaning. What Byers’s book reveals This book is a radically new account of math- is that ambiguity is always present, from the most ematical discourse and mathematical thinking. elementary to the most advanced level. In teach- It’s addressed to everyone, from a lay reader ing school mathematics, it is an unacknowledged who hasn’t met complex numbers, up to a pro- source of difficulty. At the level of research, it is fessional who appreciates Sarkovsky’s theorem often the key to growth and discovery. on cycles of iterated functions, or Goodstein’s Ambiguity can just mean vagueness. But also, number-theoretic equivalent of Gödel’s theorem it can mean, as Byers puts it, “a single situation for arithmetic with induction. No math prepara- or idea that is perceived in two self-consistent tion is presupposed, and everything is explained but mutually incompatible frames of reference.” with complete clarity, yet deep contemporary is- (p. 28) In fact, he makes a persuasive case that sues are faced with no hesitancy. The discussion ambiguity is actually what makes mathematical is free of pretentiousness or grandiosity. Byers ideas so powerful: straightforwardly explains the issues and clarifies them. Normally ambiguity in science and mathe- Starting with Imre Lakatos’ 1976 Proofs and matics is seen as something to overcome, Refutations, some writers have been turning away something that is due to an error in un- from the search for a “foundation” for mathemat- derstanding and is removed by correcting ics and instead, seeking to understand and clarify that error. The ambiguity is rarely seen as the actual practice of mathematics—what real having value in its own right, and the exis- mathematicians really do. Conferences toward this tence of ambiguity was often the very thing end have been held in Mexico, Belgium, Denmark, that spurred a particular development of Italy, Spain, Sweden, and Hungary. In particular, I mathematics and science…. The power of would mention books by Bettina Heinz, Carlo Cel- ideas reside in their ambiguity. Thus any lucci, and Alexandre Borovik. My own anthology project that would eliminate ambiguity from collects essays by mathematicians, philosophers, mathematics would destroy mathematics. (p. 24)
Reuben Hersh is professor of mathematics at the Familiar examples of ambiguity include: Negat- University of New Mexico. His email address is ing Euclid’s parallel postulate. Different sizes of [email protected]. infinite sets. Using logic to prove the limitations
1496 Notices of the AMS Volume 54, Number 11 of logic. Infinitely rough curves, self-similar on according to context. Using the simplest exam- infinitely many different scales. And on and on. ple imaginable, Byers elucidates the ambiguity No surprise that there’s ambiguity in “infinite” inherent in the notion of equality: or “infinity”. The philosophically inclined won’t When we encounter “1+1=2”, our first reac- be surprised that there’s ambiguity in “true” and tion is that the statement is clear and precise. even in “proof”. But even in the simplest, most We feel that we understand it completely and “elementary” mathematical steps, there is already that there is nothing further to be said. But deep, unacknowledged ambiguity. is that really true? The numbers “one” and An obvious example is: square roots of negative “two” are in fact extremely deep and im- numbers. It takes effort simultaneously to know portant ideas…The equation also contains that “−1 has no square root” (on the real line) and an equal sign. Equality is another very basic “it has two of them” (in the plane.) The student idea whose meaning only grows the more must switch contexts as needed. Sometimes there you think about it. Then we have the equa- is no square root, sometimes there are two. It all tion itself, which states that the fundamental depends on what are you are talking about, what concepts of unity and duality have a rela- are you are trying to do! But a while back, the same tionship with one another that we represent effort was required regarding negative numbers. by “equality”—that there is unity in duality We have forgotten that for D’Alembert or De Mor- and duality in unity. This deeper structure gan, it both made sense and didn’t make sense, to that is implicit in the equation is typical of a contemplate a quantity less than “nothing”. situation of ambiguity. Thus even the most Indeed, “zero” is ambiguous! Unlike D’Alembert elementary mathematical expressions have or DeMorgan, today we don’t say “nothing” when a profundity that may not be apparent on we mean “zero”. Zero is something—it’s a num- the surface level. (p. 27) ber. Yet, of course, “nothing” is what it means. A more advanced example of the ambigu- Zero is a something, and what is stands for, what ity of the equals sign is 1 = .999…. it means—is “nothing”! This is ambiguous, but What is the precise meaning of the “=” we math teachers have buried the ambiguity so sign? It surely does not mean that the num- deeply, that if we ever have to talk to a student ber 1 is identical to that which is meant by who is troubled by it, we can hardly understand the notation .999…. There is a problem here, what is her difficulty. and the evidence is that, in my experience, “One” is ambiguous! Frege’s famous book, most undergraduate math majors do not be- Grundlagen der Arithmetik, was motivated by lieve this statement… they all agreed that mathematicians’ inability to explain coherently .999…was veryclose to 1. Some even said“in- what they meant by “one”. Frege’s answer was: finitely close”, but they were not absolutely “one” is the “concept” of singletons. But Dedekind sure what they meant by this…. This nota- and Peano had a different answer: “one” is just tion stands both for the process of adding an undefined term, in the axioms for the relation this particular infinite sequence of fractions of “successor”. And still another answer is given and for the object, the number that is the in every elementary math classroom—“one” is a result of that process…. Now the number 1 slash or a tally mark, which can be repeated to is clearly a mathematical object, a number. make “two”, and repeated again, to make “three”. Thus the equation 1 = .999… is confusing If that’s not enough ambiguity, there’s still a because it seems to say that a process is deeper ambiguity in “one”. When we choose to equal (identical?) to an object. This appears think about all the things that belong to some to be a category error. How can a process, a system (for example, all the counting numbers) verb, be equal to an object, a noun? Verbs and regard that collection as “one” set—when and nouns are “incompatible contexts” and we make a unity out of a multiplicity—we are thus the equation is ambiguous…. I hasten committing an ambiguity. An ambiguity, indeed, to add that this ambiguity is a strength, not that is a central feature of mathematical thinking. a weakness, of our way of writing decimals. (Notice how the word “universal”, with the sense To understand infinite decimals means to be of “all-embracing”, uses the primitive root, “uni”, able to move freely from one of these points a single slash or tally mark.) of view to the other. That is, understanding In fact, the relation of “equality” in general is involves the realization that there is “one ambiguous, for the entities on the two sides of the single idea” that can be expressed as 1 or equals sign are usually not identical. (“x = x” is not as 1 = .999… that can be understood as the usually interesting.) In an interesting equation, the process of summing an infinite series or an entities on the left and the right are not identical, endless process of successive approximation so the claim that they are “equal” is necessari- as well as a concrete object, a number. This ly ambiguous, subject to different interpretations kind of creative leap is required before one
December 2007 Notices of the AMS 1497 can say that one understands a real number of the concepts of group, or graph, or solution of as an infinite decimal. (p. 41) a differential equation. Byers also discusses how students struggle with In discussions of the nature of mathematics, ideas that are less advanced than 1 = .999… For the notion of “abstraction” is often mentioned, example, here he unravels the ambiguity of the but rarely clarified or explicated. Byers has a re- “variable” x as students encounter it in the seventh markable explanation of abstraction. “Abstraction grade: consists essentially in the creation and utilization of ambiguity.” For example, when functions are Does the “x”in“x + 2 = 4” refer to any num- first introduced, either in the classroom or in ber or does it refer to the number 2? The the history of mathematics, they are active. The answer is, “Both and neither.” At the begin- function transforms one number into another. ning, x could be anything. At the end, x can Later, when we focus on differential operators, the only be 2. Yet at the end, we are saying that functions are passive. The operator transforms = every number x NOT 2 is not a solution, one function into another. So which is it? Is a so the equation is also about all numbers. function active or passive, verb or noun? “The Thus at every stage, the x stands for all initial barrier to understanding, that a function numbers but ALSO for the specific number 2. can be considered simultaneously as process and We are required to carry along this ambiguity object—as a rule that operates on numbers and throughout the entire procedure of solving as an object that is itself operated on by other the equation. It begins with something that processes—turns into the insight. That is, it is could be anything and ends with a specific precisely the ambiguous way in which a function number that could not be anything else. What is viewed which is the insight.” (p. 48) an exercise in subtle mental gymnastics this Byers doesn’t stop with mathematics itself. Not is! How could this way of thinking be called only mathematics, but even more, philosophy of merely mechanical? No wonder children have mathematics, is inextricably tied up with ambi- difficulty with algebra. The difficulty is the guity, paradox, and contradiction. “Do we create ambiguity. The resolution of the ambiguity, math or do we discover it?” “Is it in our minds, or solving the equation, does not involve elim- is it out there”? Contradictions, nicht wahr? inating the double context but rather being One deep ambiguity is the double meaning of able to keep the two contexts simultaneous- “exist”. Does it mean something is “construct- ly in mind and working within that double ed” from already “constructed” entities, by some context, jumping from one point of view to clearly understood notion of “construct”? Or does the other as the situation warrants. (p. 42) it rather mean something is contradiction-free, Mathematicians are accustomed to making use is “safe” to “postulate”, because it doesn’t crash of multiple “representations” of “the same” thing. into or interfere with other notions or facts that With the precise notion of “isomorphic equiva- we don’t want disturbed? This is just the stale lence”, we are able legitimately and smoothly to old argument between intuitionist/constructivists use different representations simultaneously. The and standard/classical mathematicians. For Byers, group of permutations on three letters is “the the point isn’t to choose sides, to decide who is same thing” as the automorphism group of the right and who is wrong. Rather, it is to perceive equilateral triangle, or the group of functions un- that this ambiguity of “exist” is intrinsic to our der composition generated by 1/x and 1−x, andso mathematical practice, and is fruitful. The clash on and so on. And any graph is equivalent to, is vir- of viewpoints arising from this ambiguity brings tually “the same”, as its adjacency matrix. And any forth interesting mathematics. solution of Laplace’s partial differential equation Speaking of the often mentioned but rarely is an integral with a Green’s function as kernel, and analyzed unreflective Platonism of the working it is simultaneously the minimal solution of a cer- mathematician, Byers writes: tain variational problem, and it is simultaneously The ambiguity of an unsolved problem is the limit of a sequence of solutions of difference mitigated somewhat by the Platonic attitude equations, and it is simultaneously the expected of the working mathematician. That is, she value of the outcome of the random motion of feels that it is objectively either true or false a Brownian particle, as well as the equilibrium and that the job of the mathematician is distribution of heat in a homogeneous medium, “merely” to discover which of these a pri- and also the potential of a distribution of grav- ori conditions applies. Psychologically, this itational mass or electrostatic charge. When we Platonic point of view brings the ambigui- make simultaneous or alternate use of “different” ty of the situation into enough control so representations or interpretations of “the same” that researchers have confidence the correct structure, we are using ambiguity in a controlled, solution exists independent of their efforts. algorithmic way—using the multiple-meaningness It moves the problem from the domain of
1498 Notices of the AMS Volume 54, Number 11 “ambiguity as vagueness” in which anything what mathematics is all about, this is the very core could happen to the sort of incompatibility and nature of mathematical thinking!” that has been discussed in this chapter where Byers finds far-reaching consequences, beyond there are two conflicting frameworks, true mathematics, for our very understanding of what or false. it means to be human. “Contradictions demand resolution!” you may Any great quest demands courage. It is a voy- say.“Torisetothenextlevelinphilosophyofmath- age into the unknown with no guaranteed ematics, we must overcome the contradiction, results. What is the nature of this courage? It resolve it, not just pooh-pooh it!” is the courage to open oneself up to the am- But Byers offers us an insight—this is the biguity of the specific situation. The whole thing may end up as a vast waste of time, way it has to be! Live with it! Life is ambigu- that is, the possibility of failure is inevitably ous and contradictory. Mathematics is part of present... Our lives also have this quality of life. Insofar as the philosophy of mathematics de- a quest, the attempt to resolve some funda- scribes the total mathematical situation—process mental but ill-posed question. In working on as well as content—naturally it’s also bound to be a mathematical conjecture, life’s ambiguities ambiguous. solidify into a concrete problem. That is, the Well, that does in fact seem to be the case. situation of doing research is isomorphic to You might say that the work of the mathe- some extent with the situation we face in matician is to drive away ambiguity. “Precision” our personal lives. This is one reason that is what mathematics is all about. “Say what you working on mathematics is so satisfying. In mean, mean what you say, nothing is there except resolving the mathematical problem we, for what is right on the page.” Byers pushes us back, a while at least, resolve that large, existential to the ambiguous situation that calls for math- problem that is consciously or unconscious- ematical explication. He makes us see that the ly always with us…. Learners need support ambiguity we insist on banishing is the source, the when they are encouraged to enter into origin, of the mathematical work. “Logic moves new unexplored ambiguities. A new learn- in one direction, the direction of clarity, coher- ing experience requires the learner to face ence and structure. Ambiguity moves in the other the unknown, to face failure. Sticking with direction, that of fluidity, openness, and release. a true learning situation requires courage Mathematics moves back and forth between these and teachers must respect the courage that two poles…. It is the interaction between these students exhibit in facing these situations. different aspects that gives mathematics its pow- Teachers should understand and sympathize er.” (p. 78) “Mathematical ideas are not right or with students’ reluctance to enter into these wrong; they are organizers of mathematical situ- murky waters. After all, the teacher’s role as ations. Ideas are not logical. In fact the inclusion authority figure is often pleasing insofar as should go the other way around—logic is not the it enables the teacher to escape temporarily absolute standard against which all ideas must be from their own ambiguitiesand vulnerability. measured. In fact logic itself is an idea.” (p. 257) Thus the value of learning potentially goes The normal mathematician—the philosopher’s beyond the specific content or technique but “working mathematician”, the ordinary mathe- in the largest sense is a lesson in life itself. matician, the “mathematician in the street”?—may (p. 57) respond with a shrug and a, “So what?” We do our This book strikes me as profound, unpreten- calculations and prove our theorems by following tious, and courageous. our noses, not by looking right or left to see where we are in the broader conceptual or “philosophic” References realm. You don’t need to know what is meant by Alexandre Borovik, Mathematics Under the Micro- ∼ “one” in order to know that one and one is two. scope, http://www.maths.manchester.ac.uk/ avb/ micromathematics/downloads. But recall the old saying of Socrates, about the Carlo Cellucci, Filosofia e Matematica, Laterza, unexamined life. Most mathematical life, like most Bari, 2002. human life in general, is unexamined. Byers pulls Bettina Heinz, Die Innenwelt der Mathematik, away the covering habit and routine, to expose Springer 2000. life-giving embarrassments hidden beneath. Reuben Hersh, Ed., 18 Unconventional Essays on the You can’t quite say that nobody has said this Nature of Mathematics, Springer, 2005. before. But nobody has said it before in this all- , What Is Mathematics, Really?, Oxford Univer- sity Press, 1997. encompassing, coherent way, and in this readable, Imre Lakatos, Proofs and Refutations, Cambridge crystal clear style. The examples are well known University Press, 1976. and familiar, but it’s something else to put them all together and say, “This is it! This is exactly
December 2007 Notices of the AMS 1499 Emma Lehmer 1906–2007
John Brillhart
Emma Lehmer, née and inviting them to her house for dinner. She T r o t s k a y a , w a s was always modest and self-effacing and seemed born on November completely satisfied in her role of providing a 6, 1906, in Samara, genial social setting on a variety of mathematical a city on the Volga occasions. She and her husband had marvelous River in Russia. senses of humor. When she was four In 1969 she and her husband founded the West years old, her fam- Coast Number Theory meeting, which has met ily moved to Harbin every year since at locations around the West. It in Manchuria, where has been of tremendous value as a meeting where she was tutored at young people in particular can come and enjoy a home in basic sub- comfortable, friendly, and informal environment jects, music, and where they can find their way into the real world languages. When she of mathematics. It remains a real tribute to the was fourteen, she fi- Lehmers. Emma Lehmer nally went to the new From the beginning, Emma was a mathematical public high school in researcher and scholar along with her husband. Harbin. During the She wrote 56 papers, 17 jointly with her husband, next four years, she saved enough money to allow 5 three-way papers with him and a second person, her to travel across the Pacific and enroll at the and one jointly with H. S. Vandiver. She was as University of California in Berkeley. knowledgable about computing in number theory Many years later she described her earlier feel- as her husband, always being there while projects ings when she was at home as follows: “I expect I were being discussed and contributing at all levels was rather insufferable in my intense desire to fly to their development. Her publications appeared from home into the great unknown.” in the primary mathematical journals. At Berkeley she had the extraordinary good for- She was also a Russian translator for the AMS tune of being hired as a student assistant to math- for whom she translated Pontryagin’s Topologi- ematics professor D. N. Lehmer and his son Dick, cal Groups and Delone and Faddeev’s work on to help them with some work in number theory. Irrationalities of the Third Degree along with Sue She soon found herself a part of the Lehmer family Ann Walker. itself, many of whose members were independent, Her papers deal with cyclotomy, character creative, artistic, and enterprising like herself. theory, difference sets, residuacity, and algebraic In 1928, when she graduated from UC Berkeley units. When she was eighty, she discovered that in mathematics, she and Dick were married, thus certain units can be gotten from Gaussian periods beginning sixty-three years of devoted marriage by translation, a good example of the kind of basic that lasted until his death in 1991. She was his number theory she and her husband had done all helpmate, mother of their two children Laura and their lives. Donald, mathematical collaborator in their re- When she became one hundred, the AMS an- search, and hostess to the vast number of visitors who came to Berkeley and stayed at the Lehmers’ nounced this fact in the December 2006 Notices. home as guests. Part of the announcement said that if people would Emma was very informal, cordial, and charm- like to celebrate her birthday with her, they might ing. She especially made a point of greeting Rus- send her a card, perhaps with flowers on it or a sian visitors in Russian to make them feel at home natural scene. Her daughter later said that cards came pouring in from all over the world for the John Brillhart is professor emeritus of mathematics at the next two months. In all, the number of signatures University of Arizona. His email address is jdb@math. on the cards was 275. How many of us would draw arizona.edu. that kind of response when we reach the age of
1500 Notices of the AMS Volume 54, Number 11 one hundred? It was most gratifying to her and to [27] (with D. H. Lehmer), The cyclotomy of Kloosterman everyone else. sums, Acta Arith. 12 (1967), 385–407. Emma passed away quietly in her sleep at her [28] On the divisors of the discriminant of the period home in Berkeley on the morning of May 7, 2007. equation, Amer. J. Math. 90 (1968), 375–379. [29] (with D. H. Lehmer), The cyclotomy of hyper– Publications of Emma Lehmer Kloosterman sums, Acta Arith. 14 (1968), 89–111. [30] (with D. H. Lehmer and D. Shanks), Integer sequences [1] A numerical function applied to cyclotomy, Bull. AMS having prescribed quadratic character, Math. Comp. 36 (1930), 291–298. 24 (1970), 433–451. [2] On a resultant connected with Fermat’s last theorem, [31] On the quadratic character of some quadratic surds, Bull. AMS 41 (1935), 864–867. J. für Math. 250 (1971), 42–48. [3] On the magnitude of the coefficients of the cyclotomic [32] On some special quartic reciprocity laws, Acta Arith. polynomial, Bull. AMS 42 (1936), 389–392. 21 (1972), 367–377. [4] On congruences involving Bernoulli numbers and the [33] Power characters of quadratic units, Proc. 1972 quotients of Fermat and Wilson, Annals of Math. 39 Number Theory Conf., Boulder Colo., 128–132. (1938), 350–360. [34] On the cubic character of quadratic units, J. Number [5] (with D. H. Lehmer), On the first case of Fermat’s last Th. 5 (1973), 385–389. theorem, Bull. AMS 47 (1941), 139–142. [35] On the quartic character of quadratic units, J. für [6] The quintic character of 2 and 3, Duke Math. J. 18 Math. 268/269 (1974), 294–301. (1951), 11–18. [36] A family of supplementary difference sets, Bull. [7] On residue difference sets, Canad. J. Math. 5 (1953), 11 425–432. Austral. Math. Soc. (1974), 1–4. [8] (with D. H. Lehmer and H. S. Vandiver), An application [37] An outcropping of combinatorics in number theory, of high speed computing to Fermat’s last theorem, Proc. Fifth S–E Conf. on Comb., Graph Theory, and Proc. Nat. Acad. 40 (1954), 25–33. Computing, Boca Raton, 1974, 75–92. [9] On cyclotomic numbers of order 16, Canad. J. Math. [38] (with D. H. Lehmer), A new factorization technique 6 (1954), 449–454. using quadratic forms, Math. Comp. 28 (1974), 625– 2 635. [10] On the number of solutions of uk D w(mod p), + = Pacific J. Math. 5 (1955), 103–118. [39] Existence criteria for supplementary difference sets, [11] Period equations applied to difference sets, Proc. J. Austral. Math. Soc. Ser. A 21 (1976), 438–446. AMS 6 (1955), 433–442. [40] Generalizations of Gauss’ Lemma, Number Theory [12] Number Theory on the SWAC, Proc. Symp. Appl. and Algebra, Academic Press, New York, 1977, 187– Math., AMS, v. 6, McGraw Hill, New York, 1956, 194. 103–108. [41] Rational reciprocity laws, Amer. Math. Monthly 85 [13] On the location of the Gauss sums, MTAC 10 (1956), (1978), 467–472. 194–202. [42] (with D. H. Lehmer), Picturesque exponential sums [14] (with H. S. Vandiver), On the computation of the I, Amer. Math. Monthly 86 (1979), 725–733. number of solutions of certain trinomial congruences, [43] (with D. H. Lehmer), Picturesque exponential sums J. Assoc. Comp. Mach. 4 (1957), 505–510. II, J. für Math. 318 (1980), 1–19. [15] Criteria for cubic and quartic residuacity, Mathema- [44] (with D. H. Lehmer), Cyclotomy with nonsquarefree tika 5 (1958), 20–29. moduli, Lecture Notes in Math., Springer–Verlag 899 [16] On Euler’s criterion, J. Austral. Math. Soc. 1 (1959/61), (1981), 276–300. 64–70. [45] (with D. H. Lehmer), Multiple sums of cyclotomic [17] (with D. H. Lehmer), On the cubes of Kloosterman numbers, Util. Math. 23 (1983), 223–239. sums, Acta Arith. 6 (1960), 15–22. [46] Patterns of power residues, J. Number Th. 17 (1983), [18] On Jacobi functions, Pacific J. Math. 10 (1960), 37–46. 887–893. [47] (with D. H. Lehmer), Properties of polynomials hav- [19] (with D. H. Lehmer), On runs of residues, Proc. AMS ing Fibonacci numbers for coefficients, Fib. Quart. 21 13 (1962), 102–106. (1983), 62–64. [20] (with D. H. Lehmer and W. H. Mills), Pairs of con- [48] (with D. H. Lehmer), Cyclotomy with short periods, secutive power residues, Canad. J. Math. 15 (1963), Math. Comp. 41 (1983), 743–758. 172–177. [49] (with D. H. Lehmer), The sextic period polynomial, [21] (with D. H. Lehmer, W. H. Mills, and J. L. Selfridge), Pacific J. Math. 111 (1984), 341–355. Machine proof of a theorem on cubic residues, Math. [50] On special primes, Pacific J. Math. 118 (1985), Comp. 16 (1962), 407–415. 471–478. [22] (with D. H. Lehmer), Heuristics, Anyone?, Studies [51] On square–separable primes, Amer. Math. Monthly Math. Anal. and Related Topics, Stanford U. Press, 92 (1985), 719–720. 1962, 202–210. [52] Cyclotomy of order twice a prime, Rocky Mountain [23] On the infinitude of Fibonacci pseudo–primes, Fib. J. 15 (1985), 481–489. Quart. 2 (1964), 229–230. [53] (with D. H. Lehmer), Cyclotomic resultants, Math. [24] (with J. Brillhart and D. H. Lehmer), Bounds for pairs Comp. 48 (1987), 211–216. of consecutive seventh and higher power residues, [54] Connection between Gaussian periods and cyclic Math. Comp. 18 (1964), 397–407. units, Math. Comp. 50 (1988), 535–541. [25] Artiads characterized, J. Math. Anal. Appl. 15 (1966), [55] An indeterminate in number theory, J. Austral. Math. 118–131. Soc. Ser. A 46 (1989), 469–472. [26] On the quadratic character of the Fibonacci root, Fib. [56] (with D. H. Lehmer), The Lehmer Project, Math. Comp. Quart. 5 (1967), 135–138. 61 (1993), 313–317.
December 2007 Notices of the AMS 1501 Remembering Philip Rabinowitz Philip J. Davis and Aviezri S. Fraenkel
The applied mathematician and numerical ana- and one programmed it in what was called “the lyst Philip (Phil, Pinchas, Pinny) Rabinowitz was four address system”. A line of code went typically born in Philadelphia on August 14, 1926, and as follows: take the number in cell 28, combine it passed away on July 21, 2006, in Jerusalem. Philip with the number in cell 37 according to standard Davis recounts reminiscences from his early operation S, store the result in cell 6 and go to cell scientific career; while Aviezri Fraenkel relates 18 to pick up the next instruction. Computations some of his activities at the Weizmann Institute were in fixed-point arithmetic so that scalings had of Science, where he began work in 1955, as well to be introduced to keep the numbers in bounds. as snapshots from earlier periods. The lines of code were first set out in pencil on standard coding sheets; these were transferred to Philip J. Davis punch cards or teletype tape, thence to magnetic wire from which they were inserted in SEAC. I had a long and fruitful friendship and collabora- In retrospect SEAC would be called a first gen- tion with Phil (Pinny) Rabinowitz that began in the eration computer. Though many numerical strate- fall of 1952 at the National Bureau of Standards gies (algorithms) had been worked out for a wide (NBS: now NIST) in Washington, D.C. When I began variety of mathematical problems in pre-electronic my employment there in the late summer of 1952, days, the new computers expanded the algorithmic Phil was already there. possibilities tremendously. But it was important Phil (I never called him Pinny) grew up in Phila- to work out by trial and error (and occasionally delphia. He got his Ph.D. degree from the Univer- by theory) which of these strategies were optimal sity of Pennsylvania in 1951 under the supervision vis-a-vis the limitations of time, storage, money, of Walter Gottschalk with a thesis titled Normal and the difficulties inherent within the algorithm Coverings and Uniform Spaces. Of course, this itself such as complexity, divergence, instability, topic in topology was irrelevant to the work of ill-posedness, etc. the bureau, and Phil was immediately pulled into The 1950s were a transitional age computation- numerical analysis, computation, programming, ally speaking. Until about 1955 or so, the electronic and running mathematical models of importance computers were still grinding out tables of Special to members of other portions of the bureau and Mathematical Functions and publishing them in of the U.S. government. bound volumes. Later, this was seen as largely At that time, the Bureau of Standards had one unnecessary; special software would be incorpo- of the very few electronic digital computers in the rated into scientific computational packages and world. It came on line in 1950 and was known as the would produce values of special functions on call SEAC (Standards Electronic Automatic Computer). and as needed. Within a very short period of time Phil became an One of Phil’s first publications (1954) was a expert programmer on SEAC. Table of Coulomb Wave Functions done jointly If I remember correctly, some of the features with Milton Abramowitz (head of the Bureau of of SEAC were as follows: It had 128 memory cells, Standards Computation Laboratory) and Carl- Philip J. Davis is emeritus professor of applied mathemat- Erik Fröberg, a numerical analyst from Lund, ics at Brown University. His email address is Philip_ Sweden. [email protected]. Shortly after I arrived in Washington, Phil Aviezri S. Fraenkel is professor of computer science worked on a project that teamed up Kenneth Cole and applied mathematics at the Weizmann Institute of of the National Institutes of Health and Henry Science. His email address is aviezri.fraenkel@ Antosiewicz of NBS. Cole was a biomathemati- weizmann.ac.il. cian who studied the Hodgkin-Huxley equations
1502 Notices of the AMS Volume 54, Number 11 of impulse transmission down a nerve fiber. If I remember correctly the H-H model consisted of a system of ordinary nonlinear differential equa- tions. Antosiewicz was an expert in that field. This very successful work was reported as “Automatic Computation of Nerve Excitation” and appeared in the Vol. 3, September 1955 issue of the Journal of the Society for Industrial and Applied Mathematics (SIAM). Some incidental gossip: SIAM was founded around 1952 essentially by Ed Block who was a Ph.D. classmate of mine and who for many years Photograph courtesy of the authors. the of courtesy Photograph was its managing director. In 1963, Alan Hodgkin and Andrew Huxley won the Nobel Prize in physiol- View of SEAC, circa 1952. ogy for their work on nerve excitation, and it seems out of favor in the days of paper-and-pencil scien- likely to me that the work of Cole, Antosiewicz, and tific computation, as the numbers involved were Rabinowitz contributed a bit towards this award. helter-skelter irrational decimals, impossible to Many years later, around 1988, my wife Hadassah remember and difficult to enter on a keyboard and I met Hodgkin and his American wife socially without error. in Cambridge, England. I told Hodgkin this NBS It was my plan to carry the computation beyond story, but I do not now remember what his reac- n 16. I suggested to Phil that we attempt the tion was. = In Washington, my friendship with Phil grew, Gaussian computation on the SEAC. He was game. and Hadassah and I grew to know Phil’s family: his I anticipated that it would be desirable to work in wife Terry and his children. One of his sons was double-precision arithmetic to about 30 decimal born in Washington, and we were invited to the places, and Phil, who was much more skillful at brit. There we met Phil’s father and his mother. His SEAC coding than I, agreed to write the code that father was a major chassidic rabbi in Philadelphia would effectuate the double precision. and “held court” there with many followers. But first I had to devise a numerical strategy. Some years later, on one of my professional The n abscissas of the Gaussian integration rules trips to Philadelphia, I was able to meet Phil’s are the roots of the Legendre polynomials of de- sister, Margola. I believe she had or was getting gree n. The weights corresponding to the abscissas a degree in philosophy from the University of can be obtained from the abscissas by a number of Pennsylvania. She showed me around tourist relatively simple formulas. I proposed to get the Philadelphia and later we took in a summer theatre Legendre polynomials pointwise by means of the production of “Amphitryon 38” (Giradoux/S.N. known three-term recursion relation. I would get Behrman) with Kitty Carlyle Hart in one of the their roots by using Newton’s iterative method, roles. In the course of our wandering, Margola told starting from good approximate values. These me quite a bit about how it was growing up in a starting values would be provided by a beautiful chassidic court in Philadelphia in the late 1940s. I asymptotic formula that had been worked out in was so amazed and intrigued by what I heard that the 1930s by the Hungarian-American mathemati- I told her she ought to do a book of reminiscences. cian Gabor Szegö. Perhaps she has. I didn’t know whether this strategy would In one of my first jobs at the NBS and as part work. It might fail for three or four different rea- of an extensive project, I was confronted with the sons. I was willing to try, and if it worked, good; necessity of doing some approximate integrations if it didn’t—well, something is always learned by in the complex plane very accurately. I worked on failure. We could give the failure some publicity, this with Phil.1 I thought a good strategy would be and other mathematicians would avoid the pitfalls to use a very subtle and accurate scheme derived in and might then be able to suggest more successful the early 1800s by the great Carl Friedrich Gauss. strategies. Prior to 1954, the Gaussian integration rules were I wrote the code and Phil wrote the double- available only up to n 16 points. The values had = precision part. I tried to anticipate what scaling been calculated on desk calculators—an extremely would be necessary. I reread my code and checked laborious task—by Lowan, Davids, and Levenson. it for bugs. Phil checked it for bugs. I (or Phil) It was also the case that the Gaussian rules were punched up the code on teletype tape and checked that out. The tape was converted automatically to 1P. Davis and P. Rabinowitz, “Some SEAC computations of subsonic fluid flows by Bergman’s method of integral a wire, and the wire cartridge was inserted in the SEAC. We manually set n 20, crossed our fingers, operators” (1953), in M. Z. v. Krzywoblocki, Bergman’s = Linear Integral Operator Method in the Theory of Com- held our breath, and pushed the button to run the pressible Fluid Flow, Springer, Vienna, 1960. program.
December 2007 Notices of the AMS 1503 Rehovot, in connection with the WEIZAC computer (1954) and the GOLEM (1964). He was hired by Chaim Pekeris who headed up the applied math group at the Weizmann Institute. Although we were now separated, our interest in producing a book on numerical integration persisted. We worked together on the book in several places; in Providence, where Phil and his family spent two semesters at Brown in 1965–66, and from February to May 1970 in Rehovot where my wife and two of our children, Ernie and Joey, spent three months. Again, in 1972, I was by myself in Rehovot for about a month, staying in the San Martin Guest House of the Weizmann Institute. With the publication of the third edition of Meth-
Photograph courtesy of the authors. the of courtesy Photograph ods of Numerical Integration in 1975, my interest P. J. Davis (left) and P. Rabinowitz (right) in the subject slackened, though I believe that Phil laying out an algorithmic strategy. Circa 1955. published papers in the topic from time to time. He also did a book A First Course in Numerical The SEAC computed and computed and com- Analysis with Anthony Ralston which has gone puted and computed. Our tension mounted. Fi- through several editions. nally, the computer started to output the Gaussian In between and in the years that followed, I abscissas and weights. Numbers purporting to be would see Phil from time to time at conferences in such started to spew out at the teletype printer. different parts of the world. In 1969 we were at a The numbers had the right look and smell about conference at the University of Lancaster. The first moon landing occurred during the conference on them. We punched in n 24 and again pushed the “run” button. Again, success.= And ditto for even July 20 and the sessions were suspended while—all higher values of n. agog—we all watched on the TV. The last day of The staff of the NBS computing lab declared the conference occurred on Tisha b’Av. Phil pre- us “Heroes of the SEAC”, a title awarded in those pared to leave the conference early and return to days to programmers whose programs ran on the London. I asked him why. He replied that sundown first try—a rare event—and for some while we had occurred earlier in London than in Lancaster and to go around wearing our “medals,” which were so he would be able to break his fast sooner. An drawn freehand in crayon on the back of used example of his humor. teletype paper. (The word “hero” was in parody 2 of the practice in the Soviet Union of declaring Aviezri S. Fraenkel persons “Heroes of the Soviet Union” for this and Pinny (I never called him Phil) grew up in Phila- that accomplishment.) delphia in a chassidic-zionist family. Since there This was the first electronic digital computa- was no Jewish day school there at the time, he tion of the Gaussian integration rules. In the years studied Jewish subjects with a private tutor who since, alternative strategies have been proposed, came to his house for a few hours on a daily basis. simplified, and sharpened (by Gautschi, Golub, and While in high school, and later at the University of others). And though all the theoretical questions Pennsylvania, he attended Talmud lessons given in that kept us guessing in 1955 have been decided various synagogues in Philadelphia. He continued positively, there are many problems as yet un- these studies until his deathbed. solved surrounding the Gauss idea. At the university he studied medicine, but at Phil and I also worked together—in an experi- the end of the first year he did not take a test that mental fashion—on the numerical solution of ellip- took place on Saturday, in order not to desecrate tic partial differential equations using expansions the sanctity of the Sabbath, so he switched to math. in orthogonal functions, and published a number He got his first, second, and third degree from of papers on that topic. the University of Pennsylvania during 1946–1951. For Phil and me, our success and our continued There was an important interlude: during 1948–9, interest in approximate integration led to numer- Pinny was chosen to go to the new Servomecha- ous papers and to a book on the topic which, over nism Laboratory at MIT, where he joined the Whirl- the years, has been widely used and referenced. wind Computer Project numerical analysis group. Our Methods of Numerical Integration, Academic There he acquired his first experience in writing Press, has gone through three editions. programs for a digital computer, interacting with Sometime in the mid-1950s Phil decided to “make aliya” to Israel. An opportunity opened up 2A shorter version of this part, in Hebrew, appeared re- for him at the Weizmann Instuitute of Science in cently in a Weizmann Institute publication.
1504 Notices of the AMS Volume 54, Number 11 infrastructure for numerical solutions of mathematical problems. In addi- tion he gave programming courses at various levels to many people who later became the leading programmers in Israel. In addition to Institute scien- tists, key personnel from government, defense, and industry participated. Pinny was the pioneer who triggered the large potential of software and high-tech industries in Israel. Pinny taught numerical analysis at the Hebrew University, Jerusalem, and Tel Aviv and Bar Ilan Universities, in ad- dition to the Weizmann Institute, and helped various colleges to establish computer science programs. In 1968 he received the annual prize of the Israeli Information Processing Society, the Israeli parallel of the U.S.-basedScience. of Institute Weizmann the of permission with reproduced Photograph The front of WEIZAC. Association for Computing Machinery. He traveled extensively, collaborating with math- people such as Alan Perlis (numerical solutions of ematicians all over the continents. A conference integral equations), J. W. Carr (2-register method “Numerical Integration”, the core of his scientific for floating point computations), Charles Adams interests, was dedicated to his sixtieth birthday. (programming languages), Alex Orden and Edgar The meeting took place in Halifax, Nova Scotia, in Reich (solution of linear equations). In Boston he August, 1986. also met Terry, whom he married shortly after get- He helped the defense establishment in writ- ting his Ph.D. in 1951. During 1951–55 he worked ing their first programs. During the tense days at the Computation Laboratory, National Bureau preceding the 6-day war, he wrote new programs of Standards, Washington, DC. and backup programs at the Institute, as fallback In 1954, the first digital computer in Israel protection in case the defense department’s main was constructed under the leadership of Jerry computer should become incapacitated. Estrin, who was a member of the team that had just finished constructing John von Neumann’s Among his students were applied mathemati- first “stored program” computer at the Institute cian Nira Dyn of Tel Aviv University and computer for Advanced Study, Princeton. Jerry later went to scientist Mira Balaban of Ben Gurion University. the Engineering Department at the University of In 1991 Nira organized an international confer- California, Los Angeles. The initiator of WEIZAC’s ence on numerical analysis at Tel Aviv University, construction was the late Chaim L. Pekeris, head to mark Pinny’s retirement. Mira is interested in of the Applied Mathematics Department at the artificial intelligence, especially computer music. Weizmann Institute. The WEIZAC project was re- She wrote her Ph.D. thesis on this topic, under the cently recognized by the Institute for Electrical and joint supervision of Pinny and Eli Shamir of Hebrew Electronics Engineers as a Milestone in the History University. This enabled Pinny to fuse his loves for of Computing. The unveiling of the plaque took science and art. place at the Institute on December 5, 2006. On that Pinny was a passionate connoisseur of the fine occasion the team members who constructed the arts, especially paintings, and a frequent visitor at machine received the WEIZAC Medal. Pinny and modern art galleries. A large collection of modern some others got it posthumously. paintings decorated every free inch of the walls Major operation times of WEIZAC were, addi- of his home. He had a sharp eye for recognizing tion: 50 microsecs; multiplication: 750 microsecs young talents, whose creations he purchased be- on the average; division: 850 microsecs. It had one fore they became famous, thus encouraging bud- of the first ferrite core memories with 4,096 words; ding talents. As a token of thanks, some of them, memory access time: 10 microsecs. A unique fea- such as Menashe Kadishman, dedicated some of ture of the machine was its word length: 40 bits. their creations to him. He loved music ardently, Input/output was via punched paper tape. especially that of Jean Sibelius. Pekeris invited Pinny to head the software devel- He also encouraged and guided young math- opment, which Pinny began in 1955, after relocat- ematical talents. David Harel began concentrat- ing in Israel. Pinny wrote the first utility programs ing on topology for his M.Sc. degree at Tel Aviv and built up the scientific software library, in the University. After one year he decided to leave his form of subroutines, which constituted the basic studies and become a programmer. Pinny advised
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