OTICES OF THE

AMERICAN MATHEMATICAL SOCIETY

The Growth of the American Mathematical Society page 781 Everett Pitcher

~~ Centennial Celebration (August 8-12) page 831

JULY/AUGUST 1988, VOLUME 35, NUMBER 6

Providence, Rhode Island, USA ISSN 0002-9920 Calendar of AMS Meetings and Conferences

This calendar lists all meetings which have been approved prior to Mathematical Society in the issue corresponding to that of the Notices the date this issue of Notices was sent to the press. The summer which contains the program of the meeting. Abstracts should be sub­ and annual meetings are joint meetings of the Mathematical Associ­ mitted on special forms which are available in many departments of ation of America and the American Mathematical Society. The meet­ mathematics and from the headquarters office of the Society. Ab­ ing dates which fall rather far in the future are subject to change; this stracts of papers to be presented at the meeting must be received is particularly true of meetings to which no numbers have been as­ at the headquarters of the Society in Providence, Rhode Island, on signed. Programs of the meetings will appear in the issues indicated or before the deadline given below for the meeting. Note that the below. First and supplementary announcements of the meetings will deadline for abstracts for consideration for presentation at special have appeared in earlier issues. sessions is usually three weeks earlier than that specified below. For Abstracts of papers presented at a meeting of the Society are pub­ additional information, consult the meeting announcements and the lished in the journal Abstracts of papers presented to the American list of organizers of special sessions. Meetings

Abstract Program Meeting# Date Place Deadline Issue 845 * October 28-30, 1988 Lawrence, Kansas August 24 October 846 * November 12-13, 1988 Claremont, California August 24 October 847 * January 11-14, 1989 Phoenix, Arizonat October 12 December (95th Annual Meeting) April 15-16, 1989 Worcester, Massachusetts January 25 March * May 19-20, 1989 Chicago, Illinois March 1 April August 7-1 0, 1989 Boulder, Colorado May 16 JulyI August (92nd Summer Meeting) October 21-22, 1989 Hoboken, New Jersey August 30 October October 27-28, 1989 Muncie, Indiana August 30 October January 17-20, 1990 Louisville, Kentucky (96th Annual Meeting) January 16-19, 1991 San Francisco, California (97th Annual Meeting) * Please refer to page 884 for listing of special sessions t Preregistration/Housing deadline is November 10

Conferences

June 4-August 11, 1988: Joint Summer Research July 18-29, 1988: AMS-SIAM Summer Seminar on Conferences in the Mathematical Sciences, Bowdoin Computational Solution of Nonlinear Systems Equations, College, Brunswick, Maine Colorado State University, Fort Collins, Colorado July 3-23, 1988: Summer Research Institute on Operator August 6-7, 1988: AMS Short Course: Chaos and Fractals: Theory /Operator Algebras and Applications, University of The mathematics behind the computer graphics New Hampshire, Durham, New Hampshire September 6-10, 1988: International Neural Network Society, 1988 Annual Meeting, Boston, Massachusetts (see news item, page 269, February issue) Deadlines

September Issue October Issue November Issue December Issue Classified Ads* Aug 1, 1988 Aug 31, 1988 Oct 3, 1988 Oct 31, 1988 News Items Aug 5, 1988 Sept 6, 1988 Oct 7, 1988 Nov 4, 1988 Meeting Announcements** Jul28, 1988 Aug 24, 1988 Sept 26, 1988 Oct 24, 1988 * Please contact AMS Advertising Department for an Advertising Rate Card for display advertising deadlines. ** For material to appear in the Mathematical Sciences Meetings and Conferences section. OTICES OF THE

AMERICAN MATHEMATICAL SOCIETY

ARTICLES DEPARTMENTS

781 The Growth of the American Mathematical Society·· 779 Letters to the Editor The Secretary of the Society examines the growth of the Society's services 810 News and Announcements and publications during its second fifty years, in response to the increased needs of the mathematical community. 815 NSF News and Reports 818 News from Washington 783 William J. LeVeque Retirts as Executive Director 822 For Your Information In honor of Dr. LeVeque's retirement in September, a biography is presented, The Dilemma of the highlighting his contributions to the Society. Tributes from George David Golden Age Mostow, President of the Society, and Irving Kaplansky, President of the 826 50 Year Anniversary Society during 1985--1986, are also included. 828 1988 AMS Elections 785 The AMS- Then, Now, and Soon 829 Meetings and Conferences In a talk presented at the 842nd meeting of the Society, William J. LeVeque, of the AMS (Listing) who will retire from his post as Executive Director of the AMS in September, 890 Mathematical Sciences reflects on the past, present, and future of the Society. Meetings and Conferences 896 New AMS Publications 790 Research Mathematicians in Mathematics Education This article, the first of a two-part series, will examind issues facing research 911 AMS Reports and mathematicians in education. The second part will describe educational Communications activities being led by members of the research community. Recent Appointments, 911 Reports of Past Meetings, 911 FEATURE COLUMNS 913 Miscellaneous Personal Items, 913 Deaths, 913 795 Computers and Mathematics Jon Barwise 914 VIsiting Mathematicians Jon Barwise, in setting the tone for his new column, has incorporated three articles into this month's offering. The articles explore various aspects of (Supplement) computing, in particular: Edward Zalta examines the philosophical questions 916 Reciprocity Agreements of whether algorithms should be patentable; Yves Nievergelt describes how 924 New Members of the AMS the HP28S can be used in the classroom; and N. Shankar examines the role that proof checkers can play in mathematics. · 929 AMS Polley on Recruitment Advertising 930 Classified Advertising 807 Inside the AMS: A Profile of the AMS Offices 955 Forms This article describes the workings of the Providence and Ann Arbor offices, highlighting the functions of each of their departments. ~ Centennial AMERICAN MATHEMATICAL SOCIETY .fA! Celebration

EDITORIAL COMMITTEE This year's Summer Meeting has been termed the Centennial Cele­ Robert J. Blattner, Ralph P. Boas bration of the American Mathematical Society, but it is more than Lucy J. Garnett, Mary Ellen Rudin Nancy K. Stanton, Steven H. Weintraub the occasion for the Society to celebrate its 100 years of service to Everett Pitcher (Chalnnan) the mathematical community. Yes, there will be a party atmosphere, with receptions, ceremonies, and happy hours for people to renew MANAGING EDITOR James A. Voytuk old friendships and start new ones. However, as with all Society meetings, the scientific program is the focal point and, in particular, ASSOCIATE EDITORS this meeting will provide the opportunity for mathematicians to Ronald L. Graham, Special Articles look ahead and to explore the future of mathematical research. The Jeffrey C. Lagarias, Special Articles highlight of the scientific program is the "Symposium on Mathemat­ SUBSCRIPTION INFORMATION ics into the Twenty-First Century." This symposium will examine a Subscription prices for Volume 35 (1988) ara spectrum of mathematical ideas, through a series of eighteen lectures $1 05 list; $84 institutional member; $63 individual by mathematicians selected for their outstanding contribution to member. (The subscription price for members is included in the annual dues.) A late charge of mathematics and with the realization that they will continue their 10% of the subscription price win be imposed mathematical activities into the next century. The speakers include upon orders received from nonmembers after four Fields Medalists, three recipients of the Waterman Award, and January 1 of the subscription year. Add for post­ age: Surface delivery outside the United States several recipients of other major mathematical honors. Their lectures and lndia---$10; to lndia-$20; expedited deliv­ will be more expository in nature than usual and will appeal to the ery to destinations in North America---$15; else­ general mathematical audience. where-$38. Subscriptions and orders for AMS publications should be addressed to the Amer­ ican Mathematical Society, P.O. Box 1571, An­ The normal practice of having Special Sessions will not apply at nex Station, Providence, Rl 02901-9930. All or­ ders must be prepaid. this meeting, so as to provide the opportunity for the expanded set of invited addresses. Symposium lectures will be given each ADVERTISING morning and afternoon, except for Monday morning which will be Notices publishes situationa wanted and classi­ devoted to the Opening Ceremonies. Each lecture will be given in the fied advertising, and display advertising for pub­ lishers and academic or scientific organizations. Providence Performing Arts Center and they have been scheduled Copyright@ 1988 by the American Mathemat­ so as to avoid conflict with other major scientific activities. At least ical Society. All rights reserved. Printed in the one contributed paper session will be held each day, and over 125 United States of America. papers will be presented at these sessions. The AMS-MAA Invited The paper used in this journal is acid-free and falls within the guidelines established to ensure Addresses, at this meeting, will be presented by Raoul Bott, Peter pennanence and durability. @ Lax, and Saunders Mac Lane and each will explore various aspects [Notices of the American Methematlcal Society is of mathematics during this century. The scientific program will published ten timas a year (January, February, be complemented by the AMS Short Course "Chaos and Fractals: March, April, MayfJune, July/August, Septem­ ber, October, November, December) by the Amer­ The Mathematics Behind the Computer Graphics" and a series of ican Mathematical Society at 201 Charles Street, MAA Minicourses. These activities will take place on Saturday and Providence, Rl 02904. Second class postage paid Sunday, prior to the Centennial Celebration. at Providence, Rl and additional mailing oflices. POSTMASTER: Send address change notices to Notices of the American Mathematical Society, All in all, the meeting will be rich in mathematics and enjoyable for Membership and Seles Department, American Mathematical Society, P. 0. Box 8248, Provi­ its social activities. dence, Rl 02940.] Publication here of the Soci­ ety's street address, and the other information in brackets above, is a technical requirement of the U. S. Postsl Service. All correspondence should be mailed to the Post omce Box, NOT the street address.

778 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY MS2000 Committee Representation community, for USSR, (in a spe­ The April 1988 Notices listed the Letters cial letter to my superior colleagues), 16 individuals who have accepted queries possibility was one of the appointments on the MS2000 Over­ to the Editor main points. sight Committee. (As many as three Maybe make it irregular, not close? other appointments are under con­ answers. Many of my queries could To tell the truth I don't like the pos­ sideration.) not pass the USSR boundary, some sibility to start correspondence with All members are either non-math­ where somewhy rejected by AMS­ AMS like that I had with Soviet Air­ ematicians or academicians. It seems nevertheless, many were published. lines for their red-tape regulations, a serious oversight that not one is And now-alas. If the decreased num­ but the closure of queries is a bad currently practicing mathematics in ber of queries is the unique reason­ precedent. Shall we open Notices with government or industry. well, to tell the truth I could (and the same fear as we here some years Stanley J. Benkoski can) personally send as many sense­ ago opened Soviet newspapers: what Daniel H. Wagner, Associates ful queries as you wish-I thought it else is made worst-without any con­ Sunnyvale, California nonpolite to send too many of them. sultation with us? (Received April 25, 1988) I'm really disappointed by the I'm not a lawyer but to my mind it is necessary that such decisions Queries Column manner in which queries column was closed: a red-tape manner recalling be made by AMS administrations I am appalled that the Notices Edi­ the worst Soviet traditions, without as a whole, after public discussion­ torial Committee has decided to dis­ any previous discussions (or even Editorial Committee must not act as continue the Queries Column, which a suggestion for discussion). When a Stalin Politbureau. has been ably and conscientiously I argued 10 years ago that review­ Excuse my possibly non polite edited by Hans Samelson. The Col­ ing is good for us, for mathematical words. umn, occupying only a single page, Vladik Ya. Kreinovich is the only regular feature of the Leningrad, USSR Notices of permanent mathematical Policy on Letters to the Editor (Received March 4, 1988) Letters interest. It is ironic and galling that submitted for publication in Notices are reviewed by the Editorial Committee, Comments on a Commentary Letter the announcement appears in an is­ whose task is to determine which ones are sue which has 30 of its 128 pages suitable for publication. The publication In "Commentary on Defense Fund­ given over to paid advertisements. schedule requires from two to four months ing" (October 1987), Mikhail Katz What is more fundamental to between receipt of the letter in Providence mentions John von Neumann as an mathematical and publication of the earliest issue of No­ example of a mathematician who inquiry than asking tices in which it could appear. was questions and getting answers to Publication decisions are ultimately willing to work on military-related questions? Leading mathematicians, made by majority vote of the Editorial Com­ projects. Katz asserts "Thurston might such as J.-P. Serre (in the current is­ mittee, with ample provision for prior dis­ well be challenged to produce a fig­ sue), have taken the pains to provide cussion by committee members, by mail or ure of von Neumann's stature on answers. How many research at meetings. Because of this discussion pe­ his side of the fence." The follow­ arti­ riod, some letters may require as much as cles have been inspired by questions seven months before a final decision is made. ing passage from the late Richard raised in the column? How much of Letters which have been, or may be, pub­ Feynman's Surely You're Joking, Mr. our teaching has been enriched by it? lished elsewhere will be considered, but the Feynman!, however, casts doubt on Is there another major journal doing Managing Editor of Notices should be in­ the use of von Neumann's credentials formed of this fact when the letter is sub­ in the context of the the same work? mitted. current debate: I urge the Editorial Committee The committee reserves the right to edit And von Neumann gave me an to reconsider its decision and restore letters. interesting idea: that you don't the Column. Notices does not ordinarily publish com­ have to be responsible for the plaints about Seymour K.ass reviews of books or articles, world that you're in. So I hav~ although University of Massachusetts, Boston rebuttals and correspondence con­ developed a very powerful sense cerning reviews in Bulletin of the Ameri­ (Received February 8, 1988) can Mathematical Society will be consid­ of social irresponsibility as a re­ A great pity that you decide to stop ered for publication. sult of von Neumann's advice. queries. It was a unique possibility to Letters should be typed and in legible (Richard Feynman, Surely You're form or they will be returned to the sender, Joking, Mr. Feynman [New York: ask questions to the whole mathemat­ possibly resulting in a delay of publication. ical community, especially useful for Letters should be mailed to the Editor Bantam Books, 1986], 115). of Notices, American us, mathematicians from countries Mathematical Soci­ Stephen J. Fromm ety, P.O. Box 6248, Providence, Rl 02940, with severe postage restrictions. I of­ Massachusetts Institute and will be acknowledged on receipt. of Technology ten used it, often obtained valuable (Received April 5, 1988)

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 779 ANNOUNCING... I MathSci™ on CD-ROM

Now you can access the American Mathematical Society's online database, MathSci, on CD­ ROM (Compact Disc-Read Only Memory). The CD version, called MathSciDisc™, will combine the searching features of online MathSci with the browsing ease of printed Mathematical Reviews (MR ). For a fixed annual fee, MathSciDisc can be used at leisure without access charges or telephone connections. Semi-Annual Issue MathSciDisc will be produced by SilverPlatter ®and will be issued semi-annually. The first MathSciDisc, available in January 1989, will contain all the reviews and abstracts from MR 1985 through 1988 and over 50,000 entries from Current Mathematical Publications (CMP). The July disc will have all the information on the January disc plus the January-June 1989 updates. Access to current information between the CD issues is available online from MathSci, which is updated monthly on DIALOG, BRS, and ESA.

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The MathSciDisc 1989 annual lease fee will include the January and the July issues of MathSciDisc, the SilverPlatter search software for the mM or the Macintosh, the SilverPlatter search manual with a MathSciDisc chapter, and a toll-free help line.

The 1989 MathSciDisc will be available at a low annual lease fee, equal to that of the printed MR: Nonmembers: List price - $3,510* AMS members & MR. subscribers: additional leases- $2,106* AMS Members: $2,808* MR subscribers: $2,106* AMS members with an MR or MathSciDisc subscription: $1,685*

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TM - MathSci and MathSciDisc are trademarks of the American Mathematical Society ®- SilverPlatter is a registered trademark of SilverPlatter International N.Y. THE GROWfH OF mE AMERICAN MAmEMATICAL SocmTY

Everett Pitcher

The American Mathematical Society has grown and ex­ The programs of the Society, distributed as separata panded during its second fifty years while remaining true until 1953, became a journal, Notices, in 1954. Beginning to its' stated purpose. In 1938 the Society was an incorpo­ in 1958, abstracts of contributed papers appeared in rated membership organization with 2139 members. Its advance of meetings in the Notices. The abstracts were purpose was stated in its article of incorporation as "the shifted to a new journal, the Abstracts, which appeared furtherance of the interests of mathematical scholarship in 1980. and research." It published two journals, the Bulletin Mathematical Tables and Aids to Computation was and the Transactions, and one book series, the Collo­ a journal published by the National Research Council quium Publications. It held about ten meetings a year, (NRC) since 1943. Beginning in 1962, it was published an Annual Meeting in December, a Summer Meeting by the Society for the NRC, the title having been changed in September, and six to eight sectional meetings in in 1960 to Mathematics ofComputation. By 1966 it was a three sections, Eastern, Western, and Far Western. Its journal of the Society with representation on the Council. headquarters was at Columbia University where it had The Society has engaged in translation of books and a staff of four including an office manager. The annual journals since 1948. The most abundant source language dues were $8.00. has been Russian. Translation from Chinese was carried The Semicentennial Meeting was by far the largest on briefly and has begun again. The most recent entry is meeting held until that time by the Society, with nearly the journal Sugaku Expositions, which is the translation seven hundred persons in attendance. Summer meetings of expository papers from the Japanese journal Sugaku. were usually larger than winter meetings, but attendance The Journal of the American Mathematical Society figures around three hundred were common. is the newest of its journals, with 1988 as its year of The world of the late 1930s was one of seventy-five inception. · Ph.D.'s in mathematics per year in the United States. The Society has been associated with the American World War II had a substantial influence, both disrup­ Journal of Mathematics since 1926, and supplies two of tive and developmental. The Zentralblatt fur Mathematik its editors, but has not otherwise subsidized the journal und Ihre Grenzgebiete almost closed because of effects of since 1975. antisemitism and war. The gap was filled by the founding On the other hand, there are distinguished publica­ of Mathematical Reviews, .which began in January 1940 tions with which the Society has been associated but is with literature from January 1939. It has been regarded now no longer. It supplied some editors for the Duke as the most significant publication of the Society. journal from its inception in 1935 to 1975. This action The need for space for publication increased after initially added support in prestige to the journal, which the war and was met by several steps. The Bulletin had outgrew the need. consisted of two kinds of issues, gray issues of papers From 1940 to 1966 the Society supplied some editors and green issues of all other material. The gray issues for The Annals ofMathematics. became the Proceedings in 1950, and the number of The Society has subsidized at least nine other journals pages increased greatly. At the same time, the Memoirs in the interval 1946-1976. The principle was to help a was started as a book series (and later as a journal with new journal with a good concept when space available back issues sold separately) by diverting longer papers for publication was constricted. from the Transactions. The latter increased greatly in size as well.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 781 Growth of the AMS

Membership has grown with time. Here are a few index, is a factor of 8.4. In those terms, the dues of figures. 1938 translate to $67.00 today. In fact, the regular dues Year Number of Members of 1988 are $88.00 for those with higher professional 1940 2314 income and $66.00 for those with lower income. The 1950 4386 change can be attributed to the increase in services and 1960 6725 the changes in labor from volunteers to employees. 1970 14197 Here are some of the new services that have been 1980 19984 instituted for the members, some self-supporting and others covered from general funds. In 1938 one did The current number is almost 21000. not apply for a position. One's thesis adviser had con­ The staff of the Society has grown as well. In 1938, versations with friends who were department heads and there was a staff of four. The Society was housed in space proposed one for a position. The Society now operates an rented from Columbia University. Mathematical Reviews employment register and publishes a journal on available was housed in Providence from the beginning. When the candidates and positions. headquarters moved to Providence in 19 51 the total was The Annual Survey, covering a variety of questions 19. An Executive Director had been in charge since the on salaries and other aspects of employment, is in its end of 1949. In 1962, the entire operation was still · thirty-second year. in Providence with about 82 employees. Mathematical The availability of fellowships and assistantships Reviews moved to Ann Arbor in 1965. In 1968, there were has been listed annually since 1959, and has appeared about 87 employees in Providence and 26 in Ann Arbor. as a regular issue of Notices since 1965. Postdoctoral The number now stands at about 157 in Providence and opportunities have been listed in the same issue. These 74 in Ann Arbor. matters are about to become a separate publication. Since leaving Columbia, the Society has been housed The annual, summer, and sectional meetings were in four different rented locations in Providence and all there was in 1938. Now the Society sponsors a Mathematical Reviews at three in Ann Arbor. The head­ summer institute in pure mathematics, a summer seminar quarters is now in a building built by the Society in 1973 in applied mathematics, up to ten summer research and Mathematical Reviews is in a building bought and conferences, and several symposia each year. remodeled by the Society in 1984. From one book series and two journals the publication The style of publication has changed materially. It was program has grown to twenty journals and eighteen book in 1938 that it was first specified that articles submitted series. to The Bulletin must be typewritten. Mathematics was The last one hundred years have witnessed an un­ composed in cold type and printed by printing houses precedented growth in the volume, sophistication, and under contract. The Society has gone through a succession diversity of mathematical research. During that time, the of technological changes; today the books and journals of American mathematical community has grown from a the Society are composed on electronic computers in the small group to a much larger one that now sets the world language lEX in the Providence and Ann Arbor offices standard for many areas of mathematics. The range of or by authors themselves. They are printed and bound in services and publications the Society provides has grown the Providence office, except for a few with the largest in response to the increased needs of the mathematical press runs. community. The burgeoning of the discipline and the People inquire gently sometimes about the cost of increasingly important role that mathematics plays in the operation. The dues in 1938 were $8.00. Inflation society will provide even greater challenges as the AMS between 1938 and 1988, measured by the consumer price enters its second century.

782 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY WILLIAM}. LEVEQUE RETIRES AS EXECUTIVE DIRECTOR

Dr. LeVeque has also been involved in a number In September of this year, William J. LeVeque will retire from his of scientific sessions at AMS meetings. He spoke at position as Executive Director of the AMS. The Society wishes the Symposium on Theory of Numbers in Pasadena in to express its appreciation for his many contributions during November 1963, and at a Special Session on Probabilistic his eleven years of dedicated service. Number Theory at Urbana in November 1970. He also organized Special Sessions on Number Theory (Houston, Jaunuary 1967), and on Distribution Modulo 1, and Biographical sketch Random Number Generation (San Francisco, January Dr. LeVeque was born on August 9, 1923 in Boulder, 1974). Colorado. He earned his B.A. degree with highest honors from the University of Colorado in 1944, and received both a master's degree in 1945 and a Ph.D. in 1947 from Cornell University. From 1947 to 1949 he was a Benjamin Peirce Instructor in Mathematics at Harvard University. Becoming an Instructor at the University of Michigan in 1949, he advanced to the rank of Professor by 1960, and from 1967 to 1970 he served as Chairman of the mathematics department. In 1970, he accepted a position as Professor at the Claremont Graduate School in Claremont, California, where he served as Chairman ofthe mathematics department in 1973 and 1974. Here­ mained at Claremont until he became Executive Director of the Society in 1977. During 1951-1952, Dr. LeVeque was a Fulbright Research Scholar in Manchester, Eng­ land. He was also a Sloan Research Fellow from 1957 to 1960 in London, England and Gottingen, Germany. Dr. LeVeque was very active within the Society even before he became Executive Director. He served on the Proceedings Editorial Committee · from 1961-1963 and was Executive Editor of Mathematical Reviews in 1965 and 1966. He also served on the MR Editorial Committee and the MR Crisis Committee. In addition, he has been a member of many other Society committees, including the Invitations and Organizing Committee for William J. LeVeque the November 1963 Symposium in Number Theory; Active in national science policy matters, Dr. the Committee to Monitor Problems in Communication LeVeque served as Chairman of the Commission on (Chairman, 1967-1969); the Selection Committee for the a National Information System in Mathematics in 1968 Cole Prize in Number Theory {1971); the Committee and 1969, and as a member of the Advisory Panel of the on a Comprehensive Subject Index {1971-1972); and the Mathematical Sciences Section of the National Science Committee on Science Policy {1975-1977). Foundation from 1969-1972 (Chairman, 1971-1972). He also was a member of the U. S. National Committee for the International Mathematical Union of the National

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 783 LeVeque Retirement

Academy of Sciences (1969-1975), a United States del­ Irving Kaplansky, President ofthe Society during 1985-1986, was egate to the 1970 and 1974 General Assemblies of the asked to provide his personal reflections on Dr. LeVeque's con­ International Mathematical Union, and Chairman of the tributions to the AMS. He responded: Conference Board of the Mathematical Sciences from Some time soon it will be appropriate to look back 1973 to 1975. · at the decade of the 1980s and ask: how is mathe­ Dr. LeVeque's research interest is in number theory, matics doing? Is the enterprise in good health? A list and he has published several books on the subject, of accomplishments might include: the classification of including Topics in Number Theory in 1956, Elementary finite simple groups, the Mordell conjecture, the Bieber­ Theory of Numbers in 1962, and Number Theory in bach conjecture, the van der Waerden conjecture on 1978. In 1978, Dr. LeVeque received an award from the permanents, the four-dimensional Poincare conjecture, Special Libraries Association for editing the six-volume the structure of smooth four-manifolds, and the new work, Reviews in Number Theory, and for his association knot polynomials. Let me add that the decade also saw with MR. The award cited his contributions to the the reconciliation of mathematics and physics moving documentation of the literature of mathematics. ahead nicely. So the verdict is that mathematics is thriv­ ing. Moreover, American mathematics is clearly at the forefront, the American Mathematical Society is the nerve center of American mathematics, and all this is happening with Bill LeVeque at the helm of the Society. Comments On the mathematical political scene, the remarkable George Daniel Mostow, President of the Society was asked to David report was followed by a degree of visibility for provide his personal reflections on Dr. LeVeque's contributions mathematics and its Washington presence that would to the AMS. He responded: have been unthinkable just a short while ago. The AMS. deserves much of the credit, in the crudest quantitative first met Bill LeVeque, then a young number theorist I sense: the lion's share of the funding of the Washington on an AMS committee. I soon learned from Michigan, presence is being provided by the Society, under Bill's closely to his comments, which invariably to listen very leadership. sense and were illuminating. That impression made good We owe other things to Bill. Not the least of these has persisted throughout all the years and has of Bill are his fine two volume treatise on number theory, his repeatedly confirmed during my term as president been collection of Reviews in Number Theory, and his years AMS. of the as Executive Editor of Mathematical Reviews. scientific societies, there is a divergence Very often, in The years 1985 and-1986 were good years to be the among the staff and the scholars. Fortu­ of viewpoints President of the Society. There was a gratifying tum for the AMS, Bill has never drifted far from the nately around of the proverbial dime, from alarming deficits to distinguished academic career as a math­ moorings of his reassuring surpluses. Bus Jaco will inherit a Society that His great vigor has consistently been directed ematician. is fiscally sound, ready to march in good order to the end right target: the advancement of mathematics. at the of the century. his stewardship, our Society has significantly During What was it like to work with Bill? In a word: great. services to mathematicians, and Bill has increased its I remember with pleasure his courtesy, his wit, and his an important role in developing the revenue to played untiring efforts to get everybody to do their jobs right. for the increased mathematical activities. pay When needed, there was also a hint of steel inside the the past two years, as the need to develop During velvet glove. Many times, when a tricky discussion was visibility for mathematics in the public forum a higher bogging down, he would wait till the right moment and recognized, Bill played a unique role in leading became then firmly point to the nub of the matter. Policy Board for Mathematics through a tense the Joint I am currently having a good taste of administration. of mounting budgets. Thanks to Bill's persuasive period Let me tell something to the readers of this piece. When JPBM finally can look forward to an and steady role, we go to a meeting and find everything running smoothly and Public Affairs which can both Office of Government with every need foreseen, we probably do not give it a our purpose and be affordable. accomplish thought. (Of course, when the opposite is true we scream stature and achievements have imparted consid­ Bill's bloody murder.) The next time this happens to you at prestige to the AMS Executive Director position. I erable an AMS meeting, think about it. Somebody has worked to appreciate this keenly during my service on the came hard to make it happen. That somebody is Bill LeVeque, search committee for his successor. ably aided by the superb staff he has assembled. the many admirers of Bill LeVeque in hailing I join Bill, I wish to give you my personal heartfelt thanks the lasting contributions he has made to the American in this public way, and wish you lots of fun in the next Society. Mathematical stage of your career.

784 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY THE AMS - THEN, NOW, AND SOON

William J. LeVeque

Walter Hayman, then of Imperial College, London, had A talk presented at the 842nd meeting of the Society in Las finished a speaking tour in this country a few months Cruces, 8 April 1988. early ' and he came to Providence too. We were put to work assigning journal articles to the vanous. reVIewers,. from what seemed to be an enormous backlog. As we How I Got Here finished assigning the articles in each issue we put the latter on a pile in the comer of the room-until the As I started thinking about this talk and how the top of that pile rose to eight feet and we had to start Society used to be, I got to thinking about how I used a new one. Of course we soon lost track of what we to be-and how I came to be talking about the AMS had assigned and to whom, and probably some reviewers instead of mathematics tonight. It occurred to me that got hundreds of articles eventually. I don't know just there were one or two anecdotes that would shed some when 'eventually' was, as not a single issue was removed light on the Society's development, and at the same time from the stack and processed further during our stay provide some clues to those of you who would prefer to there! There was a new executive editor a few months stay clear of the kind of thing I have been engaged in later (Jack Lohwater) who was capable of working 18-20 these past eleven years, or, equally, to those of you who hours a day, and within about three years he had the might be interested in getting into this kind of work. journal back on schedule. . . . My first serious brush with the AMS came almost 30 The summer of '61 didn't seem especially significant years ago. During a two-year stay in Europe a few years in my life at the time, as I was otherwise busy teaching earlier, while on a Sloan Fellowship, Harold. Davenport and doing research in number theory at Michigan. But and I had begun to talk about a rather grandiose plan to in fact it strongly influenced much of what happened prepare a sequel to Dickson's History of the Theory of to me after that. By 1964 the Board of Trustees had Numbers. This was a monumental book which covered decided that the MR editorial office had to be moved to the subject from antiquity to about 191 0; it was known a large university with an abundant supply of cons~ting and used by every number theorist in the world, I specialists. When the decision was made to move It to suppose. When I returned to Michigan, Don Lewis and Ann Arbor, I agreed to take over from Lohwater, who I developed the plan further. It was to be a splendid was by then ready to retire from the job, and to set book, with chapters by· about a dozen number theorists up the new office. I had never hired or fired anyone, of the caliber of ErdOs, Turan,. Linnik, and Mahler, nor kept regular hours, nor made or attempted to abide tracing the development of their specialties, and with by a budget, and I didn't really know much about how lists of open problems as well as complete bibliographic MR worked, so it was a plunge into unkl)own waters. I listings for the period involved. But it· turned out to be hired a secretary, a Slavic expert, a proofreader, and a too expensive and the NSF didn't fund the proposal. librarian and we all went to Providence for six weeks to At that point Gordon Walker, then executive director of learn ou; jobs. Only two staff members moved back with the AMS, invited me to work for Mathematical Revie~s us in June of 1965, but somehow all our issues went off instead, and I went to Providence for three months m to the printer at most two weeks late, and we had hired the summer of 1961. a nearly complete staff of about 20 by the end of the MR was then near death. The executive editor's summer. attention had been diverted to the problem of translating A little over a year later Sterling Berberian took over Chinese mathematics, another editor was bogged down in as editor and I went off to the Moscow Congress, and the problem of producing a 1959 subject index, already again it seemed that I was finished with AMS jobs. But two years overdue, and a third was temporarily on leave. by 1968 I found myself chairman of the newly formed

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 785 AMS- Then, Now, and Soon

Committee to Monitor Problems in Communication, or have the courage or resources to push back past the date Cornm.-Comm., as it came to be known. This must have at which MR had started. been one of the most active committees in the annals of The point in recounting these activities is to show the Society; we met quarterly for a day or two for the you how I was naturally and gradually led away from better part of my three-year term, at least. This level of my career in teaching, and research in number theory, to activity was probably due primarily to Gordon Walker, a stronger interest in the dissemination of information who helped enormously through his awareness of new in mathematics. When I was invited to succeed Gordon developments in scientific communication and technol­ Walker as director I really couldn't resist, and I went to ogy; this was his first opportunity to make practising Providence again in 1977. mathematicians realize what was going on in these areas in the other sciences. How The Society Was Then Not everything we tried was successful, of course, but some things lasted: The Society was then doing many of the things it • We organized a conference at which an entirely does now, of course. In the thirty years since the new classification scheme was developed for modem headquarters had been moved from New York City it mathematics; a slight variant of it is still used by MR had outgrown four successive sites, including a portion and Zentra/blatt fur Mathematik. (By the way, if you are of a mental hospital, and the Society had finally built unhappy with either the Dewey Decimal or the Library its own building two years before I came. The hundred of Congress classification system, don't think they have or so employees were busy producing our various books gone unnoticed; Society representatives have tried many and journals, running meetings, maintaining membership times to get them improved, without success.) records, fulfilling orders, and keeping the books, just as today. The Society's computer, a Univac Spectra, was • The Mathematical Offprint Service, or MOS, was a used for all of these functions. Its use in publishing, precursor of the online bibliographic databases common along with some other exotic equipment, was quite of its today, and I think it was the first full-blown service unusual for that time, as most publishers still sent paper kind in any scholarly discipline. Each subscriber provided copy to commercial typesetting houses where hot type a profile of his or her interests, using the classification was cast on a Linotype machine. Again, Gordon Walker system just mentioned, and each article from hundreds must be credited with envisaging the effect that modem of journals was similarly classified as to its contents. technology could have on publishing. Then the computer matched articles with subscribers, Let me tell you about some of this avant garde work. from and appropriate offprints, which had been obtained The least exotic, perhaps, but to me the most impressive 1100 the publishers, were sent out. There were about technique involved the use of typewriters. A number the subscribers, but the price had been set too low, and of the journals were then prepared on typewriters, in income. service was finally killed for lack of sufficient the interests of economy, and of course it's not easy to • Comm.-Comm. also did smaller things, such as type decent-looking mathematics. The solution to that instituting abstracts and key words in AMS journals, problem still astounds me. The machines were IBM getting the zero-backlog policy adopted for all Society Selectrics, which used type balls instead of keys. In front journals, and starting a new periodical, Contents of of each of the typists were thirteen balls, with italic, Contemporary Mathematical Journals. The latter was Cyrillic, Greek, and Roman alphabets in various sizes, an inexpensive journal consisting simply of copies of as well as balls with special mathematical symbols. tables of contents of about 300 major journals. Later, its The typists appeared to use all thirteen balls with format was changed and it was renamed as Contents of equal fluency, interchanging them rapidly and sometimes Mathematical Publications. It is still thriving. frequently, and typing with no discemable diminution in • We also tried something much more ambitious, to speed, and usually without special keyboard templates. create a so-called National Information System for the Like Chinese acrobatics, it had to be seen to be believed. Mathematical Sciences. We worked at this for over a year An experiment using more technology was still under with representatives from about ten other mathematical way when I came. By that time the Society already organizations, but it never got off the ground. owned its second, improved, model of the Photon, a I left Ann Arbor in 1970 after three stormy years dur­ pioneering optical typesetter. On this machine, when ing the Viet Nam War as chairman of the department at the typist struck a key it caused an opaque disk to Michigan. During the seven years I taught in California, rotate so as to position an aperture having the shape I finally accomplished part of one of my earlier goals of the desired character in front of a beam of light. by publishing a six-volume collection of Reviews in the Lenses focused the beam on light-sensitive film so that Theory of Numbers. But it covered only the period after the image of the character was recorded in the right 1940; the subject had simply grown too large for me to place on the film, and then the operation was repeated.

786 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS - Then, Now, and Soon

Code names in ordinary ASCII letters could be used to system to the Providence machine entailed constructing call up nonstandard characters, so the potential number a rather large and intricate database. Now we are in the of characters was very large. But the Photon had far midst of converting both Providence's and Ann Arbor's too many moving parts and mechanical components, major programs to large databases on the VAX. This will of course, and it was soon displaced by a typesetting be about a five-year project in Providence, and rather program run on a computer; the code was input in less in Ann Arbor because much of the structure of the Providence and shipped on tape to a firm on Long database has already been worked out. Island, where it was processed and the material was set Perhaps I will give you some data on the size of on a phototypesetter. the various groups involved so that you can see the Equipment was also in the office for scanning mate­ magnitude of the work the two offices engage in. Twenty­ rial into the computer from manuscripts prepared with four people are now employed in the Computer Services special typewriters having bar codes beneath the various division in Providence, including programmers, analysts, letters. It was thought that the bar codes could be more and computer operators, and there are eight more in Ann reliably read than the special fonts then in use with scan­ Arbor. ning equipment. But the equipment could never be made Two other technological developments are worth men­ sufficiently robust, and the experiment was abandoned tioning. In 1978 Don Knuth gave the Gibbs Lecture on soon after I came. Another unsuccessful experiment was mathematical typography and announced his new com­ with the Flexowriter, a machine that turned keystrokes position system for mathematical text, which he called into punched tape, which in tum was fed into a com­ TEX. Even before that the Society had become involved puter. Its use had been pretty well discontinued by the with TEX, and we have contributed strongly to its devel­ time I came, but I still heard stories of mountains of opment ever since. We hired Michael Spivak to write an tangled and tom paper tape, and of millions of little add-on package, which we call J!MS -'JEX, to simplify the round dots of paper that had been punched out and were use of TEX for mathematicians. We now use these sys­ floating around the building. tems for the composition of all our books and journals, and we publish papers from TEX input tape prepared What Has Happened Since by authors. We also provide the personnel to edit the TEX Users Group newsletter, TUGboat, and we make So that's how it was in 1977. Since then, change has available to the public several collections of characters permeated every facet of the AMS operation, and I might for 1EX output, including Fraktur, Cyrillic, and our mention some of the major events and movements. definitive collection of mathematical symbols. The Univac wasn't in great healthwhen I came, and About 30 people in Providence and five more in Ann we soon installed a DEC 20-20 and began rewriting Arbor are principally engaged in the keyboarding and almost all the hundreds of production programs then composition of text for our books and journals. in use. We thought the new computer would suffice Also on the publishing side, we decided a number for several years, but this was the first time-sharing of years ago to make MR available as an electronic machine in the office, accessible to persons who were database, which we call MathSci. You can dial in to one not computer experts. The operators couldn't install of the commercial vendors (Dialog or BRS) and search terminals fast enough to get everyone on who wanted all of MR back to 1959. Recently, we have added other to use it, so in fact the machine lasted only about six societies' databases on statistics and computer science, months, when we had to replace it with the bigger model augmenting the coverage of these subjects in MR, and 20-60. Then MR was tied in from Ann Arbor, and usage you can access all of this material through MathSci. continued to grow, so we bought a, second 20-60, thinking In your search you can use any boolean combination we would soon replace one of the two with the still larger of words from the author's name, the title of the 20-80. But then DEC announced it was discontinuing the article or book, the classification code, the journal name, entire line. Even though the machines might have lasted and the review itself. This enormously strengthens the indefinitely, there would be no new software for them access modes available in comparison with the traditional and programmers would not be interested in working author and subject indexes. The mathematical symbols on them, so after a thorough study of the options we in the reviews are in TEX input code, which is fairly purchased a VAX 8600 two or three years ago and again readable as it stands, or you can run each record through started rewriting the programs. Now we also have a VAX 'JEX and get a page that looks like the original MR 8700, and with the 20s still in place for another year or review. so, we temporarily have plenty of computing power. Speaking of MR indexes, it finally became possible, Along with hardware came software advances. MR with the help of computers and 'JEX, to produce that had already started to computerize its operations on the missing subject index from 1959 that I mentioned earlier, University of Michigan computer system, and moving the and all its brethren, so that there is finally a complete

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 787 AMS - Then, Now, and Soon set of author and subject indexes from 1940 on. This Society now owns about five million dollars worth of real grand project, which was guided by John Selfridge while estate. he was MR editor, involved some hundreds of thousands By now I've accounted for about 110 Providence of dollars in production costs. employees. The other big groups that I haven't mentioned The nearly complete computerization of Society op­ include the department that handles membership records erations, while in a way the largest effort of the past ten and order fulfillment, the fiscal department, and the years, is certainly not as visible to outsiders as some of management groups in the two offices. Altogether, there the other developments. The Notices, for example, has are now about 150 employees in Providence and about changed considerably. The Abstracts was split off as a 70 in Ann Arbor. separate journal some time ago. Several new kinds of And of course we also have had a small contingent articles were incorporated, such as short reports on re­ in Washington for the past few years, in the person of cent important mathematical developments, Dick Palais' Ken Hoffman and his staff, who are engaged in public column on technical word processing, and the annual awareness efforts. But that is really a joint activity with reports of the Secretary and myself. And there has been the MAA and SIAM. a gradual but steady improvement in the overall appear­ ance of the Notices, culminating in a complete redesign this year. Soon ... One new book series, Contemporary Mathematics, was started and now contains about 70 volumes. Another, Now let me tum to the future activities of the Society. Surveys and Monographs, was successfully rejuvenated I can't fail to mention first that this centennial year is and usually several new books now appear in it each also the year of the changing of the guard. I will be year. And 1988 saw Volume 1 Number 1 of the new replaced in September by William Jaco (or Bus, as he is Journal of the AMS. It will be followed this summer by commonly known), who is coming from Oklahoma State Sugaku Expositions, a journal of translations of Japanese University. And Everett Pitcher, the very able Secretary expository articles. of the Society for the past 21 years, retires at the end The editorial function, of readying authors' manu­ of 1988. He will be followed by Robert Fossum of the scripts for keyboarding, engages the attention of about University of Illinois. These are much younger men and 35 employees in each of the Providence and Ann Arbor they will surely bring many new ideas with them. There offices. are also a number of young and very active members The st~ady-state meetings program is very similar of the Council who can be expected to influence Society to what it was eleven years ago, except that we now policy. So I think you can look forward to a good number also run the ten Summer Research Conferences each of changes that I can't possibly predict. year. Of course, the ICM in 1986 and the Centennial I can predict some new developments, however, meeting coming up this summer are major perturbations simply because they are already in the works. One in what was already a very demanding program for the is that the Society will be concerning itself rather more staff involved. You can have no idea of the amount than in the past with mathematics edueation. The first of detailed planning that goes into a national meeting! visible movement in this direction came just this spring, There is a check list of over 1,000 items to take care when we sent off to the NSF a joint proposal, with the of, from the simple matter of learning the telephone MAA, for funding a newsletter concerned exclusively number of the local FBI office to the preparation of the with collegiate mathematics education. This will be a complete meeting schedule. The collection of instructions · bimonthly at the outset, in newspaper format, and it for everyone working at a meeting runs to more than 200 will be sent free during 1989 to all members of the pages. I doubt very much that any other society takes AMS, MAA, and AMATYC, as well as to all teachers of such pains as does the AMS to see to it that meetings are advanced placement mathematics in the country's high as pleasant and smooth as possible for the participants. schools. Also, the Executive Committee of the Council is These meetings, the SRCs, and another 15 meetings now developing a committee structure that will enable the and conferences are organized and run each year by a Society to work effectively with the MAA, the MS2000 staff of ten. project, and the Mathematical Sciences Education Board Five or six years ago we decided we could do our own on problems of common concern. The AMS has a printing and warehousing more cheaply than we could parochial interest in these matters, of course, because pay someone else to handle these functions, so we put of the predictable shortage of PhDs in mathematics to a rather large addition onto the Providence building to staff the nation's colleges and universities during the next house them. About 20 employees are required to staff decade or two. But there is also growing awareness among these operations~ And three or four years ago we bought many research mathematicians that our participation is the building occupied by MR in Ann Arbor, so the needed if the spectrum of problems in mathematics

788 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS - Then, Now, and Soon education from the elementary schools through graduate I would guess that the Society will establish an school is to be properly addressed. electronic bulletin board in the near future. Anyone with In the publications area, I recently mentioned in my a microcomputer and a modem can then dial in and annual report in the Notices that several new books series keep in touch with any of various special interest groups, have been authorized. These include books on the history such as the people interested in a certain mathematical of mathematics, reprints of important older books that specialty, a committee, an editorial board, or a collection have been allowed to go out of print by other publishers, of calculus teachers. This service may be free at first, and translations of books originally published in Chinese if Federal funding can be obtained, and in any case it or Japanese. should be quite inexpensive. Speaking of translations, we have a contract to pub­ Efforts are now being made to find ways to get math­ lish a new Soviet journal, scheduled to commence in ematical publications to Third World libraries at costs 1989. And it appears that glasnost' may enable the So­ they can afford. These may include gifts of, or large dis­ ciety to establish more direct relationships with Soviet counts on, AMS publications, as well as arrangements for mathematicians who are writing books, so that we can donations of unwanted books and journals by American copublish books in English more or less simultaneously mathematicians. with the Russian editions. As regards meetings, I think that it can be predicted The big news about MR is that it will soon be that there will have to be serious reconsideration of published in an entirely new form; as a compact disc. Summer meetings. The AMS is one of the relatively The contents of about five years' worth of issues of MR few organizations that holds two national meetings a can be digitally recorded on one CD-ROM, as this kind year, and the one in the summer regularly costs tens of of disc is called. Using appropriate indexes and search thousands of dollars more than the income generated software, also recorded on the disc, you will be able from registration fees. It may be that with the ten to search the disc just as you can MathSci, described annual summer research conferences, and all the other earlier, and see the reviews on the monitor. With a conferences, symposia, and meetings that everyone can printer attached to your microcomputer, you can search find to attend, the Summer meeting is an anachronism. I and read the reviews retrieved at your leisure. The price have no inside knowledge on the subject, but I do foresee of a subscription, in which the discs will be updated a reexamination of the Society's meetings program. semiannually or maybe even quarterly, will be only a few So these are the principal things I can see coming. hundred dollars for individuals at institutions having Both the new secretary and the next executive director subscriptions to the discs. This may sound expensive at are articulate and thoughtful mathematicians, and I am first reading, but it is extremely low by the standards of sure you will be hearing from both of them next year, the CD-ROM industry, and it will give you much more with really interesting ideas for the future. I suggest you powerful access to the mathematical literature than does stay tuned. MR on paper. Somewhat less exciting, perhaps, is that Mathematical Reviews, in collaboration with Zentralblatt jar Mathe­ matik, is preparing a revised version of the classification scheme, to be used in both journals starting in 1990.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 789 RESEARCH MAmEMATICIANS IN MAmEMATICS EDUCATION Part 1: Background & Issues

mental," and mathematicians are the ones who know Research mathematicians have traditionally had little to do with which mathematical concepts are truly fundamental. For school mathematics, but this situation appears to be changing. example, some argue that the study of geometry should This article, the first of a two-part series, will examine the back­ be replaced by a course on tilings in the plane. Whether ground and issues pertaining to the involvement ~f resea~ch or not this is an appropriate direction for school math­ mathematicians in education. The second part w111 descnbe ematics to take "is a question for mathematicians," says some educational activities now being led by members of the research community. Young. "We have to be involved in arguments like that." Other contributions that mathematicians can make to education include providing a solid mathematical train­ ing for future teachers and improving the understanding At a recent meeting of an advisory panel for a govern­ of current teachers. In addition, mathematicians are the ment agency, one university mathematician, prominent only ones who can ensure that the profunditi~s of ma~­ in the national science policy scene and a well-respected ematics are part of the teaching of the subject. Theu researcher asked for an explanation of the acronym enthusiasm and love for mathematics can also be a NCTM (National Council of Teachers of Mathematics). powerful motivator for both students and teachers. And His ignorance of this enormously influential organization finally, research mathematicians can lend legitimacy to of more than 70,000 school mathematics teachers across the entire mathematics teaching enterprise. the country exemplifies the endemic lack of communica­ Yet very few research mathematicians show active tion between research mathematicians and mathematics commitments to education: queries about such individu­ educators. als repeatedly tum up the same half dozen or so names. However, this situation appears to be improving. The main reason for this lack of involvement is that good Spurred by the declining numbers of college degrees mathematical research and a substantial commitment to awarded in mathematics, outdated mathematics curric­ education are time-consuming activities, and very few ula that neither take advantage of new technologies have the capacity to do both at the same time. Lenore nor impart a sense of the beauty and utility of the Blum, professor of mathematics and former. department subject, and the general climate of educational reform, head at Mills College in Oakland, California, has been more mathematics researchers are responding to calls involved in both education and research. She says that, to "build bridges" between the research and education to do a good job in education, "you can't do it in your communities. spare time. It takes as much time and commitment as research. It's not just what happens in the classroom, The Roles of Research Mathematicians but how you build a community of teachers that respect themselves and their profession." As the group with the deepest and most sophisticated In addition, says Blum, "educational activities are not understanding of the subject, research mathematicians afforded the prestige that research activities are." Work clearly have important contributions to make to math­ in a local high school might count as community service, ematics education. The most fundamental role is as­ but is seen as completely separate from scholarly activ­ certaining that what is taught is accurate; and that the ity. Blum says that too often mathematicians are lab~led general direction of mathematics education is appropri­ either as researchers or educators, and the mathematical ate to the discipline and the way it is changing. Gail community is not flexible enough to allow people to Young, Program Director for Materials Development ~t move between the two roles. "The mathematical com­ the National Science Foundation (NSF), says that, m munity has to take responsibility for giving innovative school mathematics, "we want to teach what is funda-

790 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Research Mathematicians educational activities on the part of mathematicians the scientists in other disciplines may be more accustomed to recognition they deserve." dealing with complex administrative structures because These problems are perpetuated by the academic many of them do research within the administrative system, in which one's research publications, not one's structure of a laboratory. "Mathematicians tend to be commitment to education and teaching, are the basis much more solitary in their work." for tenure decisions. Becoming seriously involved in In addition, Stevens says that attempts to improve educational matters, especially at the precollege level, mathematics education can have greater ramifications can imperil one's academic career. Herbert Clemens, than in science. "At the pre-high school level, there professor of mathematics at the University of Utah, is so little science instruction and there is no fixed says he believes that mathematicians have recently begun curriculum that needs to be covered," she said, so to show greater interest in education, but they lack there is more freedom to experiment. In some cases, institutional support to become more involved. If, for improvement in science education means introducing example, universities would release mathematics faculty science classes where there were none, but mathematics from some of their teaching duties to allow them to spend is generally a compulsory course of study. "Intervening time with elementary school teachers, their research need in the mathematics curriculum is more difficult because not suffer. But, says Clemens, "the universities don't show people think they know what should be learned in a willingness to do that." mathematics," Stevens notes. Because many view the existing mathematics curriculum as good and valuable, Comparison with Other Sciences "experimentation is seen as far riskier." Indeed, many-mathematicians included-have Of course, the "publish or perish" phenomenon is not viewed this curriculum as somehow preordained, and unique to mathematics-it exists in all academic disci­ have felt that substantial change is unnecessary and per­ plines. So how does the involvement of mathematicians haps unwise. However, this perception is changing, as in education compare to that of researchers in other computers and calculators threaten to render obsolete areas of science? Few have a sufficiently broad per­ much of the mathematics students learn at the precollege spective on both science and mathematics education to and even the college level. Stevens says that the intel­ give a definitive answer. However, most seem to agree lectually challenging question of what students need to that researchers in such areas as physics, chemistry, and know about mathematics in the computer age is begin­ biology have been more active in education than have ning to attract the attention of the mathematics research mathematicians. For example, the main professional so­ community. ciety for chemists, the American Chemical Society, has The use of the computer is one of the most important an educational division that focuses on chemistry edu­ influences in mathematics and science education, and cation at the college and precollege levels. One of the mathematicians have lagged in this area as well. Anthony current projects is to introduce societal issues into the Ralston, professor of computer science and mathematics high school chemistry curriculum in order to make it at the State University of New York at Buffalo, points more relevant to the students' lives. to little use of technology in mathematics classrooms as T. Christine Stevens, on leave from her position as evidence of too little concern about education on the professor of mathematics at Arkansas State University part of mathematicians. "Mathematicians at universities to serve as a a Program Director for Teacher Preparation have been less willing to use technology in teaching and Enhancement at the NSF, speculated on a number than those in other disciplines and less willing than of reasons why mathematicians are less involved in precollege educators," he says. Ralston says that because education than other scientists. For one thing, she said, mathematicians generally use computers less in their chemistry, physics, and biology are generally electives research than do scientific researchers, using computers in high school, and researchers in those disciplines in the mathematics classroom requires a dedication to know that unless they interest students in science before teaching that too few mathematicians have. college, the students' exposure to these subjects will be minimal. Because mathematics courses are usually Detachment from Societal Concerns required, exposure to the subject is automatic. Stevens also pointed out that a good school project According to Blum, one reason that mathematicians for teacher enhancement or materials development can are less involved in education than scientists is that be "a big administrative deal" involving mathematicians, mathematicians tend to be more detached from societal mathematics educators, the school system, and so on. "It concerns in general. "Professionals in other sciences can involve research mathematicians in many things they have been more astute in terms of where they fit into may not know anything about," such as school boards, the political and social structure of our society," she parent groups, and teacher unions. Stevens says that says. But she notes that in the past five years or so,

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 791 Research Mathematicians the mathematical community has been reaching out This analysis may apply to mathematics education as in various ways to forge stronger connections with the well. "The Underachieving Curriculum," a 1987 national larger scientific community and with the public. She cites report on the Second International Mathematics Study, declining research funds for mathematics and decreasing sent a tremor through the mathematical community and numbers of collegiate mathematics students a~ pressures the nation with a host of alarming statistics about the low that contributed to the change. mathematics achievement of U.S. students as compared Blum also says that women research mathematicians to their international counterparts. One of the main have outpaced men both in innovation in educational problems, the report says, is that U.S. students typically activities and in level of involvement. The reason is follow a "spiral curriculum" in mathematics, in which that the lack of women in mathematics naturally led the same topics are revisited year after year, with little women mathematicians to consider ways of attracting deepening of understanding and little progress. "The more female students to become interested in the sub­ U.S. mathematics curriculum is characterized by a great ject. "Women research mathematicians have known for deal of repetition and review," the report says. "The a long time that there is a crisis in mathematics educa­ continued dominating role of arithmetic in the junior tion because so many female students are filtered out of high school curriculum results in students entering high mathematics careers," says Blum. Now that the number school with very limited mathematical backgrounds." of male mathematics students appears to be declining, . A research mathematician hoping to contribute a more she says, the problems of mathematics education are re­ sophisticated understanding and broader perspective on ceiving more widespread attention in the mathematical the subject is likely to be disappointed by the students' community. In addition, Blum notes that many innova­ weak backgrounds. tive programs to encourage girls in mathematics have This lack of quality training in school mathematics is developed methods that are effective with all students. now being felt in the workplace. Many U.S. companies are finding that, because of advances in manufacturing and business procedures, employees' mathematical skills International View are inadequate to handle the new tasks. For example, U.S. mathematicians do not measure up well against Motorola Incorporated found its employees lacked the their international counterparts when it comes to educa­ mathematical skills to understand a new statistics-based tional activities. The greatest European mathematical re­ quality control system at one of its plants. The company searchers have a history of writing elementary textbooks responded by offering six hours of remedial training in and lecturing to school children. Extensive reforms in grade school mathematics for all of its workers. And school mathematics education in the Soviet Union were this is not an isolated situation: a recent survey found led by Kolmogorov, Pontryagin, and Vinogradov. Klein, that 30% of American companies with 10,000 or more Hilbert, Hadamard, Banach, and others were involved employees offered some sort of remedial training. in mathematics teaching and education at the precollege level. In fact, in all areas of science, it has generally been New Math true that researchers in other countries have been far more influential in educational matters than have their Many believe that one reason research mathematicians counterparts in the United States. are not more involved in education is that the "new There are a number of reasons for this difference. math" movement of the 1960s is widely perceived as a In the case of the Soviet Union, the mandated national failed attempt by research mathematicians to improve mathematics and science curriculum makes it much the school mathematics curriculum. A product of the simpler to revise the curriculum if need be. In the U.S., post-Sputnik emphasis on mathematics and science edu­ changing what is taught in the schools means dealing cation, new math was a collection of innovative school with the nation's 16,000 school boards, whose ideas of mathematics curricula that stressed understanding of the what is important and worthwhile can and do differ. underlying principles of mathematics over drill in math­ Jack Wilson, executive director of the American ematical computation. While many different curricula Association of Physics Teachers, offers one explanation came under the collective heading of new math, the best for the greater educational involvement on the part known materials were produced by the School Mathe­ of European physicists. He says that because there is matics Study Group (SMSG), funded primarily by the a much greater emphasis on physics and science in NSF. The SMSG began in the late 1950s and continued general in European schools, the students progress much its activities in one form or another into the 1970s. more rapidly and can handle more sophisticated and New math took a beating in the news media, which substantive topics capable of capturing the researchers' told stories of students who could not calculate because interest. The researchers' involvement naturally promotes of new math and of frustrated parents who could not help the continued emphasis on science in the schools. their first-graders with the esoteric homework problems

792 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Research Mathematicians they were assigned. Even the better new math curricula of the curricula "grossly underestimated" the difficulties were denounced. For example, some believe the SMSG of changing the mathematics curriculum. At the ele­ curriculum was one of the best of the new math lot, but, mentary school level, the task of training the teachers because SMSG was seen as the flagship of the new math to use the new math curricula was never adequately movement, it was ridiculed with all the rest: SMSG was addressed. While about half of the nation's 135,000 sec­ said to stand for "Some Mathematicians Sure Goofed" ondary mathematics teachers attended NSF-sponsored and "Some Mathematics, Some Garbage." institutes on new math, only a small portion of the There were several main themes common to the new nation's 1.1 million elementary school teachers received math curricula. Most of them emphasized the discovery such pedagogical instruction. In addition, elementary method of learning, which is intended to enhance the school teachers generally had rather weak mathematics students' conceptual understanding by leading them to backgrounds and were unable to properly ~tilize the discover on their own the internal logic and coherence curricula on their own. While secondary school teachers of mathematics. In addition, new math was centered were better prepared to use the new math curricula, there on the assumption that mathematics could be organized was pressure not to change the traditional mathematics around a few basic ideas, such as the field axioms. courses for college-bound students. This idea was tied to the discovery method by the As a result, the new math movement was not as belief that an understanding of the logical, deductive influential as is commonly thought. Fey was involved in nature of mathematics would elucidate the subject's a retrospective study in the mid-1970s that evaluated the structural characteristics. A third theme, prompted by success of the new math movement. "We found that, the ever-increasing pace of mathematical discovery, was for the most part, new math had far less impact than the introduction of more sophisticated mathematics to it is credited with," he says. Sowhy the perception that younger students. new math was a huge disaster? Fey claims that the truth Many believe that one of the shortcomings of new about the impact of new math simply did not make a math was that it ignored the psychological aspects of good story; far more interesting were the unflattering curricular development. Ed Dubinsky, professor of math­ caricatures prevalent in the news media at that time. ematics and education at Purdue University, says that According to John Dossey, past president of NCTM and one of the problems with new math was that all of the professor of mathematics at Illinois State University, thinking was about the mathematics. "No one asked, new math was a "press event." The few curricula that what are kids capable of?" he says. "It made sense math­ were beyond the capabilities of the students and their ematically, but can kids learn it?" For example, some of teachers were the ones that got publicity, he says. the new math curricula stressed the use of nondecimal Many education experts agree that new math had an based number systems in the middle grades. The idea overall positive effect. "That the elementary mathematics was to develop the abstract concept of a numeration curriculum is more than just arithmetic is a consequence system, but some have questioned whether students of of new math," says Fey. The inclusion of such topics as that age are capable of grasping that level of abstraction. geometry and the beginnings of probability and statistics In the early 1970s, many held the new math move­ is a result of the new math thrust. In the secondary ment responsible for declining scores in standardized schools, new math was responsible for the introduction mathematics tests and called for a "back to basics" ap­ of inequalities, the expansion of the function concept, proach, stressing drill and ~omputation. The connection and the greater prominence of algebraic and logical between new math and declining test· scores is question­ structure. able because comparable declines were observed in the New math had some positive effects beyond the verbal sections of the tests as well. This general pattern schools as well. For example, a 1965 study by the Con­ of decreasing academic achievement has been attributed ference Board of Mathematical Sciences found that 75% to such factors as television viewing and the permis­ of the departments surveyed said that it was new math siveness of the schools in the 1960s. In addition, the that made it possible to introduce calculus as a fresh­ proponents of new math argued at the time that the new man, rather than a sophomore course. Andrew Gleason, math curricula had never been properly implemented in professor of mathematics at Harvard University, was the first place, so the effectiveness of the approach was involved in the new math movement, and he says that unknown. one piece of evidence for the positive effect of new math is that the number of students now entering college Problems with Implementation having already taken calculus is far larger than before new math. Gleason also attributes the surge of Ph.D.'s in The implementation problem had several aspects. Ac­ the early 1970s in part to new math, saying that the more cording to James T. Fey, professor of mathematics ed­ sophisticated material stimulated many mathematically ucation at the University of Maryland, the formulators talented students.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 793 Research Mathematicians

Some believe that the perception of failure of new "there are papers being written and research being done math worsened the split between research mathemati­ that is not as creative as one might hope." However, he cians and mathematics educators. However, Dossey notes stresses that the quality of the research is improving. that the SMSG was composed of college and univer­ Dubinsky also emphasizes the importance of uti­ sity mathematicians, mathematics educators, "classroom lizing such research in rethinking education, for the teachers, and experts in the psychology of teaching and typical mathematical point of view may be inappropriate learning mathematics. Because SMSG's chairman, the for school mathematics. For example, the usual math­ late Edward G. Begle of Stanford University, was a ematical development treats metric space ideas before research mathematician, the group became identified topologies without metrics. Dubinsky says that there is with the mathematics research community. In addition, cognitive evidence that children grasp topological con­ Dossey says that research mathematicians and mathe­ cepts, such as whether or not a planar region has holes, matics educators alike were dissatisfied with many of the sooner than they grasp metric concepts, such as whether poor new math curricula. the region has comers. This particular piece of evidence is controversial, says Dubinsky, but it represents the kind of theoretical work that needs to be done to better under­ Research-education Split Persists stand the most effective ways of teaching sophisticated Whatever effect new math had on relations between the mathematical concepts. research and education communities, it is clear that the Indeed, many mathematicians who have become in­ split persists. Research mathematicians sometimes claim volved in education have a high regard for mathematics that their ideas to improve education are not welcomed educators. "Like many mathematicians, I thought mathe­ by education experts, who say that the mathematicians' matics education was not valuable, but I have developed ideas do not take into account current theories of a tremendous respect for those involved in it," says teaching and learning. Mathematics educators say that Blum, citing their creativity in rethinking educational the mathematicians do not recognize the validity of issues. According to Ralston, "the problems of mathe­ research in the psychological aspects of mathematics matics education are harder than mathematical research education. This cyclical pattern has prevented mutual in some ways, because mathematics education deals with respect and cooperation from developing. the human element." He also adds, "It would help if Nevertheless, there are some signs that the situation is mathematicians looked on mathematics education with beginning to improve. For example, college and univer­ a less jaundiced eye than they tend to." sity mathematics departments are starting to strengthen Young says that, until coming to the NSF, "I did not ties to education departments. Clemens says that, five realize how far off our views of mathematics educators years ago, eyes would "roll heavenward" in his depart-. are." Noting that the mathematics educators are doing ment if one of its members became involved with the "important intellectual work," he says that the proposals education department. Now, he says, there is more seri­ he has been receiving for innovative school mathematics ous interest and cooperation. Dubinsky says that Purdue materials are "way ahead of anything being done at the University is making a serious effort to ensure that its university level." For example, he cites a high school level two joint mathematics-education positions go to people program that will create a course on chaos and fractals, with strong backgrounds in both research and education. using computers to allow the students to discover new In addition, Dubinsky says, more mathematics de­ facts on their own. "This [course] will be brilliant," partments are accepting the task of teacher training. A he says, noting that almost all college courses on this 1986 report by the Carnegie Forum on Education and material are taught outside mathematics departments. the Economy, entitled "A Nation Prepared: Teachers for Young asks, "What will happen when the students who the Twenty-First Century," recommended that elemen­ take courses like this get to college?", for their college tary and secondary school teachers earn degrees in their mathematics classes may seem dated and boring in areas of specialty rather than in education; other re­ comparison. ports have made similar recommendations. According to The second part of this article will focus on these Dubinsky, "the movement within education to increase kinds of innovative programs, with an emphasis on those the content aspect and reduce the psychology-education in which research mathematicians are involved. Many component means an increased role for mathematics and changes are taking place in mathematics education at the mathematicians." precollege level, and it is crucial that the mathematical Mathematicians generally have not viewed mathe­ community be aware of them, not only because of matics education research as very helpful or profound. the contributions mathematicians can make, but also Dubinsky, who is known for his distinguished research because of the effect these changes will have on collegiate in both areas, acknowledges that while a great deal of mathematics education. excellent research is published in mathematics education,

794 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

Edited by Jon Barwise

Editorial Notes ...... Jon Barwise has enormous implications for all tirely new ways of communicat­ Are Algorithms Patentable? ...... aspects of the mathematical pro­ ing about mathematics. One new ...... Edward N. Zalta fession. This column is based on area we can already see emerging The HP-28S Brings Computations the assumption that the prediction is what has been called "exper­ and Theory Back Together in the is right, so that, as a profession, imental mathematics." At a re­ Classroom ...... Yves Nievergelt we need to start discussing its im­ cent meeting at the University of Observations on the Use of plications. The aim is to provide Wisconsin, Madison, I saw some Computers in Proof Checking ..... a forum where the mathematical exciting examples of this in the ...... N. Shankar community can discuss the impact study of 2-dimensional cellular au­ that the computer is having and tomata. The thrust of this work, by Editorial Notes will have on our profession. We David Griffeath and collaborators, Think about the role that language can also debate the prediction, if is to use the power, speed, and plays in mathematics. Written as there are doubters. graphic abilities of today's com­ opposed to spoken language, that There are two aspects to the puters to assemble and organize is. Even the most diehard Pla­ computer/writing analogy. In the massive amounts of data about tonist mathematician has to admit first place, computers greatly en­ the probabilistic behavior of large the practical necessity of writing in . hance the various powers that writ­ collections of interacting cellular mathematics, both in carrying out ing gives us in doing mathematics. automata. Watching the demon­ calculations and in keeping track This seems obvious in the cases strations was very exciting, as one of the details of all but the simplest of calculation and in the writing saw unsuspected hidden regulari­ proofs. And, of course, writing is of mathematics. (The use of hand ties emerge, regularities that cur­ crucial to being able to commu­ held calculators in calculus classes rently have no explanation. Of nicate our results to others, our is discussed in a review of the course, one hopes that such dis­ contemporaries and future gener­ HP-28S below.) So far the role of coveries will lead, in the long ations. It seems safe to say that computers organizing, checking, or run, to definitions and theorems without the invention of writing, even generating proofs, has been that explain the regularities dis­ mathematics as we know it would small. But there are some strik­ covered. Is this experimental work not exist. This is the kernel of ing exceptions, exceptions which real mathematics? It is hard to truth in the formalist philosophy suggest that this too will surely know what else to call it, but the of mathematics. change in ways we do not yet issue can be pursued in this col­ Computers promise, or perhaps imagine. (One of the articles be­ umn. threaten, to have roughly the same low discusses the present and fu­ While my general opinion is magnitude of impact on mathe­ ture role of computers in checking that mathematics will become in­ matics. It seems likely that in a mathematical proofs.) credibly richer as a result of com­ hundred years or so mathematics But computers will do more puters, there are some very seri­ without the computer will be ev­ than extend the power of writ­ ous problems on the horizon be­ ery bit as unimaginable as would ing by opening up new vistas ing brought about by computers. current mathematics without writ­ for the application of comput­ In particular, there is suddenly ing. That's a strong prediction, ers to mathematics, new areas of potential for many mathematical not because it is surprising, but mathematics, new ways of doing discoveries with algorithmic con­ because, for better or worse, it and teaching mathematics, and en- tent to have enormous financial

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 795 Computers and Mathematics

consequences as they become im­ expect the column will come to other mathematicians. Send arti­ plemented. And so it could hap­ contain. cles and letters about computers pen that mathematics will become • Reviews of mathematical soft­ and mathematics, or information poorer as mathematicians become . ware and hardware, both those about software you have devel­ richer. This possibility seriously used in research and in teach­ oped, to the address below. In threatens the free flow of ideas ing. the latter case, your announce­ which many of us take to be fun­ • Expository articles about math­ ment should be fifty words or less, damental to mathematics. Mathe­ ematical research topics in­ and should tell what system the matics is based on the idea that spired by computers. program uses, and whether a disk small, irrefutable steps, often by • Articles telling how computers should be sent to you for copying many people, taken in the large, are now being used in tradi­ the program. Also, software (or lead eventually to a body of knowl­ tional branches of mathemat­ hardware) that you think would edge that belongs to all who con­ ics. be of interest to the mathemati­ tributed to it. We all know how • Opinion pieces used to air cal community can be sent to me. impossible it is to assign credit for opinions about the computa­ I will try to find an appropriate the solution of an old problem, tional needs and desires of the reviewer. which has gradually given away. mathematical community. Professor Jon Barwise How can we protect the free flow • Other sorts of articles about Center for the Study of of ideas on which this accumula­ computers and mathematics, Language and Information tion of knowledge depends? How on special topics. Ventura Hall should we balance these concerns The column can also serve as Stanford University with rights of individuals to profit a clearinghouse for information Stanford, CA 94305 from their own labors? about software that people de­ Email can be sent to: Outside of mathematics, at velop and are willing to share with [email protected]. least, these are the questions the U.S. patent law is intended to an­ swer. The whole point of patents is to protect the rights of a discov­ erer while encouraging the discov­ erer to take his basic discovery out into the public realm. But do such Are Algorithms Patentable? laws apply to mathematics? It is usually assumed that they don't. Edward N. Zalta But if not, should they? These are difficult issues addressed in an ar­ Stanford University ticle below. This is clearly a matter where mathematicians need to be heard from, in this column, but also in congress, the patent office, and probably as expert witnesses When a colleague of mine in majority opinion in Diamond v. in law cases. (Recently, methods of the philosophy department told Diehr, said quite explicitly: resource allocation using the Kar­ me that an electrical engineering We defined 'algorithm' as a markar algorithm were patented. professor at Stanford had patented 'procedure for solving a given See News and Announcements in an algorithm, I did not believe type of mathematical problem,' this issue of Notices.) him. I figured that he had misread and we concluded that such For those of you who might the campus reports and had got­ an algorithm, or mathematical have missed it, this column was ten the facts wrong. I was teach­ formula, is like a law of nature, born in the previous issue of these ing a course on computers and which cannot be the subject of Notices. In the first number, we ethics, and everything I had read a patent. described the general plan for the indicated that algorithms were not column, and invited contributions. patentable subject matter. Indeed, Clearly, then, my colleague had Here is a brief review of what we Justice Rehnquist, writing the been misinformed.

796 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

However, as I made further in­ things should be done with attempts to satisfy §10 I in two quiries, I soon found that other certain substances, and in a ways: the word 'computer' in­ well-informed people on campus certain order. dicates that a device was being were saying the same thing. Ron patented, and §I 02 requires that the object the word 'method' Bracewell, an EE professor, had indicates that of the patent satisfy three further a process was also apparently been granted a patent being patented. The tests. It must patent in­ for his new method of computing be useful, novel, and nonobvious. cluded an abstract, drawings, in­ Fourier series. Overcome by cu­ troduction, summary, Computer programs that sat­ and a de­ riosity (and quite possibly, by a tailed isfy §102 may be granted description. Much to my measure of righteous indignation), patents, as long as they surprise, the summary began as I acquired a copy of the patent. do not claim protec­ tion for, or pre-empt, follows: It was entitled, "Computer and mathemat­ ical algorithms.* Method for the Discrete Bracewell In Gottschalk v. The present invention is a spe­ Benson, for example, Transform." One can infer a lot the Supreme cial purpose computer and Court found that from a title to a patent, and I an algorithm method of computation for per­ that converts binary immediately acquired certain ex­ code decimal forming a n-length real-number numbers to equivalent pectations. Before I describe the pure bi­ discrete transform. For a real­ nary patent any further, let me back up numbers was not patentable. valued function f(r), where r Moreover, in Parker a bit and tell you why I was led to v. Flook, the has the values 0, 1, ... , n - I, Court made these inferences and expectations. it clear that an im­ the Discrete Bracewell Trans­ proved A patent is a grant of a 17- method of calculation per form [DBT] H(v) in accor­ se, even when employed year right to exclude others from as part of dance with the present inven­ a physical process, making, using, or selling one's in­ is not patent­ tion is as follows: able. In that decision, vention, and includes the right to however, the 1 license others to make, use, or sell justices left room for patents H(v) = (n- )1::~,(/f(r) that involved algorithms it. To get patent protection, the as long cas(2nvrjn), as they did not object of the patent has to satisfy pre-empt the al­ gorithm itself. They where, §I 0 I and §1 02 of the patent law noted that "a process v = 0, 1, ... , n - 1 35 U.S.C. §IOI defines what is is not unpatentable sim­ ply because cas()= cos()+ sin 0. patentable subject matter: it contains a law of nature or a mathematical algo­ Whoever invents any new and rithm." And they backed this up In addition, a Fast Bracewell useful process, machine manu­ in Diamond v. Diehr. In Diehr, the Transform [FBT] was associated facture, or composition of mat­ Court upheld (5 to 4) a patent on with the DBT. The FBT is a ter, or any new and useful im­ a process for curing rubber even new method for arriving at the provements thereof, may ob­ though the process recited a math­ DBT, and it does so by sequen­ tain a patent, subject to the ematical formula. Unlike Flook, tially performing a series of opera­ conditions and requirements of the justices did not view the claim tions (such as permutations, sum­ this title. as an attempt to patent the for­ mations, etc.) on input that con­ mula itself. On the traditional reading .of sists of an 'ordered data set of n So with this background, §10 I, the object of a patent has to let's real numbers.' Together, the DBT return to "Computer and be either a device or a process. Method and FBT are claimed to be much for the Discrete Bracewell In the case of a device, there Trans­ more efficient than the Discrete form" (Patent #4,646,256).** has to be some hardware; in the The Fourier Transform and its associ­ title made it clear case of a process, some substance that the patent ated Fast Fourier Transform. has to be transformed. Indeed, the At this point, I began to be Supreme Court justices in Cochran * Presumably, such programs satisfy § 101 suspicious that the patent was v. Deener define: because they involve data (i.e., signal) pro­ seeking protection for both a for­ cessing, and hence, they meet the definition mula and an algorithm. Maybe A process is a mode of treat­ it of a 'process.' was to undermine such suspicions ment of certain materials to ** For a copy, send $1.50 to: Patent that the author of the patent (at­ produce a given result ..... The and Trademark Office, Washington, D.C. torney David Lovejoy) developed process requires that certain 20231. next a detailed description of the

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 797 Computers and Mathematics

'invention' around three 'hardware' he is seeking protection for. He doesn't mean that if it were chal­ diagrams. Figure 1 of the patent says: lenged in court, it would be up­ shows 'a discrete transform com­ held as valid. My analysis sug­ The present invention has been puter.' It is a simple block diagram gests that it would not. Indeed, in described employing several consisting of a 'control unit' that the symposium that I organized mathematical representations, provides timing and other control on this topic, the two law pro­ but the invention is not lim­ signals to a 'generator' and two fessors (who were specialists in ited to any particular repre­ 'processors' (the latter perform the patent law as it relates to com­ sentation. . . . While the in­ transform). Figure 2 shows an 'ap­ puter programs) seemed to agree vention has been particularly plication system' that utilizes the that if the Supreme Court remains shown and described with ref­ discrete transform computer. It, consistent, then patents that pre­ erence to preferred embodi­ too, is a simple block diagram empt algorithms will be judged ments thereof, it will be un­ in which a 'data collection de­ invalid. So everything turns on the derstood by those skilled in vice' feeds data to the 'computer,' question of whether the justices the art that the foregoing and which in tum feeds output to a will view such patents as essen­ other changes in form and de­ 'utilization device,' all subject to a tially just attempts to protect algo­ tail may be made therein with­ controlling device. Finally, Figure rithms.*** Presumably, they will out departing from the spirit 3 is a layout of 'one specific em­ have to face, at least implicitly, and scope of the invention. bodiment of a computer for per­ the questions raised in the previ­ forming transforms.' It shows that To me, this suggests that it ous paragraph. an input data sequence would be is really the mathematical equa­ Let us assume that patents such processed in four stages, each stage tion and the underlying algorithm as Bracewell's essentially claim being governed by a set of equa­ that are being protected, though of protection for the underlying al­ tions. The main body of the patent course, the author of the patent is gorithms. Should the Court re­ consists in detailed explanation of extremely careful not to say this. main consistent and declare such how these devices perform their Before we can decide what patents legally invalid? Let me tasks in accordance with the un­ the patent in fact protects, tough conclude with a few remarks on derlying mathematical equations. philosophical questions must be this question. It is usually said As I proceeded through this addressed. What are algorithms? Is that the purpose of patent law is description, I remained skeptical. there a canonical way to describe to promote the public exchange of For one thing, no particular piece them? How can one determine useful practical knowledge. In ex­ of hardware was ever described. whether a claim is really seek­ change for bringing your invention Rather, the patent described a ing protection for an algorithm, before the public, the laws grant series of possible 'embodiments,' especially when it does not em­ you certain protections you would some of which were 'preferred.' ploy the canonical descriptions? not otherwise have were you to The patent seemed to be just an Despite the fact that there are keep your invention to yourself attempt to portray what was es­ no generally accepted answers to (for example, if someone indepen­ sentially a mathematical and com­ these questions, the Patent Office dently invents the object of your putational discovery in the guise saw fit to grant the patent. And patent, he or she would not be of the hardware (abstractly con­ if you were to discover an algo­ able to profit from it). However, ceived) that would implement it. rithm that was novel, useful, and it is also said that the public has Secondly, the 'process' that was nonobvious, you could probably an interest in preventing anyone being protected did not essen­ get patent protection for it as well. from owning the building blocks tially involve signal-processing but As long as you couch your claims of science. That is why Newton's rather numerical transformations carefully, focus attention on the Laws and the Fundamental Theo­ of one sequence of numbers into a apparatus and processes required rem of Calculus are not patentable. another. So I failed to see a clear to implement your algorithm, and So if algorithms are the build­ sense in which §101 was being never specifically claim protection ing blocks of computer science, it satisfied. My skepticism became for the algorithm itself, it appears complete when I read the final that you have a good chance of section of the patent, where the being successful. *** I should mention, however, that one of the law professors thought that the Court author begins to say exactly what Unfortunately, the fact that the would overturn Benson. Patent Office grants you a patent

798 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

would seem that they, too, should effects of basic science no longer any key on their HP-15C. Rec­ not be patentable. pursued primarily for the intrinsic ognizing the usefulness of such Some would argue that even value of finding answers to myste­ computing power, other academic the building blocks of science rious questions about nature, but units, such as business schools, should be patentable. Why instead for the extrinsic (mone­ have already incorporated fancy shouldn't scientists be allowed to tary) value of finding such an­ calculators (the HP-12C, for ex­ profit from their discoveries? Such swers. In addition, many of us ample) into their mathematics and a policy wouldn't prevent oth­ think that property rights should statistics courses. Thus, there al­ ers from using the patented dis­ not extend to natural and mathe­ ready exists a certain experience coveries to make new .discoveries matical laws. They are just not the with computing devices within (patents allow experimental use right kind of thing to be counted somewhat abstract mathematics of the protected entity); it would as property, not even 'intellectual courses. Based upon such expe­ only require that if patents are to property.' Finally, it is certain that rience, the present essay suggests be granted for discoveries based there would be an administrative specific answers to some of Zorn's upon previously patented discov­ and bureaucratic nightmare were questions about computing in the eries, some arrangement would we to allow pure scientists and mathematics classroom. have to be made to divide any mathematicians to patent their 2. An Historical Perspective profits made from the application discoveries. Indeed, the individ­ of the new discovery. Presumably, uals Gottschalk, Parker, and Dia­ on Mathematical Computing the scientists involved will have an mond involved in the legal cases Some appreciation of the histor­ interest in the distribution of any described above were not ordi­ ical relation between computing profits, and though equitable ar­ nary litigants-each was serving as and mathematics will help in sep­ rangements could result from bar­ the Commissioner of Patents and arating the novelty from the fad, gaining on these issues, those who Trademarks, and each was justi­ for the evidence shows that many are cynical about human nature fiably concerned about the ability of the most elegant mathemati­ will agree that the transa<;tional of the Patent Office to process ef­ cal theories arose from concrete costs will be very high. fectively the flood of applications applied computations. Many of us, however, would that would inevitably result should Example 1. In Plato's time be uneasy about such a situa­ such a policy be undertaken. (about 400 B.C.), according to tion. We might fear the corrupting van der Waerden's sources [W2, p. 13], the Delians requested the gods' assistance against a plague; The HP-28S Brings Computations and Theory Apollo asked that they construct an altar of the same shape as Back Together in the Classroom* the existing one, but with twice the volume. Although the Greeks Yves Nievergelt knew how to duplicate the cube by intersecting two parabolae [W3, Eastern Washington University p. 26], they could not construct the cube root of two with ruler and University of Washington and compass alone, an impossibil­ ity proved later by Galois' theory [Wl, p. 199]. courses, and such answers as 1. Introduction Incidentally, in the opinion of While the capabilities of hand­ f~ 1 dx/(1 + x 2) = 1.570796327 in Charles H. Edwards [Ed, p. 11 ], held supercalculators have de­ calculus tests, have for some time the Greeks viewed ruler and com­ lighted students for years, less­ now required from students noth­ pass as their calculator for solv­ than-enthusiastic mathematics in­ ing more than the touch of ing algebraic equations in terms structors have been left in the of geometric magnitudes. Appar­ * Essay based on a talk presented by the quandary described by Paul Zorn author at the eighth Instructional Comput­ ently, then, the Greeks' "calcula­ [Zo]. For example, solutions of ing Conference, on 6 November, 1987 in tor" has corrupted neither mathe­ linear systems in linear algebra Wenatchee, Washngton. matics nor students.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 799 Computers and Mathematics

Example 2. Also according to have made such discoveries with­ mentioning that the yield rate of van der Waerden's historical out prior experiments with a com­ a one-year Treasury bill is the sources, the Renaissance Italians puter [H1]. [positive] solution of a quadratic had material incentives for solv­ Example 4. Not only mathe­ equation) helps give students mo­ ing polynomial equations. In 1344, maticians, but other scientists, too, tivation for learning the formula. Dardi of Pisa used his solution of utilize computers to investigate Then, after an algebraic deriva­ certain cubic equations to deter­ new phenomena before formulat­ tion of the formula (a neces­ mine the internal rate of return ing theories. For example, The sary step to qualify the mate­ on loans [W3, p. 49], while in Wall Street Journal reports that rial as mathematics), a numer­ 1535 the Venetian Tartaglia won astrophysicist Michael L. Norman ical example may aid students thirty banquets by discovering the is currently gaining some under­ in assimilating the concept. Be­ solution of a more general type standing of intergalactic gas clouds yond pedagogical examples with of cubic equations [W3, p. 55]. by picturing their evolution with a integer coefficients, real applica­ Their results then inspired their supercomputer [Kn]. tions require substantial compu­ successors' quest for the solution As the foregoing examples do tations. Fortunately, the HP-28S of the general polynomial equa­ not dismiss the merit of exper­ "knows" the quadratic formula: it tion, eventually leading to Galois' imental computing in theoretical solves A * XI\ 2 + B * X + C = 0 theory and Abel's proof of the im­ investigations, likewise they do not by returning (after cancelling re­ possibility of solving such equa­ appear a priori to forbid calcula­ dundancies) ( -B + s1 * .j(BA2- tions with radicals only [W1, p. tors in the mathematics classroom. 4 *A* C))/(2 *A). Numerically, 190], [W3, p. 86-88]. Before considering the teaching however, the quadratic formula Without implying that abstract potential of the latest supercalcu­ reserves a few surprises, regard­ algebra arose from the fear of lators, however, it may help to less of the model of calculator Greek gods or from the greed examine examples of such use. at hand. Consider the example of Italian financiers, these his­ 0.001X2+1000X -0.001 = 0. Sub­ torical examples demonstrate the Current Uses stituting the values of the coeffi­ 3. cients for A, B, and C, and select­ manner in which applied compu­ of Supercalculators tations have incited mathemati­ ing the respective values + 1 and cians to develop "one of the most in the Classroom -1 for the signs 1 produces the an­ important and beautiful theories" The following examples are in­ swers 0 and -1000000. Yet, 0 does [W3, p. 101]. These mathemati­ stances of typical classroom uses not solve the proposed equation. cians used not only their minds, of supercalculators during the past The class sits perplexed. Naturally, but also pen, ruler, and compass three years in the preparatory math­ the error comes from the division (but no calculator) as their com­ ematics course for the Executive by the "small" value of 2*A, which puting devices. In the same spirit, Master of Business Administra­ magnifies the round-off error in the had the HP-28S been available tion (EMBA) Program, at the Uni­ square root on the numerator. To then, they might well have enjoyed versity of Washington. Although, improve the situation, multiplying experimenting with it. Similarly, of course, senior executives in this top and bottom of the quadratic modem mathematicians have al­ program differ in many ways from formula by the conjugate quantity ready discovered new theorems college freshmen, the mathematics Q := -[B-sgn(B).j{B2-4AC)]/2 suggested to them by their com­ course for their program may serve leads to the numerical analysts' al­ puters. as one model for future calculus ternate formula, with X1 := CfQ Example 3. In 1984, David courses. In that context, classroom and X2 := QfA. Programming Hoffman and William H. Meeks experience shows that the planned this version yields X1 = 0.000001 III found new examples of min­ utilization of a calculator may instead of 0, an infinitely greater imal surfaces properly embedded help reveal to students the tight relative accuracy. in Euclidean spaces, and they es­ connections between applications, Observation 1. The surprise of tablished some of the new prop­ computations, and theory. an erroneous result from an appar­ erties of such surfaces, concern­ Example 5. Imagine the begin­ ently correctly programmed calcu­ ing ends, self-intersections, and ning of a lecture on the quadratic lator arouses students' curiosity. symmetry groups [H2]. By their formula. A brief allusion to a Then they learn that an equa­ own account, Hoffman and Meeks genuine application (for example, tion may admit several theoretical do not believe that they could

800 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

methods of solution, which may positive root, v, and then such such supercalculators as the HP- perform with varying degrees of calculators as the HP-12C or the 28S may now bring some theoret­ success in practice. This experi­ HP-28S find ical concepts closer to the not-so­ mathematically minded student. ence surpasses working rote prob­ r = 2/v- 2 = 0.0875995857108 lems, for students and instructor. Example 7. Consider the typi­ Example 6. Consider an intro­ in seconds, a result that students cal exercise of sketching the level ductory lecture on high-order poly­ may verify in the Treasury Bulletin sets of functions of two variables, nomial equations (one of Zorn's (September 1987, page 37): 8.76%. at any stage of the curriculum. "taboo" classroom topics [Zo, Furthermore, two students begin­ For example, figures 1 a and 1 b p. 920]), f(v) :=co+ c1 v + c2v2 + ning with different initial approx­ display two zero-sets plotted by · · · + CnVn = 0. imations obtain two different se­ the HP-28S, in effect two exer­ A short overview of an applica­ quences of iterations, which shows cises from Hartshorne's graduate tion may catch students' attention. how a concrete yield rate may il­ text [Ha, 5.1c & 5.1a, p. 35]. Also In finance, with v := 1/(1 + r/2), lustrate the abstract notion of two plotted by the HP-28S, figures 1c such equations apply to long-term equivalent Cauchy sequences. and 1 d show curves adapted from investments, each monomial cor­ Observation 2. The genuine ap­ William Fulton's text [Fu, p. 65 & responding to the present value, plication to finance can sustain 68]. As a homework problem, stu­ ckvk = ck/(1 + rf2)k, under some students' attention long enough to dents may try graphing the zero-set yet unknown rate of interest r, carry them through some theoret­ of y 2+(x2-4)(x4 -2x2+1), a vari­ of each semi-annual transaction ical considerations. Then, in the ant of exercise 3-2(d) in [Fu, p. amount ck. The positive root v theoretical absence of any "for­ 68]. Instead of plotting points by of the polynomial, if unique, is mula," some abstract theory cou­ hand, students may devote their the "yield rate", also called "inter­ pled with a supercalculator proves efforts to establishing the quali­ nal rate of return," of the invest­ satisfactory. Moreover, the class­ tative characteristics of the curve, ment, as those computed by Dardi room discussion focuses on ques­ while letting the calculator plot the of Pisa in 1344. (For the case tions of existence, uniqueness, and graph shown in figure 1d. Notic­ with multiple positive roots, con­ nature of the solution, three sub­ ing the regular singular point at sult [CP].) In practice, the concept jects usually considered better (1, 0), for instance, students may of yield rate may serve to justify mathematics than rote formulae. ask the HP-28S for the Taylor the use of taxpayers' money, as in polynomial of degree two about The Wall Street Journal's report that point, to which the calcula­ of the remarkable 25% annual in­ tor answers 12*(X- 1)A2. Indeed, ternal rate of return on Kentucky's 4. Potential Uses the curve has two tangent lines twenty-year investment in a Toy­ of Such Supercalculators crossing at (1, 0), with equations ota manufacturing plant [Ca]. As­ as the HP-288 y =! v'U(x - 1). Similar calcula­ sessing in advance the perfor­ tions confirm the vertical tangent mance of such a corporate in­ Up until now, the most pow­ lines at (2, 0) and (-2, 0). vestment may require solving for erful calculators surpassed other Observation 3. Students may the yield rate and comparing it hand-held devices mainly through discover the general properties of against other prevailing rates of their sophisticated numerical anal­ a curve faster with a supercal­ interest, because the great uncer­ ysis routines, which could solve culator than with a pencil only, tainty about future rates prevents' real non-linear equations and com­ and then they may spend pro­ the financier from estimating the plex linear systems, or integrate portionally more time establishing net present value of the cash-flow. real functions. Now, however, the those properties rigorously. Stu­ An example may help in de­ state-of-the-art calculators combine dents may thus get to see the mystifying such equations. For in­ all these numerical routines with mathematics of curves from a stance, the· thirty-year Treasury graphic displays and symbolic ma­ point of view similar to that of bond issued on 5 May 1987 gives nipulation programs that all share Hoffman and Meeks in their in­ the equation f(v) := -9989.50 + their data bases with one an­ vestigations of properly embedded 437.50v+437.50v2+·. +437.50v59 other. Rather than necessarily re­ minimal surfaces. + 10437.50v60 = 0. Calculus read­ ducing courses to mere keypunch­ ily shows that f has exactly one ing sessions, the availability of

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 801 Computers and Mathematics

(a) Example 8. A lecture on the will lead to one specific root rather roots of complex polynomials might than another? The question may begin with Steve Smale's topologi­ interest students enough to send cally algorithmic proof of the Fun­ them to the library to read about +i-..... - ....• __. .... damental Theorem of Algebra, fol­ Julia sets in [Pe]. Third, why does lowed by his proof of the conver­ the sequence of iterations appear gence of Newton's method [Sm, to spiral toward a root, rather than converging along a rectilin­ Y.2 4 ....• a 4 p. 96 & 101]. Having assimilated + 'r' = - :": statements and proofs, students ear trajectory? Students may try to -1. 7<:=<<1. 7 ' -.5{''('(.5 may then examine how Newton's explain this phenomenon by the method converges in practice. A absence of a Mean Value The­ few keystrokes suffice to program orem for complex functions. In Newton's formula the case of a root, r, this means (b) that, m general, f(r) - f(zk) =/; (r-zk)f'(zk), and therefore zk+i­ zk =-f(zk)f f'(zk) =/; r-zk; thus, into the HP-28S, which automati­ in general, zk+I - zk differs from cally runs it with complex numbers r- zk both in magnitude and in and identifies the complex plane direction. Fourth, does f really with the display. have roots at r1, r2, and r3, where 4 2 4 y = ::-:: - :x: The execution, however, may Newton's method appears to stop? reveal some unexpected phenom­ -2.3(. )<' Students experimenting with the HP-28S in the manner just described may become interested ~ ·:. ..~.· . .---2 ~ 2 4 2 in the following four questions. Y~ = (4-X )(\ -2X +1) First, why does the sequence of -2.3

802 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

(c) sider the map f: R2 -+ R2, (~) 1--+ and logical aspects of mathemat­ ics, which also happen to consti­ (y(l+sin(0.7x))-L2v'fxT) 0.21-x · tute that which employers want .Y.·· ... · Following Barry Martin's ex- in graduates. Indeed, The Wall ·.'If·.···~ ample [Ma], start with (x0, y0 ) := Street Journal [Du] reports Stuart (0, 0), and let the HP-28S com­ Reeves, vice president in charge pute and plot the first n itera­ of recruiting and development at 9 tions, (Xn, Yn) := fn(O, 0). Figure z - z + 9 = 0 Electronic Data Systems Co., as 3 shows the result. saying that "being able to think Z = 3.1 + 3i, Newton critically is more important than 0 (a) root = 0.649 + 1.088i [having] a computer-science de­ gree." The advent of such supercalcu­ lators as the HP-28S may thus have created a propitious situa­ tion for putting some mathematics r~+ •. n = 100 back into the popular mathematics t I I I ·'·,c~· I ,I ,I , I ,I J I I courses. 2 (b) Acknowledgement. I thank 9 Joyce d. Kehoe, Seattle writer, for z - z + 9 = 0 .. :::~ her kind assistance in editing this Z = 3.3 + 3i, Newton essay. 0 6. References root= 1.184- 0.423i n = 2000 [Ca] Eugene Carlson, "What's a Toyota plant worth to Kentucky? Figure 2. The hand-drawn ar­ Possibly plenty," The Wall Street The first hundred, No. 111 (9 rows show how the successive it­ Figure 3. (a) Journal, Vol. CXVI, two thousand iter­ erations appear on the screen. The and (b) the first June 1987), p. 33. Martin's example Pe­ sequence jumps off the screen at ations of Barry [CP] Roger E. Cannaday, 9). Pa­ the end of path 1 and reappears (see counterexample ter F. Colwell, and Hiram rates at the start of path 2. The hand­ ley, "Relevant and irrelevant Econ­ drawn crosses mark the indicated of return," The Engineering 5. Conclusions 1 (1986), p. roots. omist, Vol. 32, No. The previous examples indi­ 17-38. cate that the availability of sophis­ [Du] Paul Duke, Jr., "Jobs go devices may Observation 4. Even while per­ ticated computing unfilled as fewer students show number of not-so­ forming routine tasks with a su­ provide a larger interest in computer science," The students percalculator, students may need mathematically inclined Wall Street Journal, Vol. CXVII, opportunity to think to invoke theoretical results to val­ with a greater No. 106 (27 November 1987), p. the way idate their own findings. More­ about mathematics in 13. do, while over, students may stumble upon many mathematicians [Ed] Charles Henry Edwards, the potential of phenomena not described. in their stretching further The Historical Development of the minded textbooks, and may hence become the already theoretically Calculus, Springer-Verlag, New may interested in additional indepen­ students. Supercalculators York, 1979. and text­ dent reading, or perhaps in Smale's thus inspire instructors [Fu] William Fulton, Algebraic away from eleven open problems [Sm]. books authors to steer Curves, Benjamin-Cummings, Counterexample 9. Not every too many routine manipulations: Publishing, Reading, Massachu­ an au­ sequence of iterations of a map with today's technology, setts, 1978. pet peeve need appear to converge or di­ thor's new rote work [Ha] Robin Hartshorne, Alge­ symbol verge in a straightforward man­ may show up as a mere braic Geometry, Springer-Verlag, Instead, ner. To illustrate how a sequence on next year's calculator. New York, 1977. may of iterations may behave, con- the discussion of the material [H1] David Hoffman, "The now focus upon the more creative computer-aided discovery of new

JULY/AUGUST 1988, VOLUME 35, NUMBER 8 803 Computers and Mathematics

embedded minimal surfaces," The problem and Julia Sets," The Math­ Incompleteness Theorem was ver­ Mathematical Intel/igencer, Vol. 9, ematical Intelligencer, Vol. 6, No. ified in about eighteen months. No. 3 (1987), p. 8-21. 2 (1984), p. 11-20. One of the earliest proof checking [H2] David Hoffman and . [Sm] Steve Smale, "On the effi­ projects, de Bruijn's AUTOMATH William H. Meeks, Ill, "Proper­ ciency of algorithms of analysis," [dB] succeeded in checking all the ties of properly embedded mini­ Bulletin of the AMS, Vol. 17, No. proofs in Landau's Foundations mal surfaces of finite topology," 2 (October 1985), p. 97-121. of Analysis [vB]. Proof checkers Bulletin of the AMS, Vol. 17, No. [W1] Bartel Leenert van der are continuing to grow in sophis­ 2 (October 1987), p. 296-300. Waerden, Algebra I, Springer-Ver­ tication and there is as yet no [Kn] Dennis Kneale, "Comput­ lag, Heidelberg, 1971. conceivable premise for ruling the ers are making strides in portray­ [W2] --, Geometry and Alge­ task infeasible. ing scientists' ideas," The Wall bra in Ancient Civilizations, Proof checkers subvert the so­ Street Journal, Vol. CXVII, No. Springer-Verlag, New York, 1983. cial process by means of which 92 (6 November 1987), p. 25. [W3] --,A History ofAlgebra, mathematics is communicated and [Ma] Barry Martin, "The cock­ Springer-Verlag, New York, 1985. verified. atoo," The Mathematical Intelli­ [Zo] Paul Zorn, "Computing in This is not a reasonable fear gencer, Vol. 9, No. 1 (1987), p. undergraduate mathematics," No­ since mathematical methods and 72. tices of the AMS, Vol. 34, No. 6 proofs are often more interesting [Pe] H. 0. Peitgen, D. Saupe, (October 1987), p. 917-923. than the results themselves. The and F. v. Haesler, "Cayley's true source of the objection, how­ ever, is the misconception that proof checkers merely validate a purported proof. In actual fact, much of the time and effort spent Observations on the Use of Computers with a proof checker goes into locating and correcting mistakes, in Proof Checking and in refining arguments. A proof checker thus plays the role of N. Shankar a tireless devil's advocate, isolat­ ing errors, hinting at counterex­ Stanford University amples, clarifying intuitions, and highlighting inelegance. Through this process, the argument itself is polished, structured, and distilled Of all the sciences, mathe­ [BM] to check, in detail, a proof of to a point where it can be more matics is the one that has re­ Godel's Incompleteness Theorem thoroughly understood and more lied least on technology. Gen­ [SH]. effectively communicated. Robin­ erations of mathematicians have Let me start with some pointed son [RO] has recently proposed practiced their art and commu­ published objections [DM] to the that we design logics and proof nicated their ideas with only the use of proof checkers, and some checkers with the intent of clarify­ simplest writing implements. The responses to them. ing proofs. traditional mathematician perhaps The formal verification of seri­ Why should we trust a computer views computer proof checkers to ous mathematics is essentially in­ over our intuitions? be unnecessary diversions that are feasible. We shouldn't. A proof checker unlikely to have any significant, It is true that little current cannot serve as a surrogate for positive impact on the field. In research-level mathematics has mathematical judgment. However, this short piece, I would like to been computer proof checked. this does not mean that com­ suggest that proof checkers can However, a fair amount of college­ puter generated proofs do not have a useful role to play, not level mathematics has been veri­ merit the mathematician's atten­ just in theory, but in mathemati- fied with a feasible amount of tion. The Boyer-Moore theorem . cal practice. The observations be­ effort already, with the current prover, for example, can fill large low arise mainly from my use of unsophisticated level of technol­ gaps in an argument in surpris­ the Boyer-Moore theorem prover ogy. The entire proof of the ing and illuminating ways. Re-

804 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

searchers at the Argonne National collections, and so on. While it is This article benefited from the Laboratory have conjectured and obvious, it is worth noting that comments of a number of people solved a number of open problems checking whether given informal to whom I am extremely grateful. in Algebra and Combinatory Logic notions have been accurately for­ Bibliography by observing attempted mechani­ malized cannot be checked by a cal proofs [LW]. machine. It is also worth point­ [BM] R. S. Boyer and J S. It is worth listing some rea­ ing out that the choice of no­ Moore, A Computational Logic. sons why proof checking is such tations and representations, e.g., Academic Press, New York, 1979. a surprisingly difficult activity, in· sets vs. lists, functions vs. rela­ [dB] N. G. de Bruijn, A Survey case the above discussion conveys tions, can have a drastic impact of the Project AUTOMATH. In the opposite impression. If noth­ on any proof. Changes of repre­ "To H. B. Curry: Essays on Com­ ing else, these observations show sentation or notation, while easily binatory Logic, Lambda-Calculus the extent to which mathematical undertaken in an informal proof, and Formalism," J. P. Seldin and practice falls short of formal sci­ can be tedious in the extreme in J. R. Hindley, eds., pages 589- ence it is often taken to be in the the case of a formal proof. 606. Academic Press, New York, popular mind. Machine checking of realis­ 1980 One of the most serious ob­ tic, research-level proofs seems a [DM] R. A. De Millo and R. stacles to a computer verification daunting task at present, but there J. Lipton and A. J. Perlis, "Social of a given textbook or journal is a vast scope for improvement in processes and proofs of theorems proof is not the logical complexity the state of the technology. Proof and programs," Communications of the proof itself, but the over­ checkers have the potential to be­ of the ACM, 22(5):271-280, 1979. whelming amount of background come the mathematician's ana­ [RO] J. A. Robinson, "How knowledge that the reader is ex­ logue of an electron microscope, machine-oriented might a logic pected to have. A large part of especially in conjunction with the be?" Invited Talk at the Ninth the machine-checked proof of the other uses of computers in math­ Conference on Automated Deduc­ Incompleteness Theorem involved ematics such as computer algebra, tion, Argonne, Illinois, May 1988. facts that Godel took as common graphics, etc. On the basis of the [SH] N. Shankar, "Proof-check­ knowledge, e.g., the unique prime progress so far, it is reasonable ing Metamathematics," Ph.D. The­ factorization theorem. to speculate that research in com­ sis, Computer Science Department, Experience with computer proof puter proof checking will even­ The University of Texas at Austin, checking shows that informal math­ tually shed considerable light on 1986. ematical exposition makes subtle the nature of mathematical expla­ [vB] L. S. van Benthem Jut­ but pervasive use of informal con­ nation and notation, and provide ting, "Checking Landau's Grund­ cepts, concepts that have to be for­ fruitful sources of new conjectures /agen in the AUTOMATH sys­ malized prior to machine verifica­ and techniques. The pace of future tem," Technical Report. Mathe­ tion. For example, a formalization progress will largely depend on the matical Centre, Amsterdam, 1979. of a statement of the unique prime willingness of students, teachers, [LW] L. Wos, "Open Questions factorization theorem would in­ and researchers in mathematics to for Research," Tutorial presented volve definitions of collections of use and develop this technology. at the Ninth Conference on Au­ primes, equality between collec­ tomated Deduction, Argonne, Illi­ tions of primes, products of such nois, May 1988.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 805 AMERICAN MATHEMATICAL Celebrate SOCIETY 1888-1988 MATHEMA11CS theAMS 100 YEARS OF AMERICAN Centennial!

Now you can have a remembrance of the American Mathematical Society's 1OOth anniversary for your own -- a beautiful commemorative poster for your home, office, or department.

This striking poster is a photographic reproduction of the sculpture ''Torus with Cross-cap and Vector Field" by Helaman Rolfe Pratt Ferguson of Brigham Young University. The sculpture is a gift from the Mathematical Association of America to the American Mathematical Society on the occasion of its Centennial.

The poster is printed on museum quality glossy paper. The sculpture is white and photographed on a rich blue background. Torus with Cross-cap and Vector Field To order, please specify POSTER/N List $10 each, plus $2 shipping and handling per order

To order your AMS anniversary remembrance, simply send a check or money order for $10 each (plus $2 shipping and handling per order) to: American Mathematical Society Annex Station, P. 0. Box 1571 Providence, RI 02901-9930 USA

Send MasterCard or VISA orders to: American Mathematical Society P. 0. Box 6248 Providence, RI 02940 or callS00-556-7774 to order by phone in the continental U.S.

For your convenience, an order form is available at the back of this issue. Inside the AMS

A Profile of the AMS Offices

Since 1951, when the Society headquarters moved is responsible for staffing at the Providence office; and from its home at Columbia University to Providence, the Database Services Department, which manages Rhode Island, the Society's membership has grown MathSci, the online database providing an electronic from about 4400 to over 21,000. One of the reasons for index to the world's literature on mathematics, statis­ this growth is the rise of the American mathematical tics, computer science, and their applications in a wide community to a position of leadership in the world range of disciplines. Database Serivces now also offers mathematical scene. At the time of the move, the MathDisc, the CD-ROM version of MathSci. AMS had only about 20 employees, and Mathematical Reviews (MR) was only 11 years old. The Society now has a total of about 250 employees in its Providence headquarters office and the MR office in Ann Arbor, Michigan. The Society's efficient organization, dedicated staff, and excellent facilities ensure that the AMS will continue to effectively serve its membership as the AMS enters its second century. Most members are familiar with the many services the AMS provides, but many are unaware of the magnitude of this enterprise. This article describes the way the two AMS offices work by highlighting the functions of each of their departments.

The Providence Headquarters Office The AMS headquarters office is located near down­ town Providence and near Brown University. The single-level, modem design office was built for the Society in 1973. The buiiding contains facilities for almost all Society activities (except MR), including Providence a major computer operation, a complete printshop, The Finance and Administration Division is grouped and a large warehouse. The Providence office employs into three departments: Fiscal, Distribution, and Plant. about 150 people who work in various departments As its name implies, the Fiscal Department is responsi­ corresponding to different Society functions. ble for managing such functions as budget preparation The Executive Director Department is headed by and control, cash forecasting and investment, payroll, three Ph.D. mathematicians who engaged in teach­ and maintenance of financial records. The Distribu­ ing and research before becoming employees of the tion Department handles all shipping and mailing Society: the Executive Director and two Associate for the AMS, maintains the warehouse, and man­ Executive Directors. The directors are responsible for ages inventory. All purchasing, building, operations, implementing policies set by the Board of Trustees upkeep, and maintenance are provided by the Plant and the Council. This department also houses two Department. other organizational units: the Personnel Office, which

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 807 Inside the AMS

The Computer Services Division manages the com­ The Ann Arbor MR Office headquarters and puter needs of both the Providence The MR editorial offices are within walking distance Analysis Depart­ the MR office. The Programming and of downtown Ann Arbor and the Central Campus of programming, ment supplies system analysis, design, the University of Michigan. The nineteenth century, and troubleshoot­ documentation manuals, training, red brick MR building was originally constructed as ing. Computer Operations is responsible for operating a brewery. In 1984, the AMS purchased the building computer and maintaining the Society's substantial from the University of Michigan and renovated it process of chang­ facilties. The Society is now in the to accomodate the 7 5 MR staff members in the its two DEC-20 computers to its two VAX ing from departments described below. Both MR and Current are linked to the 8600 computers. The computers Mathematical Publications ( CMP) are prepared in this office by a dedicated line. The computer Ann Arbor office. facilities are an integral part of the both the AMS headquarters and MR offices, and almost every em­ ployee in both offices has a terminal on his or her desk. The Publications Division is responsible for the So­ ciety's extensive mathematical publishing enterprise, one of the largest in the world. Five departments han­ dle various aspects of book production. The Editorial Department oversees a number of functions, such as copyediting, proofreading, preparing front matter, and overall production coordination, among other tasks. The Translations Department is responsible for ob­ taining, editing, and translating various books and journals from other languages (primarily Russian) into English. Compostion Services comprises two sec­ tions: Keyboarding, which uses lEX, the computer typesetting system; and Publication Makeup, which is reponsible for finalizing pages in camera-ready form for printing. The Printing Department produces most of the Society's journals and all of its softcover books. The Marketing Department produces in-house cata­ logs, brochures, and other descriptive literature on Ann Arbor AMS products and activities. The Administration Department is responsible for The Membership and Sales Department (M&S) the management and support of the Ann Arbor op­ has four major roles. First, the department main­ eration. The Executive Editor, the Managing Editor, tains membership records for all the individual and and the Associate Executive Editor represent MR at institutional members of the Society. Second, M&S meetings of the Board of Trustees, Executive Com­ handles all orders and payments and transmits them mittee, and Council and oversee all aspects of the to the Distribution Department for shipping and han­ production processes of MR products. Also included dling. Third, it is responsible for dues collection and in this department are the Office Manager and the customer inquiries. Finally, M&S helps to prepare clerical support staff for MR. camera copy for several directories for the mathe­ The Editorial Department is staffed by professional matical sciences, such as the Combined Membership mathematicians who are responsible for selecting items List. for review in MR, for assigning the items to thousands The Meetings Department is responsible for orga­ of MR reviewers worldwide, and for editing and classi­ nizing all meetings and conferences sponsored by the fying the reviews. Special language and mathematical AMS, including the annual meeting in January, the expertise are provided by several editorial consultants, summer meeting in August, eight sectional meetings who work closely with MR editors. each year, summer institutes, seminars, research con­ The Library Department acquires and catalogs ferences, short courses, and various symposia. The mathematical publications for review in MR and Meetings Department also coordinates the Employ­ prepares the items in a form suitable for entry into ment Register during the annual meeting. the database. The Library also handles MathDoc, a

808 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Inside the AMS

document delivery service that provides copies of The System Support Department offers technical original articles to users of MR, CMP, and MathSci. assistance and advice on computer operations and MathDoc specializes in providing papers that would hardware manintenance. Its main functions are de­ otherwise be difficult to obtain. veloping and maintaining new database systems as The Slavic Languages Department translates, from necessary and working on programming projects and Russian and other Slavic languages into English, titles, system design or improvements. The Department also summaries, reviews of mathematical literature, and supports MathSci and computer communication net­ correspondence. The department also serves as a works. source of information on Slavic languages and Soviet The Production Department oversees the computer publications. operations necessary for producing MR, CMP, Math­ The Reviewer Services Department handles comu­ Sci, various indexes, review volumes, and special nication between MR and its thousands of reviewers. projects. This department also keys corrections to the The department mails all material for reviews, pro­ review text galleys, provides assistance to the Systems cesses returned reviews, and updates the reviewer and Department to run printers and manage queues, han­ paper files in the MR database. Much of the initial dles production scheduling, and coordinates deadlines. preparation of monthly MR issues is done in this The Copy Editors Department is responsible for department. copyediting and proofreading the contents of MR and The Author Records Department maintains the au­ CMP. The copyediting process includes marking math­ thor file, which contains information about all authors ematical notation, noting inconsistencies, checking for of papers that MR has reviewed or is planning to re­ grammar and adherence to MR style, and verifying view. The department's other responsibilities include references, among other tasks. work with the online database MathSci, supervising production of the MR and CMP indexes, and special Allyn Jackson projects. Staff Writer

simple equational formalism, Closely logic; semantics; first-order logics with A FORMALIZATION OF related to the formalism of the theory of finitely many variables; definability and relation algebras. There are no variables, axiomatizability questions in set theory, SET THEORY WITHOUT quantifiers, or sentential connectives. Peano arithmetic, and real number VARIABLES Predicates are constructed from two theory; representation and decision atomic binary predicates (which denote problems in the theory of relation Alfred Tarski and Steven Givant the relations of identity and set-theoretic algebras; and decision problems in membership) by repeated applications equational logic. Completed in 1983, this work culminates of four operators that are analogues of nearly half a century of the late Alfred the well-known operations of relative Tarski's foundational studies in logic, 1980 Mathematics Subject Classification: 03 product, conversion, Boolean addition, ISBN 0-8218-1041-3, LC 86-22168 mathematics, and the philosophy of and complementation. All mathematical ISSN 0065-9258 science. Written in collaboration with statements are expressed as equations 280 pages (hardcover), January 1988 Steven Givant, the book appeals to List price $60, Institutional member $48, between predicates. There are ten logical a very broad audience, and requires Individual member $36 axiom schemata and just one rule of To order, please specify COLL/41NA only a familiarity with first-order · inference: the one of replacing equals by logic. It is of great interest to logicians equals, familiar from high school algebra. and mathematicians interested in the foundations of mathematics, but also Though such a simple formalism to philosophers interested in logic, may appear limited in its powers of semantics, algebraic logic, or the expression and proof, this book proves methodology of the deductive sciences, quite the opposite. The authors show and to computer scientists interested that it provides a framework for the in developing very simple computer formalization of practically all known Shipping/Handling: 1st book $2, each languages rich enough for mathematical systems of set theory, and hence for the add'l $1, $25 max. By air, 1st book $5, and scientific applications. development of all classical mathematics. each add'l $3, $100 max. Prepayment required. Order from AMS, The authors show that set theory and The book contains numerous applications P.O. Box 1571, Annex Station. Providence, number theory can be developed within of the main results to diverse areas of Rl 02901-9930, or call 800-556-7774 to use the framework of a new, different, and foundational research: propositional VISA or MasterCard.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 809 News and Announcements

AMS Centennial Research Ph.D. from Massachusetts Institute over two years, spending the first at Fellowships Awarded of Technology in 1980 under the di­ the University of California, Los An­ rection of Norberto L. M. Kerzman. geles. His primary research interests The Society has awarded three Cen­ He taught at Princeton University are number theory and automorphic tennial Research Fellowships for 1988- from 1980 to 1985 and at Purdue forms. 1989. The recipients are Steve Bell University from 1985 to the present. of Purdue University; Don Blasius He was an NSF Postdoctoral Fellow of Bernard M. Baruch College, City in 1980 and a Sloan Fellow in 1984. University of New York; and David He plans to use his AMS Fellowship Gabai of California Institute ofTech­ over a period of two years to teach nology. half-time.

DAVID 0ABAI received his Ph.D. with Bill Thurston at Princeton in 1980. He subsequently has worked at Harvard University, the Institute for Advanced Study, the University of Pennsylvania, the Mathematical Sci­ STEVE BELL is a complex ana­ DON BLASIUS received his Ph.D. ences Research Institute, the Institute lyst who specializes in problems on in 1981 from Princeton University des Hautes Etudes Scientifiques, and holomorphic mappings between do­ where he studied under G. Shimura. the California Institute of Technol­ mains in complex euclidean space. Since then he has taught at Columbia ogy, where he is currently a Professor He has used the theory of the inho­ University (1981-1985), Yale Uni­ of Mathematics. He was the recipi­ mogeneous Cauchy-Riemann equa­ versity ( 1985-1986), and Baruch Col­ ent of NSF Postdoctoral and Sloan tions to study the boundary regularity lege ( 1987-1988), where he is cur­ Foundation fellowships. of biholomorphic mappings between rently an Associate Professor. He He plans to use this award at the domains in n-dimensional complex was at the Institute for Advanced California Institute of Technology space (n > I). Study in Fall 1981 and at the Math­ during the 1988-1989 academic year Professor Bell received a B.S. in ematical Sciences Research Institute to continue his investigations on Applied Mathematics from the Uni­ ( 1986-1987) on an MSRI Fellowship. the topology of 3-dimensional mani­ versity of Michigan in 1976 and a He plans to use his AMS Fellowship folds.

810 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY - News and Announcements

AMS Centennial 1987-1988, and three fellowships for year awards, one accompanying de­ Research Fellowships 1988-1989; however, in the several pendent; a monthly maintenance al­ Invitation for Applications, preceding years, it was not fman­ lowance; and incidental allowances 1989-1990 cially possible to award more than for travel, books, and services es­ Deadline December 1, 1988 one fellowship. sential to the assignment. The host The deadline for receipt of appli­ institution is expected to share some These fellowships ate open to in­ cations is December 1, 1988. Awards costs in the form of supplementary d"viduals five to ten years past the will be announced in February 1989, funding, as well as support for hous­ p~.D. degree (or equivalent), regar~­ or earlier if possible. ing. less of age, but below the academtc For application forms, write to The deadline for receipt of pro­ rank of professor. Applicants should Executive Director, American Math­ posals is November 1, 1988. De­ have received the Ph.D. degree be­ ematical Society, P.O. Box 6248, tailed program guidelines and pro­ tween January 1, 1979, and Decem­ Providence, RI 02940. (It should be posal forms may be obtained from ber 31, 1984. Moreover, the vita noted that completed application and the Council for International Ex­ must include the equivalent of at reference forms should NOT be sent change of Scholars, Eleven Dupont least three full years postdoctoral to this address, but to the address Circle N. W., Suite 300, Washington, teaching or industrial experience, i.e., given on the forms.) D. C. 20036-1257. Telephone: 202- non-fellowship years. The Selection 939-5401. Committee may give preference to The 1989-1990 Fulbright -Council for International applicants who have not had exten­ Scholar-in-Residence Award Exchange of Scholars sive postdoctoral research support. News Release The stipend has been set by the Opportunities for American colleges Trustees of the Society at $32,000 for and universities to host a visiting Crafoord Prize nine months of full-time research or scholar from abroad for all or part Recipients Announced its equivalent. In addition, there will of the 1989-1990 academic year are- ·,. - be an expense allowance of $1,000. available through the Fulbright Scholar- the Royal Swedish Academy of Sci­ Applicants must be citizens or per­ in-Residence Program. Institutions __ ences announced in April that it manent residents of a country in are invited to submit proposals for had conferred the prestigious Crafo- North America. Fellowships may be visiting scholars in the humanities ord Prize on two mathematicians, held at any institution the Fellow and social sciences, or in scientific Alexandre Grothendieck of the Uni- selects or at more than one in succes­ or professional specializations with a versity of Montpellier in France, and sion. There is flexibility in the choice strong international focus. Of partie- Pierre Deligne of the Institute for of time interval(s) and manner in ular interest for the 1989-1990 pro- Advanced Study in Princeton, New which the Fellow may draw funds. gram year will be proposals to bring Jersey. Shortly after the announce- For instance, given the opportunity, scholars in American literature, his- ment, Grothendieck turned down his a Fellow may elect to hold a half­ tory, or politics; professionals from half of the $210,000 prize. time academic appointment with a the media or government; or special- Grothendieck explained his rea- teaching responsibility not exceeding ists in constitutional law or politics. sons in a letter to the Academy, one course per term while holding A Fulbright Scholar-in-Residence published in the May 4 edition of the fellowship at one-half stipend may teach regular courses from a the French newspaper Le Monde. over a two-year period. The Fellow foreign area perspective, serve as a He said that his salary was already should consult with the Secretary of . resource person in interdisciplinary "much more than sufficient for the the Society to learn whether the ar­ courses, assist in developing new material needs of myself and those rangement proposed is acceptable to courses, or participate in special sem- I am responsible for." Asking, "is the Society. inars. An institution hosting a scholar- it not obvious that the excesses en- The number of fellowships to be in-residence would be expected to joyed by some can only come about awarded is small and depends on share the scholar's expertise with at the expense of the needs of oth- the amount of· money contributed a wide range of departments and ers?", Grothendieck stated that top to the program. The Trustees have with neighboring institutions, involve researchers who receive such prizes arranged a matching program from him/her in community activities and generally possess a great deal of rna- general funds in such fashion that professional organizations, and pro- terial wealth and scientific prestige funds for at least one fellowship are vide opportunities for the visitor to already. ~uaranteed. Because of the generos­ pursue personal research interests. Grothendieck, who received the Ity ~f the AMS membership it was The program provides round-trip Fields medal in 1966, is best known posstble to award two fellowships for travel for the grantee and, for full- for his fundamental advances in al-

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 811 News and Announcements gebraic geometry. He also introduced ing that his resignation was in protest dividuals and groups for distinction the idea of K -theory and revolution­ of increasing military support for in arts, sciences, and other areas. ized homological algebra. Deligne, a mathematical research. In his letter former student of Grothendieck, was to the Academy, Grothendieck said AMS Awards Prizes honored with a Fields medal in 1978. he believes that "the only decisive at the International Science Deligne provided the solution of the proof of the fertility of ideas or of and Engineering Fair three Weil Conjectures concerning a new vision is that of time. Fertil­ generalizations of the Riemann hy­ ity is recognizable by offspring, not The 39th International Science and pothesis to finite fields, and his work by honors." Grothendieck also stated Engineering Fair (ISEF) was held on has done much to unify algebraic ge­ that he did not wish to condone May 8-14, 1988 in Knoxville, Ten­ ometry and algebraic number theory. the declining ethical standards of the nessee. For the first time, the AMS The award cited their "fundamen­ mathematical community by accept­ was represented, awarding seven tal research in algebraic geometry, es­ ing the prize. prizes totalling $3,000. The AMS pecially for the introduction of etale judges Peter Doyle, David Minda, cohomology [Grothendieck] and its MSJ Announces Prizes and Carol Wood came to Knoxville application to various fields of math­ on May 11 to meet the exhibitors ematics [Grothendieck and Deligne] The Mathematical Society of Japan and to select the prize winners. The including the proof of the Weil Con­ has announced the awarding of sev­ AMS' participation was due largely jectures." eral prizes. to the initiative of Bill Thurston, Holger Crafoord, a Swedish in­ The first Spring Prize of the Japan who became interested through dis­ dustrialist who made a fortune in Mathematical Society 1988 was cussions with Westinghouse Science pulp and paper products and in arti­ awarded to KAzUYA KATo, Univer­ Talent Search contestants and with ficial kidneys, established the Holger sity of Tokyo, in recognition of his Dorothy Schriver of Science Service, and Anna-Greta Crafoord Prizes for outstanding work on Class Field The­ the Program Director of ISEF. basic research in several fields not ory for fields finitely generated over Most of the 721 entrants in Knox­ covered by the Nobel Prizes: math­ the prime fields. ville were U.S. high school students ematics, astronomy, the geosciences, The Mathematical Society of who arrived via regional science fairs. and the biosciences. Each year, the Japan's Geometry Prize of 1988 was The ISEF is subdivided by topic into Academy designates an area within awarded to HIROTAKA FUJIMQTQ of 13 categories; the AMS judges fo­ the field of science being honored Kanazawa University, Japan, iii recog­ cused on the 38 math exhibits. Peter that year and chooses individuals nition of his complete solution of Doyle scouted other categories for who have been influential in that a long standing conjecture on the entries with mathematical content, area as prize winners. The entire Gauss maps of complete minimal and he eventually found one winner. award amount is $423,000, half of surfaces. Each entrant met with at least one which goes to the prize winners, and The 1988 Japan Academy Prize of the judges; the winners met with half of which goes to support Swedish was awarded to MASAKI KASHIWARA, at least two judges. The judges' role researchers in the chosen area of sci­ Research Institute for Mathematical was two-fold: to select as winners ence. Sciences, Kyoto, Japan, for his out­ those students showing the highest The mathematics prize is given standing contributions to algebraic mathematical achievement, and to every 7 years. The first mathemat­ analysis. He, partly joined with M. offer encouragement and advice to ics award was presented in 1982 to Sato and T. Kawai, has originated all mathematics students. Louis Nirenberg of the Courant In­ microlocal analysis and developed The following received cash awards stitute of the Mathematical Sciences the theory of linear differential equa­ and certificates from the AMS: First and V. I. Arnold of the University tions, in particular, holonomic sys­ Place ($1000): CHRISTOPHER SKIN­ of Moscow. Deligne will receive his tems, using this new approach. NER, Little Rock, Arkansas, for "The award in Stockholm on September The ASAHI Prize 1988 was award­ Diophantine Equation x 2 = 4qn - 21. According to a representative of ed to MASAKI KAsHIWARA and TAKA­ 4q + 1." the Academy, Grothendieck's por­ HIRO KAWAI, Research Institute for Two Second Places ($500): BRIAN tion of the prize is expected to be Mathematical Sciences, Kyoto, Japan, CoNRAD, Sleden, New York, for "In­ returned to the general prize fund. for their outstanding contribution to verted Continued Fractions," and Grothendieck has been known for algebraic analysis. The ASAHI Prize Laura Magde, San Diego, Califor­ his exacting ethical standards. In is given annually by the Asahi Shim­ nia, for the physics entry, "Viscous 1970, he left his position at the Insti­ bun, a major Japanese newspaper Fingers as Fractals." tut des Hautes Etudes Scientifiques at company, to a certain number of in- Four Third Places ($250): YuN Bures-sur-Yvette outside Paris, say- BEOM CHOI, The Bronx, New York,

812 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY News and Announcements

for "Unitary Half Perfect Numbers"; been intended for college students, ican Academy of Arts and Sciences: ALLISON HuGHEs, Polson, Montana, the contest application never specif­ RUTHERFORD ARIS, University of for "Fibonacci-Generalized"; WEs­ ically stated that a team must be Minnesota; JoHN W. PRATT, Har­ LEY McDERMOTT, Lake Worth, from a college or university. When it vard University; RICHARD ScHOEN, Florida, for "Mathematical optimiza­ was revealed that one of the winning Stanford University; RICHARD P. tion of school district boundaries"; papers was written by high school STANLEY, Massachusetts Institute of VAMSI MooTHA, Beaumont, Texas, students, contest officials concluded Technology; and SRINIV ASA VARAD­ for "Study of P,-sets." that the students are technically un­ HAN, New York University. dergraduates and let the ruling stand. Winners Announced in The high school team produced National Academy Math Modeling Contest one of two outstanding solutions to of Engineering's the drug runner problem; the other New Foreign Secretary Six student teams from across the was from a team from Drake Uni­ country have been recognized for versity. Four teams presenting so­ Gerald P. Dinneen, Vice President their outstanding solutions in the lutions to the railroad car problem for Science and Technology at Hon­ Mathematical Contest in Modeling. received outstanding ratings: the Op­ eywell, Inc., Minneapolis, Minnesota, And this year there was a new twist: erations Research Department at the was recently elected the National the contest has traditionally been for University of California, Berkeley; Academy of Engineering's foreign college students, but one of the win­ Harvard University; the University secretary. During his three-year term, ning teams was a group of high school of Toronto; and the U.S. Military he will oversee international activi­ students. Academy at West Point. ties of the NAE and coordinate its The contest allows students to Two teams selected from the win­ contacts with engineering academies put their mathematical acumen to ners for each problem received an in other countries. He succeeds H. work on real-world applications of all-expense paid trip to the national Guyford Stever, former president of mathematics. The teams, consisting meeting of the Operational Research Carnegie Mellon University. of up to three students, have three Society of America in April, where days in which to produce an in­ they presented their results. In ad­ Patents Issued depth analysis of one of two mod­ dition, the Harvard University team to Mathematicians eling problems. Participation in the received a new award from the Soci­ contest has increased by more than ety for Industrial and Applied Math­ Researchers at AT&T Bell Laborato­ 30% per year since its inception in ematics (SIAM), in recognition of ries have been awarded three patents 1985, and this year's total of 204 exceptional performance in the con­ for methods of resource allocation participating teams represents a 37% test. The three team members will utilizing the Karmarkar algorithm for increase over the number in 1987. receive cash awards and a trip to the linear programming. The contest gives the teams a annual SIAM meeting in July. Narendra K. Karmarkar, devel­ choice of two problems, one "dis­ The contest is administered by the oper of the algorithm, received one crete" and one "continuous." This Consortium for Mathematics and Its of the patents for a method for allo­ year, the discrete problem asked how Applications (COMAP) and directed cating telecommunications and other to load a flatcar so as to minimize by Ben Fusaro of Salisbury State resources. He was also named on wasted floor space, given the dimen-. University. The problems are mailed another patent, together with Jeffrey sions and weights of each of seven to team advisers, who tum the prob­ C. Lagarias, also of Bell Labs, and types of crates and some other re­ lems over to the students on a speci­ David A. Bayer of Columbia Univer­ strictions. The continuous problem · fied Friday morning in February. The sity. The second patent covers an en­ sought an optimal search strategy for teams then choose one of the prob­ hancement that accelerates the con­ a helicopter trying to locate a drug lems and are permitted to use any vergence of the Karmarkar algorithm smuggler in a power boat, given an inanimate source they wish-books, by using power series approximation. approximate initial position of the journals, computers, etc. Solutions The third patent went to Robert J. boat relative to the helicopter and must be mailed to COMAP by the Vanderbei, also of Bell Labs, for the speeds of each. following Monday. methods of effectively stopping and The three high school students, starting the process of carrying out from the North Carolina School of Newly Elected Members the algorithm. Science and Mathematics, expressed of the AAAS It was not the Karmarkar algo­ "complete disbelief" at being chosen rithm itself, nor the computer code as one of the outstanding teams. Al­ The following mathematical scien­ utilizing the algorithm that were though the contest traditionally has tists have been elected to the Amer- patented. So what exactly was patent-

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 813 News and Announcements ed? Henry Brendzel, a patent attor­ projected telephone traffic in the net­ be given to projects in disciplines to ney for Bell Labs, says that the patent work and where the optimal places which research in China can make covers "the method and apparatus are to install additional facilities for a unique contribution and to those of allocating resources in a physi­ increased demand. Garey says that which are of mutual benefit to Amer­ cal environment." He explained that, using the K.armarkar algorithm in ican and Chinese scientists. The pro­ like a law of nature, the algorithm applications like this will save AT&T gram also supports the nomination is an abstract concept. "The carry­ millions of dollars each year. of Chinese scholars in all disciplines ing out of the concept to achieve a For a more general discussion of by American scholars. Priority will specific end result in a physical envi­ patents and mathematics, see the ar­ be given to Chinese who have not ronment is what we have patented," ticle by Edward N. Zalta in the Com­ visited the U.S. recently and to those he said. The environment might be puters and Mathematics column in whose visits will contribute to fu­ a commercial enterprise, a military this issue of Notices. ture academic exchanges. For both strategy, or a geological exploration American and Chinese components, scheme. For example, Bell Labs re­ AWM Speakers' women, minorities, scholars begin­ cently produced a system, consist­ Bureau Directory ning their careers, and scholars based ing of both computer hardware and at or visiting provincial institutions software, which uses the patented The Association for Women in Math­ in China, are especially encouraged. method to optimize allocation of a ematics is preparing a new edition of The deadline for applications to the company's personnel. the AWM Speakers' Bureau Direc­ Visiting Scholar Exchange Program The three patents were originally tory, which lists women and men who is November 15, 1988. filed in 1985. According to Michael are available to speak at high schools The Graduate Program supports R. Garey, Director of the Mathe­ and/or colleges about a wide variety individuals enrolled in a graduate matics Center at Bell Laboratories, of subjects related to women and program in social sciences or human­ the delay in obtaining the patent is mathematics. Topics include pure ities to do coursework or dissertation standard. and applied mathematics, careers in research at a Chinese university. The The K.armarkar algorithm is based mathematics, and women's contribu­ Research Program supports individ­ on interior methods for linear pro­ tions to mathematics. AWM cov­ uals in the social sciences and hu­ gramming, in which the number of ers the speaker's travel expenses. manities who hold a Ph.D. or equiv­ steps in the algorithm is reduced Those interested in speaking for the alent at the time of application to by passing through the interior of Bureau can contact AWM at Box do in-depth research on China, the the regions in the linear program­ 178, Wellesley College, Wellesley, Chinese portion of a comparative ming scheme, rather than passing MA 02181. Telephone: 617-235-0320. study, or an exploratory survey of an only along the edges. In the past aspect of contemporary China. Re­ forty or fifty years, many researchers CSCPRC 1989-1990 search grantees normally spend from have attempted to use this idea, but China Programs two months to one year in the PRC, K.armarkar was the .first to make it beginning no earlier than July 1989 work. The Committee on Scholarly Com­ and ending no later than December Garey described one application munication with the People's Re­ 1990. The deadline for application to for which AT&T uses the patented public of China (CSCPRC) has an­ the Graduate and Research Programs method. Every three months AT&T nounced its 1989-1990 National Pro­ is October 15, 1988. assembles a collection of predictions gram of Scholarly Exchanges with For application information on of the number of telephone calls for China. The program includes the fol­ all the above programs, write to the coming three months over major lowing: the Committee on Scholarly Com­ calling routes. Usually the demand The Visiting Scholar Program sup­ munication with the People's Repub­ is growing, but there might be de­ ports one to three month visits for lic of China, National Academy of creasing demand in some places. For American and Chinese scholars in all Sciences, 2101 Construction Avenue, example, if an industrial plant closes, disciplines between September 1989 Washington, D.C. 20418. telephone use may drop in that area. and August 1990. For Americans in -CSCPRC News Release The task is to figure out how to route social sciences and humanities, the calls generally over the network to program supports scholars outside meet the demand with existing tele­ the China studies field to initiate and phone trunks and to decide if new conduct research. For Americans in telephone trunks are needed. natural sciences and engineering, the The K.armarkar algorithm is used program supports scholars to initiate to figure out how best to route the and conduct research. Priority will

814 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY National Science Foundation News & Reports

Mathematical Sciences Mathematical Society at 401-272-9500. her lectures. The second stipend is Postdoctoral Research Fellowships The deadline for applications is dependent on approval of the man­ November 15, 1988. uscript by NSF. These monographs The NSF Mathematical Sciences Post­ will be published in various series doctoral Research Fellowship pro­ by one of three organizations: the gram is designed to permit recipients NSF Seeks Proposals American Mathematical Society, the to choose research environments that for the 1990 Society for Industrial and Applied will have maximal impact on their fu­ Regional Conference Series Mathematics, and, in conjunction, ture scientific development. Awards in Mathematical Sciences the American Statistical Association will be made for appropriate research and the Institute of Advanced Stud­ in pure mathematics, applied mathe­ The National Science Foundation is ies. matics and operations research, and seeking proposals from host institu­ Twenty copies of each proposal statistics at an appropriate nonprofit tions in the U. S. for five-day regional to be considered should be sent di­ United States institution. research conferences. Each confer­ rectly to the Data Support Service The Fellowships will be offered ence will feature a distinguished lec­ Station, National Science Founda­ only to persons who 1. are U.S. cit­ turer who will deliver 10 lectures on tion, 1800 G Street, NW, Washing­ izens or nationals as of January 1, a subject of current interest in the ton, DC 20550. Proposals must fol­ 1989; 2. will have earned, by the be­ mathematical sciences. The lecturer low the guidelines issued by CBMS. ginning of their fellowship tenure, a will give two lectures per day dur­ The deadline for proposals for 1990 doctoral degree in one of the mathe­ ing the five days, with the remaining conferences is April 1, 1989. For in­ matical sciences; 3. will have held the time available for study, informal formation sheet and guidelines write doctorate for no more than five years discussion, and exchange of ideas. or call The Conference Board of the as of January 1, 1989; and 4. will not Institutions having at least some Mathematical Sciences, 1529 18th previously have held any other NSF research competence in the area of Street, NW, Washington, DC 20036; postdoctoral fellowship. The evalu­ their proposals are eligible and en­ tel. 202-293-1170. ation of applicants will be based, couraged to apply. The conference in part, on ability as evidenced by should be planned for a summer National Science Board past research work and letters of rec­ week in 1990 (not earlier than May Posts Announced ommendation, likely impact on the 1), or held during a recess in the future scientific development of the succeeding academic year. Mary L. Good and Thomas B. Day applicant, and scientific quality of The host institution is responsible have been elected Chair and Vice the research likely to emerge. Appli­ for arranging conference details, ho­ Chair, respectively, of the National cants' qualifications will be evaluated tels, and social events during the five­ Science Board (NSB), the policymak­ by a panel of mathematical scientists. day conference. Participants in the ing body of the NSF. Also, Warren For copies of the application bro­ conference will receive allowances J. Baker has been reappointed to the chure or further information, contact for travel and subsistence under the NSB. Good and Day will serve in the Special Projects Program, Divi­ host institution's grant from NSF. their new positions until May 1990, sion of Mathematical Sciences, Na­ The principal lecturer will re­ and Baker will serve until May 1994. tional Science Foundation, 1800 G ceive $1,500 for delivering his or In addition, President Reagan has Street, NW, Washington, DC 20550; her lectures, and $2,500 for produc­ announced his intention to nominate tel. 202-357-3453; or the American ing an expository monograph for the Charles L. Hosler, Jr., and Roland W. NSF /CBMS Series based on his or Schmitt as members of the NSB for

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 815 NSF News & Reports terms expiring May 10, 1994. These Information Science and Engineer­ The fellowships are aimed pri­ are reappointments. ing. Wulf, who began the position in marily at researchers at the gradu­ Good is currently President-Engi­ May, was AT&T Professor of Engi­ ate or postdoctoral level, but senior neered Materials Research, Allied neering and Applied Science at the researchers, including researchers in Signal Corporation. She was Boyd University of Virginia. industry, are also eligible. Applicants Professor of Materials Science at Wulfs main areas of research must be U.S. citizens or nationals. Louisiana State University, Baton are programming systems and com­ There are two kinds of awards, Rouge (1978-1980) and Boyd Pro­ puter architecture. He is particu­ and they may be combined in one fessor of Chemistry at the University larly interested in the construction application. of New Orleans (1974-1978). of systems-programming languages, Academic program fellowships pro­ A physicist, Day has been Presi­ compilers, operating systems-and vide tuition and stipend for regular dent of San Diego State University the computer architecture that ex­ university Japanese language courses. since 1978. Before that, he was Vice ecutes them efficiently. The amount of the stipend will be Chancellor for Academic Planning Wulf received his Ph.D. in com­ based on the amount of NSF Grad­ and Policy and Special Assistant to puter science in 1968 from the Uni­ uate Fellowships (currently $12,300), the Chancellor at the University of versity of Virginia. He came to the and on the proportion that the Maryland, College Park. faculty of Carnegie-Mellon Univer­ courses represent in one's regular Baker has been President of Cal­ sity in 1968, and in 1981 founded course load. The stipend for post­ ifornia Polytechnic State University Tartan Laboratories Incorporated, a doctoral researchers is twice that f()f since 1979. A civil engineer, he also Pittsburgh company. He joined the graduate students. Deadlines ate May served as Vice President of Academic faculty of the University of Virginia 15 for programs to start the followinj Affairs at the University of Detroit in early 1988. spring (January), and December 15 (1976-1979). The directorate that Wulf heads for those starting the following fall Hosler has been Vice President is concerned with computer research, (September). for Research and Dean of the Grad­ information science and technology, Intensive-study fellowships pro­ uate School at Pennsylvania State advanced scientific computing, and vide tuition and fees as required for a University since 1985. Prior to this, computer engineering and communi­ suitable intensive Japanese language he was Dean of the College of Earth cations/signal processing. Wulf suc­ course, a cost-of-living allowance of and Mineral Sciences at Pennsylva­ ceeds C. Gordon Bell who left the $1,025 to $2,050 per month, and a nia State University (1966-1985). NSF in late 1987. travel allowance, if necessary. The Schmitt is currently President of deadline is October lS for courses Rensselaer Polytechnic Institute. Fellowships for to begin six months or more after Prior to this, he was with Gen­ Japanese Language Study application. eral Electric from 1951-1988 serving For more information, contact: most recently as Senior Vice Pres­ A new program in the Division of Japanese Language Fellowships, Di­ ident for Science and Technology International Programs will award vision of International Programs, (1986-1988) and Senior Vice Pres­ fellowships to enable researchers in National Science Foundation, 1800 ident for Corporate Research and science and engineering to develop G Street, NW, Washington, DC 20550; Development (1980-1986). Japanese language skills. Forty to tel. 202-357-9558; ARPANET ad­ One of the Board members, fifty awards will be made each year. dress [email protected]. William F. Miller, a computer sci­ The program is part of a set of entist and President and Chief Ex­ Japan initiatives developed to fa­ Grant Policy Information on ecutive Officer of SRI International, cilitate science and technology ex­ Electronic Bulletin Board is the only one who is a member of changes between the U.S. and Japan. an active mathematical organization. The initiatives are designed to en­ The Division of Grants and Con­ However, since 1986, the 24-member courage U.S. researchers to make tracts (DGC) recently set up an board has had no representatives long-term visits to Japan, to pro­ electronic bulletin board to facili­ from the mathematical sciences. mote the use of Japanese technical tate discussion on grants and general publications, and to expand collabo­ grants policy among grantees and the William Wulf rative research. The fellowships for NSF. The bulletin board is being Named Head of CISE Japanese language study are intended used to post notices of interest and to help remove language and cultural serve as a forum for sponsored re­ William A. Wulf, an experimental barriers to U.S.-Japan science and search administration. As of Decem­ computer scientist, is the new head technology exchanges. ber 1987, more than forty institu­ of the Directorate for Computer and tions had signed up to participate in

816 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY NSF News & Reports the bulletin board. Recent postings Users of BITNET who would like ing list should address requests to: on the bulletin board included an to be added to the electronic mail­ [email protected]. outline of the updated NSF F.L. 200, ing list should address requests to: INTERNET users will post no­ Grant General Conditions, which ap­ grants-r@NSF. tices to the bulletin board by sending plies to all grant awards issued since BITNET users will post notices to: [email protected]. late November 1987; a summary of to the bulletin board by sending to: For additional information, con­ the substantive changes in the revised grants@NSF. tact the NSF, Division of Grants F. L. 200; and a discussion of various INTERNET users (ARPANET, and Contracts Policy Office, 1800 G topics raised by research administra­ CSNET, MILNET) who would like Street, NW, Washington, DC 20550; tors at participating institutions. to be added to the electronic mail- tel. 202-357-7880. -NSF News Release

NINETEEN PAPERS CLASSIFICATION OF JACOBIAN IDEALS ON ALGEBRAIC SEMIGROUPS INVARIANT BY s/(2,C) ACTIONS Stephan S.-T. Yau (Memoirs of the AMS, Number 384) A. Ya. Aizenshtat, A. E. Evseev, N. E. Podran, I. S. Ponlzovskii, B. M. Shain (Boris M. Schein), E. G. Shutov, and Yu. M. Vazhenin Given M~. the space of homogeneous polynomials of degree (American Mathematical Society Translations, Series 2, Volume 139) k in n variables, one can examine Sn, which is the direct sum, over k, of those subspaces of M~ on which s/(2, C) acts This volume contains papers selected by leading specialists trivially. The central object of invariant theory is to describe in algebraic semigroups in the U.S., the United Kingdom, explicitly Sn. a surprisingly difficult task. The main result of and Australia. Many of the papers strongly influenced the this work concerns the vector subspace /(f) spanned by then development of algebraic semigroups, but most were virtually partial derivatives off, a homogeneous polynomial of degree unavailable outside the U.S.S.R. Written by some of the most k+ 1 inn variables. The author shows that, when n ~ 5, /(f) prominent Soviet researchers in the field,' the papers have a is an s/(2,C)-submodule if and only if /(f) is the same as the particular emphasis on semigroups of transformations. Boris gradient space of some s/(2,C)-invariant polynomial g. Schein of the University of Arkansas is the translator.

1980 Mathematics Subject Classifications: 20; 94, 04, 05, 06, 08 and 1980 Mathematics Subject Classifications: 11; 14 others ISBN 0-8218-2447-3, LC 88-990 ISBN 0-8218-3115-1, LC 88-10352 ISSN 0065-9266 ISSN 0065-9290 183 pages (softcover), March 1988 224 pages (hardcover), June 1988 Individual member $11, List price $19, Individual member $41, List price $69, Institutional member $15 Institutional member $55 To order, please specify MEM0/384NA To order, please specify TRANS2/139NA Shipping/Handling: 1st book $2. each add'l $1. $25 max. By Shipping/Handling: 1at book S2, each add'! Sl, S25 max. By air, 1st book $5, each add'l $3, $100 max. air, lat book $5, each add'l $3, SlOO max. Prepayment Prepayment required. Order from AMS, P.O. Box 1571, Annex required. Order from AMS, P.O. Box 1571, Annex Station, Station, Providence, Rl 02901-9930, or call 800-556-7774 to Providence, Rl 02901-9930, or call 800-556-7774 to uae VISA use VISA or MasterCard. or MasterCard.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 817 News from Washington

James Gleick segment of the American Mathe­ examinations, will be held in Can­ Awarded Math matics Competitions. The national, berra, Australia on July 9-20. Communications Prize three-stage competition involved more The annual American Mathemat­ than 389,000 students. ics Competitions, administered by The Joint Policy Board for Mathe­ The eight USAMO winners are: the Mathematical Association of matics (JPBM) will present its first First Place: Ravi D. Vakil, Toron­ America, comprise three events. The Communications Award to James to, Ontario, Martingrove Collegiate competitions begin with the Amer­ Gleick on July 13, during the annual Institute ican High School Mathematics Ex­ meeting of the Society for Industrial Second Place: Jordan S. Ellen­ amination, a 1 1/2-hour, multiple­ and Applied Mathematics (SIAM) berg, Potomac, Maryland, Winston choice test of 30 questions based in Minneapolis, Minnesota. Gleick, Churchill High School on non-calculus mathematics. More a science writer for the New York Third Place: Tal N. Kubo, Brook­ than 389,000 students participate in Times, is also the author of Chaos, line, Massachusetts, Brookline High this first round. Three weeks later, which was published in 1987 and School 1,000-3,000 of the top scorers com­ which traces the history and progress Fourth Place (tie takes up Fifth pete in the American Invitational of mathematical work on chaos. and Sixth Places:) Joshua B. Fis­ Mathematics Examination, which pre­ The JPBM Communications chman, Bethesda, Maryland, Mont­ sents 15 short-answer problems to Award is being presented as part of gomery Blair HS; Hubert L. Bray, be completed in 3 hours. The U.S. the "1 00 Years of American Math­ Houston, Texas, Westbury High competition culminates with the US­ ematics" festivities which are tak­ School; Nhat Nguyen, Columbus, AMO, a 3 1/2-hour examination con­ ing place throughout this year. The Ohio, Columbus Academy sisting of 5 essay questions. award recognizes Gleick's sustained Seventh Place (tie takes up Eighth Romania initiated the IMO in and outstanding contributions in com­ Place:) EricK. Wepsic, Boston, Mas­ 1959, and the U.S. has participated municating mathematics to the gen­ sachusetts, Boston Latin School; in the contest since 1974. Typically, eral public. Gleick will receive $1,000 David M. Patrick, Batavia, New 30-35 nations send teams to the IMO. as part of the prize, and will also be York, Batavia High School The U.S. team has usually placed honored by representatives of JPBM On June 7, the winners were hon­ among the top 3 nations in this pres­ member societies at an awards lun­ ored in Washington, DC at the US­ tigious competition, and, in 1977, cheon. AMO Awards Ceremony and Din­ 1981, and 1986, it earned first-place The award is sponsored by the ner, held at the National Academy honors. AMS, the Mathematical Association of Sciences and the U.S. Department of America, and SIAM and is sup­ of State. In the next phase of the Putnam Examination ported in part by a grant from Hon­ competition, these winners and 16 Winners Announced eywell, Inc. other students who did well in the USAMO will participate in an inten­ The forty-eighth annual William Low­ Winners of U.S. sive four-week training session at the ell Putnam Mathematical Competi­ Math Olympiad U.S. Naval Academy in Annapolis, tion was held on December 5, 1987. Maryland. The training session will Administered by the Mathematical Eight top students have won Olymp­ produce a U.S. team of 6 students for Association of America (MAA), the iad Medals in the seventeenth USA the 1988 International Mathematical prestigious competition drew 21 70 Mathematical Olympiad (USAMO), Olympiad (IMO). The IMO, which contestants from 359 institutions the final round of the high school consists of two 4 1/2-hour written

818 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY News from Washington from across the United States and to $12,000 plus tuition at Harvard opportunities in the mathematical Canada. University. sciences; collegiate and university The competition consists of an The next five top scorers, re­ mathematical sciences education and examination, given in two three-hour ceiving a $250 prize, are: DANIEL the Mathematical Sciences in the sessions. Designed to test original­ J. BERNSTEIN, Princeton University; Year 2000 (MS2000) project; and de­ ity as well as technical competence, CONSTANTINE N. COSTES, Harvard velopments and directions at the re­ the examination covers undergradu­ University; JEREMY A. KAHN, Har­ search funding agencies of ICEMAP ate mathematics through differential vard University; RAv KuMAR RA­ (Interagency Commission for Extra­ equations. The examination may in­ MAKRISHNA, Cornell University; mural Programs in Mathematics). clude questions that cut across the Japheth Wood, Washington Univer­ Under development is a work­ bounds of various disciplines, as well sity, St. Louis. shop on experiences with, and the as self-contained questions that do William Lowell Putnam, a mem­ development of proposals for, "ex­ not fit into the usual categories. ber of the Harvard class of 1882, traordinary" methods of organizing An institution with at least three began the competition in 1938. The and funding mathematical sciences registered participants obtains a team idea grew out of Putnam's profound research (e.g., science and technol­ rank based on the rank of three des­ belief in the value of organized team ogy centers, special years). Each year, ignated individual contestants. How­ competition in college studies. In the Chairs' Colloquium provides a ever, each contestant works indepen­ 1927, Putnam's wife created a trust unique opportunity for mathemati­ dently on the examination, even if fund to support such activities. The cal sciences department chairs and designated as a team member. first competition supported by the other interested parties to meet and The five winning teams are listed fund was in the field of English and discuss issues of mutual concern in a below, in order of their ranking. (In a few years later a second compe­ pleasant and stimulating setting. all of the lists that follow, the con­ tition was held in mathematics. It The registration fee is $175 and testants' names are in alphabetical was not until after Mrs. Putnam's covers the cost of handouts and order.) Harvard University: DAVID J. death in 1935 that the examination meals (including a conference recep­ MOEWS, BJORN M. POONEN, MICHAEL assumed its present form and was tion and banquet). Space is limited REID; Princeton University: DANIEL administered by the MAA. and advance registration is required. J. BERNSTEIN, DAVID J. GRABINER, Registration and program informa­ MATTHEW D. MULLIN; Carnegie Annual Department tion will be mailed in July. If you are Mellon University: PETROS I. HADJI­ Chairs' Colloquium considering first-time attendance and COSTAS, JOSEPH G. KEANE, KARL M. would like to be added to the mail­ WESTERBERG; University of Califor­ Each year, the Board on Mathemat­ ing list, or if you require additional nia, Berkeley: DAVID P. MOULTON, ical Sciences of the National Re­ information, please contact: Board JONATHAN E. SHAPIRO, CHRISTOPHER search Council presents the Depart­ on Mathematical Sciences, National S. WELTY; Massachusetts Institute of ment Chairs' Colloquium, for the Research Council, 2101 Constitution Technology: DAVID T. BLACKSTON, heads of mathematics, statistics and Ave. NW, Room NAS 312, Washing­ JAMES P. FERRY, WALDEMAR P. other mathematical sciences depart­ ton, DC 20418; telephone 202-334- HORWAT. ments. This year's colloquium, to 2421. The six highest ranking individ­ be held on October 14-15; 1988 at uals receive the Putnam Fellow dis­ the Washington Marriott Hotel in National Academy tinction and $500 award. Their names Washington, DC, will offer a suite of Sciences Election and institutions are: DAVID J. GRA­ ·of sessions on Computers in Mathe­ BINER, Princeton University; DAVID. matical Sciences Research and Train­ The National Academy of Sciences J. MoEws, Harvard University; BJORN ing. Planned are presentations and recently announced the election of 61 M. POONEN, Harvard University; discussion sessions which examine new members and 15 foreign asso­ MICHAEL REID, Harvard University; the interplay between mathematical ciates from 8 countries in recognition CONSTANTIN S. TELEMAN, Harvard sciences research and computational of their distinguished and continuing University; JOHN S. TILLINGHAST, methods, computer graphics and ge­ achievements in original research. University of California, Davis. ometry, supercomputers, and issues The election was held on April One of the Putnam Fellows will involved in organizing and support­ 26, during the business session of also receive the annual Putnam Prize ing computational facilities for math­ the 125th annual meeting of the Scholarship, to be used immediately ematics and statistics research. Academy. Election to membership or on completion of undergraduate Also anticipated are updates on: in the Academy is considered one study. The scholarship provides up plans for a second David Report of the highest honors that can be dealing with infrastructure and accorded an American scientist or

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 819 News from Washington engineer. Those recently elected bring The 34-member Board is a unique at all levels. The report, a joint effon the total number of current members coalition of teachers and supervi­ by the MSEB and the BMS, will high. to 1,540. Foreign associates are non­ sors, college and university math­ light weaknesses in U.S. educational voting members of the Academy with ematicians, educational administra­ practices and outline the national citizenship outisde the United States. tors, and representatives of govern­ mobilization needed to ensure U.S. The recent election brings the total ment, business, and industry. Shirley competitiveness. November 1988 is number of foreign associates to 257. A. Hill, professor of mathematics the target date for publication. The newly elected members and and education at the University of Curriculum report completed. At their affiliations at the time of the Missouri at Kansas City, is chairman its meeting last January, the MSEB election are as follows: MITCHELL J. of the Board, and Marcia P. Sward received the report of its Curriculum FEIGENBAUM, Professor of Physics, is executive director. Framework Task Force, chaired by Rockefeller University; THEODORE The MSEB's coordinated program Anthony Ralston of the State Univer­ E. HARRIS, Professor of Mathematics encompasses a wide range of activi­ sity of New York at Buffalo. There­ and Electrical Engineering, Univer­ ties. Some of the highlights include: port, prepared over an 18-month pe­ sity of Southern California; HERBERT Study of employers' needs. MSEB riod, addresses the forces for change A. HAUPTMAN, Research Professor, has embarked upon an plan to study in school mathematics and points to State University of New York, Buf­ the kinds of mathematical skills em­ specific directions of change in cur­ falo and President and Research Di­ ployers will need in the workplace riculum and instruction. The report rector, Medical Foundation of Buf­ of the future and to engage these is now being revised and prepared falo; JosEPH J. KoHN, Professor of employers in the revitalization of for publication in February 1989 as Mathematics, Princeton University; mathematics education. A "kickoff" an MSEB curriculum framework. JOHNN. MATHER, ProfessorofMath­ symposium, to be held November Sponsors' meetings. The first semi­ ematics, Princeton University; DANA 17-18 1988 at the new NAS Beck­ annual Sponsors' Meeting was held S. ScoTT, University Professor of man Center in Irvine, California, will last September at the National Acad­ Computer Science and Mathemati­ focus on the connections between emy of Sciences. These meetings are cal Logic, Carnegie Mellon Univer­ quality mathematics education and designed to inform the agencies that sity; EDWARD WITTEN, Professor of the ability of business and industry sponsor MSEB about the Board's ac­ Physics, Princeton University and to compete internationally. John D. tivities, and to stimulate discussions Professor of Physics, The Institute Macomber, former Chief Executive of basic issues in mathematics ed­ for Advanced Study. Officer of Celanese Corporation, has ucation. The second meeting, held The newly elected foreign asso­ been selected as head of the steering in May, focused on the new stan­ ciate and his affiliation at the time committee for the symposium. dards for school mathematics cur­ of the election is as follows; country MS2000 project launched. MS2000 rently being developed by the Na­ of citizenship is in parentheses: SIR (Mathematical Sciences in the Year tional Council of Teachers of Math­ DAVID R. Cox, Professor of Statis­ 2000: Assessment for Renewal in ematics (NCTM). tics, Imperial College of Science and U.S. Colleges and Universities) is Outreach program. The success of Technology (United Kingdom). a joint project of the MSEB and the the MSEB's overall effort depends on NRC's Board on Mathematical Sci­ establishing and maintaining working News from the ences (BMS). Over a 3-year period, relationships with a large number of Mathematical Sciences MS2000 will develop a national plan groups and organizations represent­ Education Board for enhancing the flow of mathemat­ ing academia, business and industry, ical talent, renewing faculty, rein­ and the public. Three professional In 1985, the National Research Coun­ vigorating teaching and scholarship, consultants are working on develop­ cil (NRC), at the request of the math­ and making fundamental changes in ment and implementation of a com­ ematical community, established the the curriculum at the college and prehensive plan for MSEB's outreach Mathematical Sciences Education university level. The chairman of and public information activities. Board (MSEB), to provide national the MS2000 steering committee is Review of school mathematics leadership in mathematical sciences J. Fred Bucy, former chief executive standards. The MSEB has conducted education from kindergarten through officer of Texas Instruments, Incor­ a review of the standards for school college. The MSEB is designed to porated. mathematics being developed by the provide sustained attention over Steen commissioned to write re­ NCTM. Reactions were solicited from many years to the major issues af­ port. Lynn Arthur Steen has been state officials, administrators, busi­ fecting the quality of instruction and commissioned to write the NRC's ness and industry representatives, learning in the mathematical sci­ first Report to the Nation on the parents, and others. The findings ences. state of U.S. mathematics education are intended to identify possible

820 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY News from Washington

as chairman, the panel stumbling blocks in the acceptance National advisory panel. The Uni­ of Georgia meeting on June 18. of the standards and to assist NCTM versity of Wisconsin's National Cen­ held its first staff has now grown and MSEB in planning ways to pro­ ter for Research in Mathematical The MSEB to handle the in­ mote widespread adoption of the Sciences Education is one of four to ten members and diversity of the standards by school districts and discipline-based educational research creased scope Last November, the states. A summary report has been centers established last fall by the Board's projects. permanent home presented to NCTM, and an analysis Department of Education. At the MSEB moved to its Avenue, NW, of the various reactions to the stan­ university's invitation, the MSEB ap­ at 818 Connecticut DC 20006; dards will be prepared this summer. pointed an 8-member panel on the Suite 500, Washington, research agenda for the center. With telephone 202-334-3294. Visitors are Jeremy Kilpatrick of the University always welcome.

LATTICES WITH UNIQUE COMPLEMENTS INTRODUCTION TO V.N.Salii ANALYTIC NUMBER THEORY (Translations of Mathematical Monographs, Volume 69) A. G. Postnikov Volume 68) The class of uniquely complemented lattices properly contains (Translations of Mathematical Monographs, all Boolean lattices. However, no explicit example of a non-Boolean lattice of this class has been found. In addition, Aimed at a level between textbooks and the latest research the question of whether this class contains any complete monographs, this book is directed at researchers, teachers, non-Boolean lattices remains unanswered. This book focuses and graduate students interested in number theory and its on these classical problems of lattice theory and the various connections with other branches of science. The author attempts to solve them. Requiring no specialized knowledge, has attempted to give as broad a picture as possible of the book is directed at researchers and students interested in the problems of analytic number theory while avoiding general algebra and mathematical logic. specialization and those topics already sufficiently covered in the literature. In particular, this book focuses on general 06 1980 Mathematics Subject Classification: additive number theory and the concept of a numerical ISBN 0-8218-4522-5 gives a systematic discussion of these ISSN 0065-9282 semigroup-and 128 pages (hardcover), March 1988 topics. Individual member $31, List price $51, 1980 Mathematics Subject Classification: 11 Institutional member $41 ISBN 0-8218-4521-7 To order, please specify MMON0/69NA ISSN 0065-9282 336 pages (hardcover), March 1988 List price $114, 1st book $2, each add'l $1, $25 max. By Individual member $68, Shipping/Handling: Institutional member $91 air, 1st book $5, each add'l $3, $100 max. To order, please specify MMON0/68 NA Prepayment required. Order from AMS, P.O. Box 1571, Annex Station, Providence, Rl 02901-9930, or call 800-556-7774 to By use VISA or MasterCard. Shipping/Handling: 1st book S2, each add'l Sl, $25 max. air, 1st book $5, each add'l $3, $100 max. Prepayment required. Order from AMS, P.O. Box 1571, Annex Station, Providence, Rl 02901-9930, or call 800-556-7774 to use VISA [ or MasterCard.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 821 For Your Information

The ~ilemma of the Golden Age

Frank Press

inscription in the dome of the Great Hall: "To science, The remarks below are based on a presentation by Frank pilot of industry, conqueror of disease, multiplier of Press, President of the National Academy of Sciences, on the harvest..." Indeed, throughout history the patrons the occasion of the 125th Annual Meeting of the National · of science have understood the potential of new scien­ Academy of Sciences, April 26, 1988. tific knowledge as an instrument of public welfare, of power, of national security, and of national economic In recent years, I have used this occasion to dis­ strength. cuss the Academy-its projects, its influence, and its Science has been faithful to that compact-that the relations with the government and our society. Today, American people for their support of science could in however, I want to talk with you about the state of sci­ time expect a better life and a stronger nation. And we ence and the scientific community in America. These continue to honor that compact, so much so that this are confused and troubling times for us. Limits on can indeed be called the "Golden Age of Science"-a resources have made visible serious differences within time of unprecedented progress in scientific discovery our community, differences that may weaken the na­ and its applications. tion's scientific enterprise if they are not resolved A confluence of circumstances accounts for the without acrimony. golden age: the large number of scientists at work all We face the dilemma of living in both the best and over the world; the high level of financial support by worst of times. In all fields of science, the journals governments and industries; the availability of small and professional meetings are filled with exciting and and large instruments of extraordinary sensitivity challenging reports of new discoveries, new ideas, new that make possible experiments of a precision and applications. At the same time, president Reagan's scale previously unachievable; new mathematical and proposed budget for 1989 is the strongest budget in theoretical tools; and easy access to computers and support of science and technology in recent history. electronic information transfer, facilitating the rapid Yet, that budget is in difficulty with an essentially acquisition and analysis of large amounts of data. pro-science congress, and it has divided the scien­ These tools and support drive an ever-changing tific community. The United States supports more science. New scientific fields are being created as scientific research than Western Europe and Japan traditional fields merge or are redefined. We've seen combined, and our system of universities, and na­ that with molecular biology, materials science, pho­ tional, and industrial laboratories is the envy of the tochemistry, and microelectronics. We see that with world. Why then is our community in an unprece­ emergent disciplines-in the mathematics of chaos, in dented state of stress and internal dissension? That is optoelectronics, in the exploitation of quantum effects the issue I want to confront. That is the dilemma that by surface scientists. we as an Academy must confront. Whatever their professional origins, physicists, The Dilemma of the Golden Age. We all believe chemists, biologists, and engineers, and their ideas that scientific exploration has intrinsic cultural and come together to pursue the new ideas. Boundaries intellectual values of the deepest kind, and deserves between basic and applied research are eroding and widespread support on these grounds. However, we in many fields the time between a discovery and its also know that the large resources allocated by gov­ commercialization is now measured in years, rather ernments to scientific research find their justification than decades. It is a sign of our times and of the state on more pragmatic grounds so well symbolized in the of science that industrial scientists have won Nobel

822 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY For Your Information

Prizes in physics two years in a row, and university tions with members. We scientists, who recommend professors have become successful entrepreneurs. the rationality and orderly process of our profession The changing nature of science leads to experimen­ to government policy makers, are fast losing our cred­ tation with new institutions for conducting scientific ibility for being balanced, fair, and analytical. Our and engineering research, such as interdisciplinary internal dissension and the mixed, conflicting, and science centers and technology centers involving aca­ self-serving advice emanating from our community demic and industrial participants on university cam­ are threatening our ability to inform wise policy mak­ puses. Shared facilities housing large and expensive ing. I can think of no more important set of issues to equipment are becoming more common, attracting address to the annual meeting this year. researchers from across the nation. The Reagan Budget Proposal. The concerns over The dilemma lies in that very exuberance-in that the future of American science are joined in the golden age of discovery and advance. Our scientists preparation and approval of the federal budget. The are submitting, in record numbers, proposals of the issues of allocating resources and setting priorities highest quality, with enormous intellectual and ma­ cannot be understood without an appreciation of the terial potential. We have also laid on the budget federal budget process. So, let us look at the Reagan table very large and very expensive new ventures-in budget proposed for FY 1989. The President is to be multiple fields from high-energy physics to molecular commended for the statement of support for science biology, whose time in the progress of science has implicit in his substantial budget initiatives for civil arrived. The proposals-small and large-are superb in science and technology. He proposes a 29 percent quality, but unprecedented in overall cost. And the increase over 1988 for budget function 250 which reality is that these proposals come at a time of record includes the National Science Foundation, NASA's budget deficits. Space Research and Technology programs, and the There is the heart of the dilemma. It is not the Department of Energy's general science program. He lack of political support for science. Political decision requests a 5.1 percent increase for the National In­ makers in the executive branch and Congress no longer stitutes of Health. The budget also signals approval need convincing that leadership of American science and initial funding for the sse, the genome sequenc­ and technology is vital to our nation's future. The real ing project, and the space station. Whether or not political issue is what does science most urgently need one agrees with the specific details, the proposals to retain its strength and its excellence. carry the positive message that support of science and The issues are funding levels and priorities. Our technology is essential to this nation's future. political leadership has no way of gauging the amount The President's budget adheres to the bipartisan of resources necessary to maintain the strength of budget summit agreement reached after last year's American science and technology. What it does see stock market collapse. That agreement limits increases is that the inevitable competition for funds leads to in total non-defense discretionary spending to $3.1 conflicting advice from within the scientific commu­ billion. That is a 2 percent increase, a virtually nity. It learns of caustic debates among scientists in static budget. The President's budget hits this limit our journals and in the press. And it sees issues at by allocating almost all of the allowable increase times framed simplistically, as in the arguments of to science, space, and technology, rather than to "big science," as embodied in the superconducting social programs, such as housing and community supercollider and the genome sequencing project, and development. "small science" as represented by scientists working But that decision tests political reality. It leaves alone or in small teams .... We see confrontation and Congress in the unenviable position of deciding competition bordering on the unseemly between basic between a budget that enhances America's long-term and applied work, between traditional and new fields, competitiveness and one more responsive to near-term between modes of doing research, from the single in­ humanitarian needs and other domestic projects .... vestigator to centers. At a time when we should revel I believe that it is feckless and destructive for in dazzling progress in almost every field of science, the scientific community to argue "for science at the this sniping and carping among scientists is disturbing expense of the homeless," as one congressman put it. and destructive. It is also unrealistic to argue for generous funding of The seemingly intractable problem of setting prior­ both science and social programs until the deficit is ities in the allocation of R & D funds has dominated brought under control. Nevertheless, I believe that we the discussions at our Academy's regional meetings have to make positive and responsible proposals that and it consumes my correspondence and conversa- will strengthen American science and technology ....

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 823 For Your Information

Here too, I will offer proposals that, I believe, are not DOE allocations that reach individuals and small politically stillborn, and that build on the good will of groups, fellowships and assistantships, NASA space public officials towards science. science grants, DOE basic energy sciences, mission Criteria for. Priorities. Our colleagues are divided agency research. on the issue of establishing priorities across fields. At 2. National crises, e.g., R & D for AIDS, renewing regional meetings of the Academy we are told by some the nation's space launch capacity. participants that no one in the scientific community is 3. Extraordinary scientific breakthroughs, e.g., high wise enough to set priorities among fields. Many urge temperature superconductivity. us not to enter what they feel can only be a quagmire. Category 2. Large projects, with important national Some believe that we should support all of the science or scientific goals, to be authorized now. If full funding initiatives on the table, that to propose a list of must be delayed beyond the present because of the priorities will only serve to divide our community and budget deficit crisis, let it be so. However, sufficient to insure a reduced budget. Some argue that large funding should be made available to maintain the science initiatives have historically been accompanied project until such time as large-scale commitments by increases in the entire science budget-small and can be made. Examples: large. Some propose that tradeoff's with other national 1. sse. needs, such as social programs and national security, 2. Mapping and sequencing the human genome. should not enter into our considerations and our Political Category. Prerogatives of the political sys- strategy. And, finally, there are-protagonists for specific tem. Priority has traditionally been decided on the projects or categories-the sse, space station, genome basis of value judgments by elected officials in the sequencing, superconductivity, small science projects executive branch and Congress. Examples include: etc.-who argue for their priority at the expense of 1. The DOD R & D budget and national security. others. 2. The Space Station I say that we have to do better and I would like to 3. Regional economic development, and employ­ propose criteria for the allocation of resources that are ment. appropriate for the unprecedented federal deficit, and 4. Projects that enhance the U.S. image like manned the state of science and technology. They are politically space flight, or projects with significant contributions realistic and responsive to congressional requests for by foreign governments. advice. My soundings of the scientific community 5. Initiatives to enhance U.S. "competitiveness" lead me to believe that they are the least divisive. e.g., education, training, civil sector R & D. They are based on two premises. One is that we It may be wise for huge multibillion-dollar projects must seek to maintain American leadership in science like the space station to be left for major funding and technology. The other is that we will steadily decisions by the next President, who will be in office in overcome the current national budget deficit crisis, a scant eight months, and will have the responsibility as demanded by all thoughtful leaders in national for seeing them done. government, industry, finance, education, and state The rationale for these categories is, I believe, government. straightforward. Above all, the science base must be Category 1. To be funded now with highest priority­ maintained and even strengthened. And that means at a time of budgetary constraint when all agency support for our ablest men and women with the best budgets will be limited. I include in that: ideas. We must confront national crises such as AIDS 1. Preserving the human resource base and the and our sudden incapacity in space. And we must pipeline* for science and technology. This means not be left behind when a major breakthrough occurs, absolute priority for training and research grants such as with superconductivity. reaching the largest number of scientists, engineers, Finally, although we may have to slow them down clinical researchers. This encompasses the funding temporarily, we cannot shy away from major scientific of research universities and national laboratories in opportunities simply because they cost a lot. categories reaching the largest number of investigators Basic to these criteria for establishing priorities is and graduate students. Examples include NIH, NSF, my belief that a great nation like the United States can and should undertake all of these initiatives-and can do so in the near term. I also believe, with * It is especially important to attract more Americans now that others, that the next administration, with bipartisan a substantial fraction of our graduate students in science and engineering are no longer American. support, must recommend a mix of budgetary cuts, reallocations, and revenue increases over a period

824 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY For Your Information

of a few years. Those actions hopefully will remove and technology budget. This function would have to the deficit crisis as a factor limiting the resources be carried out, as so many have said, as an instrument that can be committed to education, science, and of presidential policy, as a resource to OMB, and not technology. The argument in favor of such support is as special pleading for scientists. irresistible-the creation of wealth required to increase It is preferable-in fact, it may be the only way living standards and attend to other national needs to make the system work-for a President to direct can only flow from increased productivity, of the his science advisor to assume this function as a part kind generated by new knowledge and a well-trained of the organization of the White House staff, rather workforce. And if the trends initiated by President than seeking to mandate this function by statute of Reagan in the relationship between the superpowers Congress. To be effective in such a role, a science continue, future Presidents may even be able to justify advisor must have authority and standing; elsewhere substantial reallocation of resources from the military I have proposed that the science advisor be named to the civil sector. a member of the Cabinet without portfolio or a full Government Organization for Science and Technol­ Assistant to the President to carry out these and other ogy Budgeting. Part of the difficulty with the budget duties of like importance. and appropriation process can be attributed to the dis­ In recent weeks some members of the Congress, array of the federal government's system for develop­ struggling with the President's budget, have proposed ing the science and technology budget. It is astounding that the two budget committees in Congress provide but true that nowhere in the federal budget-making an overview for the science and technology budget. process is there an evaluation of the complete federal By statute, the budget committees play a critical role budget for science and technology and its overall ratio­ in the budget process. They issue a joint resolution nale in terms of national goals. Funds are requested by that specifies budget totals as well as allocations to some fifteen federal departments and agencies without each budget function. However, ultimate decisions coordination. Requests appear in the budget process are made by appropriations committees and disagree­ as fourteen separate budget functions, spread over at ments typically occur.... It is a reasonable extension least six divisions of the Office of Management and of their mandate for the budget committees to provide Budget, and requiring approval by nine appropriations Congress with a cross-cutting evaluation of the entire committees in Congress, with input from a number of science and technology budget, with recommendations budget and authorization committees. for allocations and priorities. It will be necessary, and I do not argue for a centralized Department of is natural, for Congress to seek outside advice if Science and Technology. That would be a disaster. it takes on this task; and it has been proposed by Nor do I propose a politically unachievable change some that the National Academies, as congressionally in the complex structure of congressional budget, au­ chartered advisers, respond if called upon. thorization, and appropriation committees. However, In a recent report the Congressional Research Ser­ I believe that with the growth and impact of R & D vice stated that "Without...advice [from the Academies expenditures that now amount to $62 billion, there or the Office of Technology Assessment] the Congress is now a need to assure a cross-cutting review of the will have little choice but to make ... decisions without budget proposal, in both the executive branch and formal guidance on priorities from experts in the sci­ the Congress. At present. the process examines how entific community, and the scientific community will each agency's science and technology budget meets the have no choice but to accept the results." agency's mission. It is now necessary to review how the That is the problem. And that is the challenge. aggregate science and technology budget serves such I believe that we cannot refuse such a call. In national goals as competitiveness, health, security, doing so, however, we must take care not to subsume and world position. It is also important to understand a governmental role. However, we can analyze and the impact of the overall budget on training and on comment on the impact of the overall budget for equipment and facility needs. science and technology. And we must also be willing, It is feasible to do so within the existing gov­ for the first time, to propose priorities across scientific ernmental structure. I agree with those who propose fields, if the .times call for it. We can do so in a manner that in the future the President's science advisor, that is knowledgeable, responsible, and useful. working with the OMB director, should be given the We should accept this new challenge in this, our coordinating role within the executive branch, and 125th year of public service. should assume the responsibility for providing the rationale for a coherent and adequate federal science

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 825 50 Year Anniversary

On the occasion of the Society's Centennial, we wish to recognize those individuals who have supported the AMS, through their membership, for 50 years or more.

V. W. Adkisson Maria Castellani K. S. Ghent E. C. Klipple L. V. Ahlfors Josephine H. Chanler James Wallace Givens T. L. Koehler R. Lucile Anderson Alonzo Church Michael Goldberg F. W. Kokomoor Ralph G. Archibald Helen E. Clarkson Herman H. Goldstine E. R. Kolchin Bradford H. Arnold Mary Dean Clement Saul Gorn Carl F. Kossack Hubert A. Arnold Alfred H. Clifford Sallie P. Grant Harry L. Krall Leo A. Aroian Leon W. Cohen John W. Green Everett H. Larguier Harold M. Bacon Teresa Cohen Thomas N. E. Greville Derrick H. Lehmer Lloyd C. Bagby Nancy Cole Harriet M. Griffin Richard A. Leibler George A. Baker Oliver C. Collins Margaret Gurney Walter Leighton N. Hansen Ball Esther Comegys Wayne W. Gutzman Bela A. Lengyel Donald H. Ballou Edwin H. Comfort Marshall Hall Howard Levi S. F. Barber Nelson B. Conkwright Paul R. Halmos Jack Levine Mabel S. Barnes H. S. MacDonald Coxeter Eugene H. Hanson Daniel C. Lewis Robert C. F. Bartels Mahlon M. Day Walter J. Harrington HansLewy Ralph Beatley John De Cicco Orville G. Harrold Edgar R. Lorch Marjorie Heckel Beaty Luther J. Deck Philip Hartman Lee Lorch Ross A. Beaumont Carl H. Denbow Bernard A. Hausmann Clarence I. Lubin May M. Beenken Douglas Derry E. K. Haviland C. F. Luther E. Maurice Beesley Ainsley H. Diamond Ernest Hawkins Saunders Mac Lane Philip 0. Bell R. P. Dilworth Nola A. Haynes William G. Madow Peter G. Bergmann Joseph L. Doob Gustav A. Hedlund Ralph Mansfield Ebon E. Betz Harold L. Dorwart Anna S. Henriques R. J. Marcou Garrett Birkhoff Francis G. Dressel Fritz Herzog Morris Marden Z. William Birnbaum Daniel M. Dribin Magnus R. Hestenes May H. Maria Gertrude Blanch Richard J. Duffin Arthur 0. Hickson M. H. Martin Ralph P. Boas Dewey C. Duncan Elizabeth S. Hirschfelder William T. Martin Samuel Borofsky William L. Duren Cecil T. Holmes W. J. Martyn Julia W. Bower Paul S. Dwyer Alston S. Householder John R. Mayor Oliver K. Bower Leaman A. Dye Warren G. Hubert Edward D. McCarthy Evelyn Hull Boyle John Dyer W. N. Huff A. J. McConnell G. V. Brady J. J. Eachus M. Gweneth Humphreys Dorothy McCoy Leo Branovan Madeline L. Early W. R. Hutcherson N.H. McCoy Joel L. Brenner Edgar D. Eaves Donald H. Hyers Donald L. McDonough H. W. Brinkmann Theodore M. Edison Robert F. Jackson W.H. McEwen John C. Brixey Margaret C. Eide Stanley B. Jackson L. H. McFarlan Foster Brooks Carolyn Eisele Nathan Jacobson Ruth S. McKee Arthur B. Brown George M. Ewing Emerson D. Jenkins Audrey W. McMillan Mildred C. Brunschwig A. D. Fialkow Borge Jessen Brockway McMillan Herbert Busemann F.A. Ficken Fritz John Edward J. McShane Jewell H. Bushey W. W. Flexner L. Wayne Johnson Florence M. Mears Ethel B. Callahan Karl W. Folley Roberta F. Johnson Albert E. Meder Edward A. Cameron A. L. Foster F. Burton Jones Lyle E. Mehlenbacher Robert H. Cameron Ronald M. Foster Wilfred .Kaplan Russell J. Michel Howard H. Campaigne J. Sutherland Frame Edward S. Kennedy Donald S. Miller Leonard Carlitz Aline H. Frink Lawrence S. Kennison Harry L. Miller Charles L. Carroll Thornton C. Fry Pierce W. Ketchum W. I. Miller Albert B. Carson A. S. Galbraith Stephen C. Kleene Alfred K. Mitchell C. Ronald Cassity Henry L. Garabedian Morris Kline Elmer B. Mode

826 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY SO Year Anniversary

T. Ruley Jose Trevino Morris Monsky Ralph S. Phillips Bryce Albert W. Tucker Harriet F. Montague Everett Pitcher Edward A. Saibel Annita Tuller Deane Montgomery Emily C. Pixley Rafael Sanchez Atwell R. Turquette Marvin G. Moore Theresa L. Podmele Sergei A. Schelkunoff R. Schneckenburger Hugh L. Turrittin Charles W. Moran Harry Polachek Edith Edmund H. Umberger David C. Morrow Hillel Poritsky Isaac J. Schoenberg Sedgewick F. A. Valentine David Moskovitz Theodore H. Preiser Charles H. W. Elbridge P. Vance Thirza A. Mossman Walter Prenowitz Irving E. Segal Herbert E. Vaughan Thomas W. Mullikin G. Baley Price Swarupchand M. Shah R. W. Wagner David C. Murdoch W. T. Puckett Isador M. Sheffer Robert J. Walker Francis J. Murray Edwin J. Purcell Darrell R. Shreve R. M. Walter Cecil J. Nesbitt John W. Querry Aubrey H. Smith Lewis E. Ward Carroll V. Newsom Grace S. Quinn Frank Smithies J. F. Wardwell Abba V. Newton John F. Randolph Robert H. Sorgenfrey Stefan E. Warschawski Ivan Niven Adrienne S. Rayl Emory P. Starke W. J. Webber Edward A. Nordhaus MinaS. Rees David W. Starr John V. Wehausen C. 0. Oakley P. K. Rees F. H. Steen Louis Weisner Edwin N. Oberg Francis Regan Rothwell Stephens Everett T. Welmers E. J. Oglesby Reingold Reingold James J. Stoker H. Wheeler Arthur Ollivier Karlem Riess Marshall H. Stone Charles Albert L. Whiteman P. S. Olmstead Fred D. Rigby Dirk J. Stroik Whitney John M. H. Olmsted Robert F. Rinehart Eric A. Sturley Hassler V. Widder A. C. Olshen John Riordan Alvin C. Sugar David E. P. Wigner Izaak Opatowski Eugene K. Ritter Sister M. Helen Sullivan Wilcox Alexander Oppenheim J. H. Roberts Charles S. Sutton L. R. Woodbridge John C. Oxtoby Malcolm S. Robertson J. L. Synge Margaret Y. W. Wrench Theodore P. Palmer G. B. de Robinson Abraham H. Taub John R. Wylie R. Smith Park Robin Robinson Olga Taussky Qarence E. Zilmer Edson R. Peck Raphael M. Robinson Angus E. Taylor Delbert Sallie E. Pence E. B. Roessler George B. Thomas M.A.Zom Paul M. Pepper Robert A. Rosenbaum Janet D. Thomas Sam Perlis Arnold E. Ross William Rae Thompson George W. Petrie J. Barkley Rosser Robert M. Thrall

ZEROS OF BERNOULLI, GENERALIZED BERNOULLI, AND EULER POLYNOMIALS Karl Dilcher D 386) (Memoirs of the AMS, Number maximum modulus of the The three classes of polynomials studied in this book have proofs are based on results on the under investigation. important applications in the theory of finite differences, zeros of polynomials related to those of real zeros of number theory, and classical analysis. The most significant Finally, the author studies the distribution The results contribution of this book is a proof that the Bernoulli and generalized Bernoulli and Euler polynomials. the classical Bernoulli Euler polynomials and the generalized Bernoulli polynomial are similar to known results about background at the level of associated with certain quadratic characters have no zeros polynomials. The book assumes a and, in particular, inside a parabolic region if the degree is sufficiently large. The an undergraduate education in mathematics author also finds zero-free regions for individual polynomials requires knowledge of basic complex analysis. and for the partial sums for the sine and the cosine. The

S2. each add'l Sl. 1980 Mathematics Subject Classifications: 11, 30 Shipping/Handling: 1st book ISBN 0-8218-2449-X, LC 88-6356 S25 max. By air. 1st book $5. each add'l $3, ISSN 0065-9266 SlOO max. Prepayment required. Order from 104 pages (softcover), April1988 AMS. P.O. Box 1571, Annex Station. member $8, List price $13, Individual Providence. Rl 02901-9930. or call Institutional member $1 0 /l 800-556-7774 to use VISA or MasterCard. L_j D To order, please specify MEM0/386NA

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 827 1988 AMS Elections

Council Nominations President's Candidates Vice-President and Members-at-Large Nominating Committee 1989 and 1990 One vice-president and five members-at-large of the Council Four members of the Nominating Committee are to be elected will be elected by the Society in a contested election in the in the fall of 1988. Continuing members are: fall of 1988. Roger C. Alperin Jane P. Gilman The vice-president will serve for a term of two years effec­ Ronald A. DeVore Leonard L. Scott tive January I, 1989. The Council has nominated one can­ President G. D. Mostow has named six of the eight can- didate for the position, namely: didates for the other four places. They are: Sun-Yung Alice Chang John B. Garnett Andy Roy Magid The Council plans to name a second candidate. Victor K.lee James D. Stasheff The five members-at-large will serve for a term of three Ray A. Kunze Alan D. Weinstein years. The Council nominated seven candidates. They are: If nominations by petition have not appeared bringing Jonathan L. Alperin George R. Sell the total number of candidates to at least eight, it will be Fan R. K. Chung William Yslas Velez brought up to eight by the President. Lawrence J. Corwin Robert J. Zimmer Hugo Rossi Everett Pitcher, Secretary Bethlehem, Pennsylvania The Council plans to name additional candidates for member-at-large to bring their number to at least ten. The deadline for petitions proposing additional nomina­ tions is July 6. Such proposals will not reach the Council for action by mail ballot until after that date.

such as matrices and coordinate systems, research in superalgebras, invariant INVARIANT THEORY and may lead to an extension of the theory, straightening algorithms, Young AND SUPERALGEBRAS entire apparatus of linear algebra to bitableaux, and Grassmann's calculus "signed" modules. The authors also of extension, the book starts from Frank Grosshans, present the symbolic method for the basic principles and should therefore be Gian-Carlo Rota, and invariant theory of symmetric and of accessible to those who have completed Joel A. Stein skew-symmetric tensors. In both cases, the standard graduate level courses in the invariants are obtained from the algebra and/or combinatorics. symbolic representation by applying This book brings the reader to the what the authors call the umbral Contents: The superalgebra super [A); frontiers of research in some topics in operator. This operator can be used to Laplace pairings; The standard basis superalgebras and symbolic method systematically develop anticommutative theorem; Invariant theory; Examples. in invariant theory. Superalgebras are analogs of concepts of algebraic geometry, algebras containing positively-signed and 1980 MathematicB Subject OlaBBificationB: and such results may ultimately turn 15, 16, 20 negatively-signed variables. One of the out to be the main byproduct of this ISBN 0-8218-0719-6, LC 87-21146 book's major results is an extension of the investigation. ISSN 0160-7642 standard basis theorem to superalgebras. 104 pages (softcover), October 1987 This extension requires a rethinking of While it will be of special interest to List price $16, all individuals $10 some basic concepts of linear algebra, mathematicians and physicists doing To order, please specify CBMS/69NA

Use the order fonn in the back of this issue or call 800-556-7774 to use VISA or MasterCard. Shipping and handling charges will be added.

828 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings and Conferences of the AMS

FUTURE MEETINGS PU · Centennial Celebration 1')!11 Providence, Rhode Island, August 8-12 831

Lawrence, Kansas October 28-30 880 :

Claremont, California November 12-13 882

Invited Speakers and Special Sessions 884

FUTURE CONFERENCES

Call for Topics 887

JULY/ AUGUST1988, VOLUME 35, NUMBER 6 829 Coming Events

We are happy to report that preregistration for the Centennial Celebra­ tion is running ahead of what is normal for a summer meeting, and so it appears that attendance at this very special event will be exceptional. The scientific program which appears on the following pages promises to provide a rich and rewarding experience for everyone. The social program offers a variety of events, designed to appeal to all partici­ pants. So, if you have not yet made up your mind about attending, we hope you will decide to join us in this celebration, which happens only once in a lifetime!

The deadline for preregistration and housing requests for the January 11-14, 1989 meeting in Phoenix, Arizona is November 10. In order to be eligible for the drawing for free sleeping rooms, forms must be received in Providence no later than October 31. Since the form will appear for the first time in the October issue of the Notices (mailed from the printer on September 30), individuals interested in the room lottery should return their form as soon as they receive the October issue.

Mark your calendar now for the August 7-10, 1989 meeting at the Uni­ versity of Colorado, Boulder! Plans are now being made for a real West­ em Style Hoe Down, featuring local cuisine and entertainment. Also, early arrivals will have the opportunity to take a bus tour through the beautiful Rocky Mountain National Park and visit an authentic mining town.

830 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Centennial Celebration Providence, RI August 8-12, 1988

Program

The Centennial Celebration of the American Mathematical The Centennial Committee was assisted by a number of Society will be held August 8- 12, 1988 (Monday- Friday). subcommittees: Sessions will take place in the Providence Performing Arts The members of the Centennial Program Committee are Center (PPAC), the Omni Biltmore Hotel, the Holiday Inn­ HYMAN BAss, Columbia University; FELIX E. BROWDER, Chair­ Providence Downtown, and at the Rhode Island School of man; PHILLIP A. GRIFFITHS, Duke University; JOHN W. MIL­ Design. NOR, Institute for Advanced Study; CATHLEEN S. MoRAWETZ, The members of the Centennial Committee are FELIX E. Courant Institute of Mathematical Sciences, New York Uni­ BROWDER, Rutgers University; HAROLD M. EDWARDS, Courant versity. FRANK T. BIRTEL of Tulane University is in charge of Institute of Mathematical Sciences, New York University; the AMS contributed paper sessions. ANDREW M. GLEASON, Harvard University, former President of The members of the Centennial Public Information Com­ the American Mathematical Society; GEORGE DANIEL MosTow, mittee are JoHN W. ADDISON, JR., University of California, Yale University, President of the American Mathematical Berkeley; YousEF ALAVI, Chairman, Western Michigan Uni­ Society; and EVERETT PITCHER, Chairman, Lehigh University. versity; WILLIAM G. CHINN, San Francisco, California; RoNALD R. COIFMAN, Yale University; RONALD L. GRAHAM, AT&T Bell Laboratories; PETER J. HILTON, SUNY at Binghamton; DoN R. WHERE TO FIND IT PAGE LICK, Eastern Michigan University; JEAN J. PEDERSEN, Santa HOW TO OBTAIN HOUSING 840, 841 Clara University; and CLIFFORD TAUBES, Harvard University. The members of the Committee on Travel Grants for the AMS OPENING CEREMONIES AND KEYNOTE ADDRESS 832 Centennial are RICHARD W. BEALS, Chairman, Yale University; SYMPOSIUM ON MATHEMATICS INTO THE JoHN W. BuNcE, University of Kansas; ANTHONY W. HAGER, TWENTY -FIRST CENTURY 832 Wesleyan University; WILLIAM F. LucAs, Claremont Graduate AMS-MAA INVITED ADDRESSES 836 School; P. EMERY THOMAS, University of California, Berkeley; SUMMER MEETING OF THE AMS 837 and FRANK UHLIG, Auburn University. The Society wishes to thank the Mathematical Association Prizes, Contributed Papers, Council, Business Meeting of America for relinquishing its usual summer meeting program AMS SHORT COURSE 879 in deference to the expanded requirements of the Society on OTHER ORGANIZATIONS 837 this special occasion. Sessions of the Association will be held AWM, JPBM, MAA, TIME August 6 and 7 (Saturday- Sunday), for the most part. The Society is pleased TIMETABLE 857 to continue the tradition of joint invited addresses with the Association. The members of the OTHER EVENTS OF INTEREST 840 Joint AMS-MAA Program Committee are HUGH L. MONT- Book Sales, AMS Members' Information Booth, Special Exhibits, Commercial Exhibits, Information Table, Petition Table REGISTRATION AT THE MEETING 845 IMPORTANT DEADLINES Fees, Dates, Times, Locations, Services AMS Abstracts Of contributed papers Expired MISCELLANEOUS INFORMATION 848 Centennial Preregistration and Housing Expired Athletic Facilities, Book Stores, Camping, Child Care, Crib MAA Minicourse Preregistration Expired Rental, Handicapped, Libraries, Local Information, Medical Motions for AMS Business Meeting July12 Clambake caoceUations (50% refund) until Jniy 25 Services, Parking, Smoking, Special Events, Souvenirs, Tours, MAA and 1tME Banquets (50% refund) until Jniy 25 Travel, Videotapes, Weather Preregistration canceUations (50% refund) until August 4 MAPS 842, 844 Airport transfer cancellations (50% refund) after August 4 Changes to residence ball packages after August 4 PRESENTERS OF PAPERS 878 Housing cancellations (90% refund) until August 4 Housing canceUation penalty (10% plus one night) after August 4 PROGRAM OF THE SESSIONS 869 Tours (50% refund) until August 4

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 831 Meetings

GOMERY, Chairman, University of Michigan, Ann Arbor; M. year of the Society. This quarterly journal contains research SusAN MoNTGOMERY, University of Southern California; IVAN articles of the highest quality in all areas of pure and applied NIVEN, University of Oregon; and RICHARDS. PALAIS, Brandeis mathematics. Editors of the journal are: Michael Artin, H. University. Blaine Lawson, Jr., Richard Melrose, Wilfried Schmid, and On Monday, August 8, there will be a private ceremony Robert E. Tarjan. at the Society's headquarters office at which the officers of Also, in recognition of the Centennial Celebration of the the Association will formally present a gift of sculpture to Society, the Council has approved changing the name of the the Society on the occasion of its Centennial. This 547 research fellowship program to the American Mathematical pound sculpture of Carrara marble titled Torus with cross­ Society Centennial Research Fellowship. cap and vector field is by HELAMAN ROLFE PRATT FERGUSON, mathematician and sculptor, of Provo, Utah. The sculpture resides in the lobby of the Society's head­ Opening Ceremonies quarters building, and can be viewed during the hours set aside The Opening Ceremonies will take place from 9:00 a.m. for Open House, described later in this announcement. Also, to 10:30 a.m. on Monday, August 8, in the PPAC. the sculpture is depicted on a commemorative poster avail­ able for purchase at the Souvenirs Section of the Centennial Representatives from the London Mathematical Society, Registration Desk. the Mathematical Association of America, the Society for The Centennial Committee thanks the National Science Industrial and Applied Mathematics, Brown University, Foundation for its support of the symposium on Mathematics city and state government, will address the assembly. A into the twenty-first century (Grant #DMS8716887), and the brief oral history of the Society will be presented by Department of Energy (Grant #DE-FG02-88ER25056), the the Secretary. President Mostow will serve as Master of Office of Naval Research (Grant #N00014-88-J-1096), and the Ceremonies. United States Army Research Office (Grant #DAAL03-88-G- 0022) for its grant supporting travel and subsistence for young Keynote Address mathematicians attending the Centennial. The Centennial Committee acknowledges with thanks the EDWARD E. DAVID, JR., President of EED, Inc. and assistance of THOMAS F. BANCHOFF of Brown University in the former Science Advisor to the President of the United design of the Centennial logo. States, will speak at 11: 15 a.m. on Monday, August 8, Some very special things have occurred in connection with on Renewing U.S. mathematics: An agenda to begin the the Centennial that participants may find of interest. First, second century. Dr. David will be introduced by GEORGE the National Academy of Sciences Award in Mathematics, DANIEL MOSTOW. sponsored by the Society in commemoration of its Centennial, in the amount of $5,000, has been awarded for the first time in 1988 and will be awarded every four years hereafter. More Symposium on Mathematics into the information on this prize can be found on page 509 of the Twenty-First Century April issue of Notices. This symposium was organized by the Centennial Pro­ The Journal of the American Mathematical Society began publication in January 1988 in conjunction with the Centennial gram Committee and is directed toward the future, not the past, of the American mathematical community, toward the creative innovation and new achievements foreseeable in terms of the ideas and the ongoing work of the generation of American mathematicians active today. The speakers include four Fields Medalists, three winners of the Waterman Award, and several winners of other major mathematical prizes. The speakers have been asked to give talks of a nature that can be understood by a general mathematical audience, surveying the motives, paradigms, and prospects of major areas of contemporary mathematical research. It is hoped that this symposium will provide an active stimulus toward developing a broader consciousness among American mathematicians of the unifying trends of mathematical research uniting not only the classical central areas of pure mathematics, but uniting them as well with some of the most vital concerns of mathematical applications. The names of the speakers, their affiliations, the titles, Edward E. David, Jr., Keynote Speaker and the times and days of their talks follow:

832 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY MICHAEL AscHBACHER CHARLES L. FEFFERMAN Professor of Mathematics Professor of Mathematics California Institute of Princeton University Technology Ph.D., Princeton University, Ph.D., University of 1969 Wisconsin, 1969 9:45a.m. 2:00p.m. Tuesday, August 9 Monday, August 8

Representations offinite groups as permutation groups Problems from mathematical physics The classification of the finite simple groups in 1981 changed This talk will cover two problems in mathematical physics. The the landscape of finite group theory and led to an increased first is from quantum mechanics and concerns the question of effort to describe the structure and representations of the simple how large numbers of electrons combine with large numbers of groups. Together with the classification, this effort has made protons to form large numbers of atoms. The second is from possible unexpected applications of finite group theory in other general relativity and concerns a proof that some small initial branches of mathematics. disturbance will not concentrate and become a black hole. Introduced by DANIEL GORENSTEIN. Introduced by FELIX E. BROWDER.

LUIS A. CAFFARELLI MICHAEL H. FREEDMAN Professor of Mathematics Charles Lee Powell Institute for Advanced Chair Professor Study University of California, Ph.D., University of San Diego Buenos Aires, 1972 Ph.D., Princeton University, 1973 3:15p.m. Monday, August 8 2:00p.m. Tuesday, August 9 The geometry of solutions to nonlinear problems Working and playing with the two-dimensional disk This talk will discuss geometric techniques to study the shape and regularity of solutions to nonlinear elliptic equations and The conformal structure of the disk is useful in studying the their level surfaces. · topology of (real) surfaces. A more combinatorial-topological study of maps of a disk has illuminated the study of three­ Introduced by Lours NIRENBERG. dimensional manifolds. This talk will briefly survey the role of the disk in the theory of high dimensional manifolds, and PERSI DIACONIS go on to address the special problems of a disk mapped into Professor of Mathematics a four-dimensional manifold. This is the point at which the Harvard University topological and smooth theories diverge, and some discussion Ph.D., Harvard University, of the disparities between them will be given. 1974 Introduced by WILLIAM BROWDER, President-Elect of the AMS. 8:30a.m. Tuesday, August 9 HARVEY M. FRIEDMAN Professor of Mathematics Ohio State University Sufficiency as statistical symmetry Ph.D., Massachusetts To judge what parts of a data set are worth saving, statisticians Institute of Technology, have developed a useful tool called sufficiency, which can be 1967 seen as an extension of the invariants of a group. Sufficiency allows a unified construction of statistical models, sheds light on the factorization of generating functions in combinatorics, 3:15p.m. and provides the underpinnings for recent work in statistical Tuesday, August 9 mechanics. This talk will explain the concept of sufficiency and survey these applications. Introduced by GIAN CARLO RoT A.

continued

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The incompleteness phenomena talk will describe these stages in the evolution of the subject an( By 1922, the formalization of mathematics in terms of ax­ indicate how they arose as outgrowths of classical problems. iomatic set theory had emerged. The axioms and rules of in­ Introduced by PHILLIP A. GRIFFITHS. ference of this formalism are collectively known as Zermelo Frankel set theory with the axiom of choice (ZFC). The in­ RoGER E. HowE completeness phenomena-assertions which cannot be proved Professor of Mathematics or refuted with ZFC- have not yet necessitated a reassessment Yale University of ZFC, but the twenty-first century may see debate on which Ph.D., University of axioms and rules of inference should be allowed. This talk will California, Berkeley, 1969 provide a historical perspective on the incompleteness phenom­ ena. 2:00p.m. President of Introduced by SAUNDERS MAc LANE, former Wednesday, August 10 the AMS.

BENEDICT H. GROSS A century of Lie theory Professor of Mathematics Harvard University The subject called Lie theory (the study of Lie groups, Lie alge­ Ph.D., Harvard University, bras, algebraic groups, and their applications) is, like the AMS, just about one hundred years old. In that century, Lie theory has 1978 established itself as a central area of mathematics, using tools from many sources and having implications for many other 8:30a.m. fields. This talk will attempt to give a feeling for the diversity Wednesday, August 10 of applications of Lie theory and for the rich internal structure that supports the applications. Introduced by GEORGE MACKEY. Modular forms and elliptic curves This talk will survey some major developments in the theory VAUGHAN F. R. JONES of elliptic curves. The theory of elliptic functions and modu­ Professor of Mathematics lar forms, created in the 19th century, concerns the real and University of California, complex solutions of cubic equations and their moduli. In the Berkeley last fifty years, the original arithmetic viewpoint has once again Ph.D., Universite de of solutions emerged. The problem of counting the number Geneve, Switzerland, 1979 (mod p) to equations with integral coefficients is related to certain Fourier expansions in the classical theory of modular forms. This relationship has led to some progress on the prob­ 3:15p.m. lem of constructing rational points. Wednesday, August 10 Introduced by JoHN T. TATE. A von Neumann algebra excursion: From quantum theory to knot theory and back JOSEPH HARRIS A surprising result in von Neumann algebras suggested repre­ Visiting Scholar in sentations of the braid group into an abstract algebra discov­ Mathematics ered in statistical mechanics. The result allows one to associate Harvard University to each braid a number which turns out to depend only on the Ph.D., Harvard University, knot obtained by closing the braid. The resulting new knot in­ 1977 variant stimulated the discovery of many more such invariants. These invariants are being used to study the way enzymes "un­ 9:45a.m. tie" knotted strands of DNA in the process of replication. Wednesday, August 10 Introduced by JOAN S. BIRMAN.

VICTOR G. KA.c Developments in algebraic geometry Professor of Mathematics One of the oldest branches of mathematics, algebraic geometry Massachusetts Institute of curves, surfaces and higher­ is concerned with the geometry of Technology dimensional objects defined by polynomial equations-conic Moscow State sections, quadric surfaces, and so on. Over the last two cen­ Ph.D., turies, algebraic geometry has undergone a series of transforma­ University, 1968 tions in which its basic objects of study were redefined, the most recent being the introduction of the concept of "schemes." This 4:30p.m. Wednesday, August 10 continued

834 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

Modular invariance in mathematics and physics DENNIS P. SULLIVAN This talk will focus on some beautiful, recently discovered con­ Professor of Mathematics nections between the representation theory of infinite dimen­ Graduate School and sional Lie algebras and the theory of modular functions, and University Center, on related progress in theoretical physics. The basic examples City University covered will be: affine Kac-Moody algebra, the central exten­ of New York, sion of the loop group of a compact Lie group; and Virasoro Ph.D., Princeton algebra, the central extension of the Lie algebra of vector fields University, 1966 on the circle. The "modular invariant" representations of these in recent devel­ algebras have been playing a fundamental role 2:00p.m. ments of conformally invariant quantum field theories and op Thursday, August 11 in string theory. Introduced by NATHAN JACOBSON, former President of Progress on the renormalization the AMS. conjectures in dynamical systems Computation has led theoretical physicists to the discovery that, ANDREW J. MAJDA in certain dynamical systems, the geometrical structure at suc­ Professor of Mathematics cessively smaller scales is asymptotically constant. Moreover, Princeton University the structure is universal in the sense that inequivalent systems Ph.D., Stanford have the same limiting structure. This talk will summarize the University, 1973 progress in the theoretical understanding of this numerical dis­ covery. 8:30a.m. Introduced by STEPHEN SMALE. Thursday, August 11 ROBERT E. TARJAN James S. McDonnell Mathematical fluid dynamics: The interaction of Distinguished University nonlinear analysis and modern applied mathematics Professor of Computer The rapid evolution of applied mathematics through large-scale Science computation reveals new fluid flow phenomena that are far Princeton University and beyond the capability of experimental measures. To explain Distinguished Member and control these complex phenomena, new mathematical ideas of Technical Staff from nonlinear analysis, differential equations, probability the­ AT&T Bell Laboratories and geometry must interact with computational methods ory, Ph.D., Stanford University, and more traditional tools of applied mathematics. This talk will present a survey of several examples of this new mode of 1972 interdisciplinary research in mathematical fluid mechanics. Introduced by PETER D. LAx, former President of the AMS. 8:30a.m. Friday, August 12 CHARLES S. PESKIN Mathematics in computer science Professor of Mathematics This talk will explore the interdependencies between mathemat­ Courant Institute of ics and computer science as illustrated in the variety of mathe­ Mathematical Sciences, matical ideas used to derive results in computer science theory New York University, and the use of computation in the proof of mathematical theo­ Ph.D., Albert Einstein rems. College of Medicine, 1972 Introduced by RONALD L. GRAHAM.

9:45a.m. WILLIAM P. THURSTON Thursday, August 11 Professor of Mathematics Princeton University Mathematics and computing in physiology and medicine: Ph.D., University of Examples from the past, present, and future California, Berkeley, 1972 The examples considered are the Hodgkin-Huxley equations for the nerve impulse, computed tomography, a mathematical 9:45a.m. model for blood flow in the heart, and the robotics of large bi­ Friday, August 12 ological molecules. Computation is a key ingredient in all of these examples, and future success is tied to the development of large-scale computers and efficient numerical algorithms. Introduced by CATHLEEN S. MoRAWETZ. continued

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 835 Meetings

Three-dimensional geometry and topology AMS-MAA Invited Addresses Three dimensions is the crossroad for geometry and topology. In By invitation of the AMS-MAA Joint Program dimensions higher than 3, topology becomes much more arbi­ Commit­ tee trary, while geometry becomes much more restricted and rigid. , the following speakers will speak on the history and In dimensions lower than 3, topology is more limited, while ge­ development of mathematics. ometric constructions are more flexible. This talk will describe RAouL H . BoTT several instances of the close match between the geometry and William Caspar Graustein topology of 3-dimensional objects, including the theory of poly­ Professor of Mathematics hedra, the theory of knots, and the theory of 3-dimensional manifolds. Harvard University D.Sc., Carnegie Institute Introduced by LIPMAN BERS, former President of the AMS. of Technology, 1949 KAREN K. UHLENBECK Professor of Mathematics 11:00 a.m. Tuesday University of Texas , August 9 at Austin The topological constraints on analysis Ph.D., Brandeis This topic has been at University, 1968 the center of one of the two great Amer­ ican schools of topology. Some of its achievements during this century will be discussed. 11:00 a.m. Introduced by ANDREW M. GLEASON. Friday, August 12 Instantons and their relatives PETER D. LAX Ins tan tons are geometric objects which were discovered by theo­ Professor of Mathematics retical high energy physicists as a result of failed attempts to un­ Ph.D., New York derstand strong interactions. The instanton equation-of which University, 1949 instantons are solutions-derives from the nonlinear version Maxwell's equations formulated by Yang and Mills in 1954. 11:00 a.m. The importance of the instanton equation in mathematics was Wednesday, August 10 recognized only in the past decade. Vortices and monopoles are only two of the many related geometric objects having elegant, interesting, and useful mathematical properties. This talk will attempt to describe some of the more colorful properties and Mathematics: Applied and pure uses of instantons and some conjectures for the future. In this century, some have viewed mathematics as separated Introduced by SHIING S. CHERN. into pure and applied. Today more and more mathematicians realize that mathematics does not "trickle down" to applica­ EowARD WITTEN tion areas, but is an equal partner with other sciences. Modern Professor of Physics computers have linked mathematics with other sciences. Institute for Advanced Introduced by GEORGE DANIEL MosTow. Study Ph.D., Princeton SAUNDERS MAC LANE University, 1976 Professor Emeritus, University of Chicago 2:00p.m. Ph.D., University of Friday, August 12 Gottingen, 1934

Quantum field theory and Donaldson polynomials 11:00 a.m. When Simon Donaldson initiated a program of using the self­ Thursday, August 11 dual Yang-Mills equations to study smooth four-manifolds, the relationship of his work to physical ideas was something of an enigma. Since then, it has become clear that relativistic quan­ Some major research departments of mathematics tum field theory provides a very natural setting for understand­ In the last century, the development of mathematics has been ing Donaldson theory and its relationship to Floer theory, el­ led by a number of outstanding research departments. The tra­ liptic cohomology, conformal field theory, and possibly to other dition was developed in the U.S. by Moore, Birkhoff, Veblen, subjects, including string theory and the Jones polynomial. This Stone, and others. This talk will describe several mathematics talk will survey some of these developments. research departments. Introduced by CLIFFORD T AUBES. Introduced by LEONARD GILLMAN, President of the MAA.

836 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

91st Summer Meeting of the AMS Activities of Other Organizations August 8-12, 1988 Association for Women in Mathematics (AWM) Prizes The AWM Membership Meeting will take place at The 1988 Leroy P. Steele Prizes will be awarded at 3:15 7:30 p.m. on Tuesday, August 9. AWM will sponsor p.m. on Friday, August 12. a panel discussion at 8:00 p.m. on Tuesday, on Cen­ tennial reflections on women in American mathematics. Panelists include MABEL S. BARNES, Professor Emeritus, Contributed Papers Occidental College; JUDY GREEN, Rutgers University, There will be sessions for contributed papers on Monday, Camden; JEANNE LADUKE, DePaul University; VIVI­ Tuesday, Wednesday, and Friday afternoons. ENNE MALONE-MAYES, Baylor University; and OLGA TAussKY-TODD, California Institute of Technology. An Council Meeting open reception is being planned by AWM to follow the The Council of the Society will meet at 5:00 p.m. on panel. Sunday, August 7. Business Meeting Joint Policy Board for Mathematics (JPBM) The Business Meeting of the Society will take place The JPBM Committee for Mathematics Department immediately following the award of the Steele Prizes at Heads has organized a National Meeting of Department 3:15 p.m. on Friday, August 12. The secretary notes the Heads at 7:00p.m. on Wednesday, August 10. This ses­ following resolution of the Council: Each person who sion will feature a panel being organized by SAMUEL M. attends a Business Meeting of the Society shall be willing RANKIN, III, Worcester Polytechnic Institute, on MS2000 and able to identify himself as a member of the Society. Panel on courses and curricula for mathematics master's In further explanation, it is noted that each person who is degrees. This will be followed at 8:00 p.m. by Birds-of­ to vote at a meeting is thereby identifying himself as and a-Feather sessions on Academic employment of master's claiming to be a member of the American Mathematical degree mathematicians in larger departments moderated Society. For additional information on the Business by DONALD F. REYNOLDS, Indiana State University. Meeting, please refer to the box titled Committee on the Panelists include ALPHONSE H. BAARTMANS, West Vir­ Agenda for Business Meetings. ginia University; ALBERTO R. GALMARINO, Northeast­ em University; and DONALD R. WHITAKER, Ball State Committee on the Agenda University. The second concurrent session is on Aca­ for Business Meetings demic employment of master's degree mathematicians in The Society has a Committee on the Agenda for Business Meetings. smaller departments, moderated by DAVID W. BALLEW, The purpose is to make Business Meetings orderly and effective. The committee does not have legal or administrative power. It is Western Illinois University. Panelists include RoNALD intended that the committee consider what may be called "quasi­ M. DAvis, Northern Virginia Community College and political" motions. The committee has several possible courses of KAREN L. WHITEHEAD, South Dakota School of Mines. action on a proposed motion, including but not restricted to (a) doing nothing; . The third concurrent session is on Industrial employment (b) conferring with supporters and opponents to amve at a mu­ of master's degree mathematicians moderated by ToM tually accepted amended version to be circulated in advance of the TROTTER, Arizona State University. meeting; . (c) recommending and planning a format for debate to suggest to a Business Meeting; (d) recommending referral to a committee; 67th Summer Meeting of the (e) recommending debate followed by referral to a committee. Mathematical Association of America (MAA) There is no mechanism that requires automatic submission of a motion to the committee. However, if a motion has not been sub­ Minicourses mitted through the committee, it may be thought reasonable by a Business Meeting to refer it rather than to act on it without benefit Seven Minicourses are being offered by the MAA, to of the advice of the committee. be held on Saturday and Sunday, August 6 and 7. The The committee consists of M. Salah Baouendi, Everett Pitcher names and affiliations of the organizers, the topics, the (chairman), and Carol L. Walker. In order that a motion for the Business Meeting of August 12, dates and times of their meetings, and the enrollment 1988, receive the service offered by the committee in the most effec­ limitations of each are as follows: tive manner, it should be in the hands of the Secretary by July 12, Minicourse #1: EXP, EXPTest, and the creation of 1988. testbanks is being organized by PETER FRISK, Rock Valley Everett Pitcher, Secretary College. Part A is scheduled from 8:00 a.m. to 10:00 a.m. and Part B from 2:00 p.m. to 4:00 p.m. on Saturday, August 6. Enrollment is limited to 30.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 837 Meetings

The scientific word processor, EXP, easily produces to experiment with a representative collection of the complex mathematical and scientific expressions, mak­ better programs in calculus, linear algebra, differential ing it ideal for writing examinations in mathematics. equations, and other mathematical subjects. A variety A related program, EXPTEST, enables instructors to of classroom applications will be demonstrated from a create, edit, and print examinations created ·by selecting number of different computer packages. At the end of questions from prepared testbanks. This workshop will each two hour session, the participants will have the op­ discuss the basics of EXP itself, including its editing portunity for "hands-on" use of some of these programs. commands, special fonts and mathematical symbols, and Handouts will provide information on other educational keyboard macros. This is basic to the full discussion of software on the market or under development. No com­ EXPTEST, which will include the creation of testbanks, puter experience is required. the writing of various types of questions those banks Minicourse #4: Coloring and path following algo­ may contain, and the automatic generation of different, rithms for approximating roots and fixed points is being but equivalent, tests. organized by WILLIAM F. LucAs, Claremont Graduate Minicourse #2: Contributions of algebraic coding the­ School. Part A is scheduled from 10:30 a.m. to 12:30 ory to finite geometry is being organized by E. F. AssMus, p.m. and Part B from 4:30p.m. to 6:30p.m. on Saturday, JR., Lehigh University and J. D. KEY, who is Emmy August 6. Enrollment is limited to 80. Noether Lecturer at Bryn Mawr and on leave from the Cayley (1879) found that Newton's method for ap­ University of Birmingham. Part A is scheduled from proximating complex roots of a polynomial equation 8:00 a.m. to 10:00 a.m. and Part B from 2:00 p.m. to could lead to complications. (See Science News, February 4:00 p.m. on Saturday, August 6. Enrollment is limited 28, 1987, regarding regions with chaotic boundaries.) to 80. H. W. Kuhn (1974) has provided an elementary path After the appearance of Shannon's fundamental paper following algorithm in the plane for finding such roots. in 1948 engineers and mathematicians set to work The roots are triple points in a simple three coloring of to implement the ideas by constructing, via algebraic the plane as was already evident in a geometric view methods, so-called "error-correcting codes." Some of provided in Gauss' thesis ( 1799). the very first attempts used certain geometries over The fundamental combinatorial lemmas by E. Spemer finite fields to construct codes. During the next three ( 1928) and A. W. Tucker ( 1946) for labeling (or col­ decades an increasingly sophisticated body of results was oring) the vertices of an n-simplex or n-octahedron fashioned and the theory became known as "algebraic are the discrete analogues of the Brouwer fixed point coding theory." Late in the 1960s it became apparent theorem and Borsuk-Ulam antipodal points theorems, that this theory could repay its debt to pure mathematics respectively. These provide the basis for the path fol­ by making significant contributions to the field of finite lowing algorithms of Scarf ( 1967) and others for finding geometry, which will be the subject of this Minicourse. approximate fixed points. Applications include the com­ Although a most exciting aspect of this inter-rela­ puting of equilibrium points or prices in game theory tionship is the one deriving from the Gappa codes and economics. and their generalizations, that area requires a rather These topics can be included at various levels in deep knowledge of algebraic geometry. The course will undergraduate courses on discrete mathematics, and do restrict itself to those aspects of the subject that can be not assume any specialized prerequisites. comprehended with only minimal knowledge of group Minicourse #5: Teaching calculus with an HP-28 sym­ theory and finite geometry, but with a strong background bol manipulating calculator is being organized by JoHN in linear algebra, and a firm acquaintance with finite W. KENELLY, Clemson University. Part A is scheduled fields. After a brief historical introduction, the course will for 8:00 a.m. to 10:00 a.m. and Part B from 2:00 p.m. develop the necessary material from coding theory and to 4:00 p.m. on Sunday, August 7. Enrollment is limited finite geometry and conclude with a detailed description to 40. of some of the contributions of coding theory to finite After briefly surveying the capabilities of currently geometry. available graphic calculators, the Minicourse will intro­ Minicourse: #3: A survey of educational software is duce participants, hands on, to the HP-28. Graphing, being organized by DAVID P. KRAINES, Duke University symbol manipulating, differentiation, equation solving, and VIVIAN KRAINES, Meredith College. Part A is sched­ Taylor polynomials and (time permitting) matrix opera­ uled from 10:30 a.m. to 12:30 p.m. and Part B from 4:30 tions will be viewed. p.m. to 6:30 p.m. on Saturday, August 6. Enrollment is There will be a discussion of the use of the HP-28 limited to 30. in calculus instruction, of how its use will change the The variety and the quality of software for IBM treatment of current topics and how it will make possible compatible computers has been increasing steadily. The the introduction of new topics in calculus. objective of this Minicourse is to allow the participants

838 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

Minicourse #6: An introduction to MATLAB is being law changes will be submitted to the membership. This organized by DAVID R. HILL, Temple University. Part meeting is open to all members of the Association. A is scheduled from 10:30 a.m. to 12:30 p.m. and Part B from 4:30 p.m. to 6:30 p.m. on Sunday, August 7. Board of Governors Enrollment is limited to 30. The MAA Board of Governors will meet at 8:30a.m. on MATLAB is an interactive software package that has Sunday, August 7. This meeting is open to all members wide application for mathematics instruction and student of the Association. use. Originally developed at a MATrix LABoratory, the Section Officers software has evolved into a versatile environment for use in linear algebra, numerical analysis, calculus, discrete There will be a Section Officers' meeting at 4:30 p.m. on mathematics, and statistics. The availability of easily Monday, August 8. accessible graphics in· two and three dimensions, a rich MAABanquet set of powerful commands, and the easy development of The MAA is planning its thirteenth annual banquet for your own extensions provide a flexible tool for instruction individuals who have been members of the Association and problem solving. for twenty-five years or more. The banquet will be This course will provide: 1) An introduction to MAT­ held in the Bacchante Room of the Omni Biltmore on LAB's command set. 2) A "hands-on" opportunity to Wednesday, August 10. Dinner will be served at 7:00 explore topics including solving linear systems, plotting, p.m. interpolation, least squares, and elementary statistics. 3) Tickets are $21 each; the price includes gratuity. The Examples of courseware for student self-practice. The menu includes consomme royale, spinach salad, roast format will be informal and self-paced with opportuni­ leg of veal dijon, potato and vegetable, warm Stanford ties for discussion and an exchange of ideas. No formal rolls and butter, strawberries Romanoff, coffee, tea, and prerequisites. decaffeinated coffee. There will be a cash bar. Those Minicourse #7: Groups, graphs, and computing is being who did not purchase tickets through preregistration organized by EuGENE M. LuKs, University of Oregon. should check at the Tickets/Tours desk to see if any Part A is scheduled from 10:30 a.m. to 12:30 p.m. and are still available before the close of registration on Part B from 4:30 p.m. to 6:30 p.m. on Sunday, August Monday, August 8. If tickets were purchased through 7. Enrollment is limited to 80. preregistration and must be cancelled, a 50% refund of This Minicourse will explore exciting applications of the amount paid for the ticket will be made if notification group theory in theoretical computer science. Some cen­ is received in Providence by July 25. After that date, no tral issues in computational complexity theory are illumi­ refund can be given. nated in an exploration of problems that require manip­ ulation of large permutation groups. Rubik's cube alone suggests both easy ("polynomial-time") and probably­ Pi Mu Epsilon {liME) hard ("NP-complete") problems. But it inspires, as well, liME will hold its annual meeting on Tuesday and problems that have defied such categorization. We shall Wednesday, August 9 and 10. The Council will meet at discuss divide-and-conquer algorithms that underlie the noon on Tuesday, August 9. On Wednesday, August 10, best-known approach to such problems. These algorithms the Dutch Treat Breakfast will be at 6:30 a.m. There also provide the machinery behind the most efficient at­ will also be sessions for contributed papers on Tuesday tacks on the important computational problem of testing evening. isomorphism between graphs. Participants interested in attending any of the MAA J. Sutherland Frame Lecture Minicourses should have completed the MAA Minicourse The J. Sutherland Frame Lecture will be given at Preregistration Form and sent it directly to the MAA 8:30 p.m. on Wednesday, August 10, by DoRIS W. office at the address given on the form so as to arrive ScHATTSCHNEIDER, Moravian College, on You, too, can prior to the June 1 deadline. Please note that these MAA tile the Conway way. Professor Schattschneider will be Minicourses are NOT the AMS Short Course. the first woman mathematician to deliver this distin­ Prize Session and Business Meeting guished lecture. The MAA Prize Session and Business Meeting is sched­ Banquet uled from 4:30 p.m. to 5:30 p.m. on Tuesday, August The liME Banquet will take place on Wednesday, August 9. The 1988 Carl B. Allendoerfer, Lester R. Ford, and 10, at 6:30 p.m. The banquet will be held in the George P6lya Awards will be presented. Certificates of Conference Room in the Chamber of Commerce Center Meritorious Service will also be presented. Some by- in the Union Station Complex.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 839 Meetings

Tickets are $8 each; the price includes gratuity. The Gloria Heath, include magic squares, polyhedrons, stars, menu includes boneless breast of chicken with fresh and solids. This exhibit can be viewed Monday through tomato and basil, creamy baked pasta, summer green Friday, August 8-12, from 9:00a.m. to 5:00p.m. salad, crusty Italian bread, fresh fruit compo~e, brownies, There will be a display of rare mathematical books coffee and punch. Those who did not purchase tickets in the John Hay Library at Brown University Monday through preregistration should check at the Tickets/Tours through Friday, August 8-12, 9:00a.m. to 5:00p.m. desk to see if any are still available before the close of registration on Monday, August 8. If tickets were Commercial Exhibits purchased through preregistration and must be cancelled, a 50% refund of the amount paid for the ticket will be The book and educational media exhibits will be located made if notification is received in Providence by July in the Grand Ballroom in the Omni Biltmore, and will be 25. After that date, no refund can be given. open Monday through Friday, August 8-12. The hours they will be open are 1:00 p.m. to 5:00p.m. on Monday, 9:00 a.m. to 5:00 p.m. Tuesday and Wednesday, 9:00 Other Events of Interest a.m. to 1:30 p.m. on Thursday, and 9:00a.m. to noon Book Sales on Friday. All participants are encouraged to visit the exhibits during the meeting. Books published by the AMS and MAA will be sold at discounted prices somewhat below the cost for the same books purchased by mail. These discounts will be How to Obtain Hotel available only to registered participants wearing the official Accommodations badge. VIsA and MASTERCARD credit cards will meeting Reservations at these hotels cannot be made by calling be accepted for book sale purchases at the meeting. The the hotel directly until after July 15, 1988. After July 15, book sales will be open the same days and hours as the 1988, the rates below may not apply. and are located in the Grand Ballroom in the exhibits In all cases "single" refers to one person in one bed; Omni Biltmore. "double" refers to two persons in one bed; "twin" refers to two persons in two twin beds; and "twin double" AMS Members' Information Booth refers to two persons in two double beds. A rollaway cot for an extra person can be added to a room; however, Please visit the AMS Membership booth in the Grand not all hotels are able to do so and for those that do, Ballroom of the Omni Biltmore exhibit area during the number of cots available is limited and given on a the Centennial Celebration. Complimentary coffee and first-come, first-served basis. tea will be served for Centennial participants. Carol­ Participants should be aware that it is general hotel Blackwood, the Head of the Membership & Sales Ann practice in most cities to ·hold a nonguaranteed reser­ Department of the Society, will be at the membership vation until 6:00 p.m. only. When one guarantees a to meet members personally and distribute a booth reservation by paying a deposit or submitting a credit special gift. Bring Mrs. Blackwood your comments and card number as guarantee in advance, however, the hotel compliments about member services. usually will honor this reservation up until checkout time the following day. If the individual holding the Special Exhibits reservation has not checked in by that time, the room is then released for sale, and the hotel retains the deposit An exhibit of selected materials from the archives of the or applies one night's room charge to the credit card Society, featuring memorabilia from the Semicentennial number submitted. Celebration held at Columbia University in 1938 will be If you hold a guaranteed reservation at a hotel, but on display in the lobby of the John D. Rockef~ller, Jr. are informed upon arrival that there is no room for you, Library at Brown University on Sunday, August 7, noon there are certain things you can request the hotel do. to 5:00p.m.; Monday through Thursday, August 8-11, First, they should provide for a room at another hotel in 9:00a.m. to 9:00p.m.; Friday, August 12, 9:00a.m. to town for that evening, at no charge. (You have already 5:00p.m.; and Saturday, August 13, 10:00 a.m. to 5:00 paid for the first night when you made your deposit.) p.m. They should pay for taxi fares to the other hotel that An exhibit of selected mathematical drawings by evening, and back to the meetings the following morning. Royal Vale Heath (author of Mathemagic, 1932) will be They should also pay for one telephone toll call so that mounted in the Bell Gallery in the List Art Building you can let people know you are not at the hotel you at Brown University. The drawings, which make up expected. They should make every effort to find a room the bulk of the collection donated to Brown by Miss

840 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

for you in th~ir hotel the following day, and if successful, How to Obtain Residence Hall pay your taxi fares to and from the second hotel so that Accommodations you can pick up your baggage and bring it to the first Centennial participants may occupy residence hall rooms hotel. Not all hotels in all cities follow this practice, so your request for these services may bring mixed results, at Brown University during the period August 5 to August 13 only. All must check out by noon on August 13. or none at all. Please make all changes to or cancellations of hotel All rooms on campus are offered through a room/board reservations with the Mathematics Meetings Housing Bu­ package only. Only a very limited number of rooms on reau in Providence before August 4, 1988. The telephone campus will be available for those participants who do number in Providence is 401-272-9500 (extension 290). not preregister but plan on attending the Centennial and After that date, changes should be made directly with registering on site. These rooms will be assigned at the the hotel. Cancellations must be made directly with the Brown check-in desk where participants will be given a hotel 48 hours prior to date of arrival in order to receive housing slip which should be taken to the Housing Section refunds of deposits. of the Centennial Registration Desk and payment made The hotels listed below accept American Express, at the participant's earliest convenience. These payments MasterCard, Visa, Carte Blanche, and Diners' Club can be made only during the hours the registration desk credit cards, personal checks with identification, and is open and can be made via personal checks, travelers' travelers' checks as payment for room charges. Rates are checks, or credit cards (Visa and MasterCard only). subject to a 10% state and room tax. Rates quoted are Participants requesting housing on the Brown Uni­ versity campus will be assigned to one of six residence firm. halls: Wayland Hall, Harkness Hall, Olney Hall, Diman Omni Biltmore (Headquarters) Hall, Hegemann Hall, or the Graduate Center. (Please Kennedy Plaza refer to the section below titled Room and Board Rates.) Providence, RI 02903 Families with children will be allowed to stay in Telephone: 401-421-0700 the dormitories; however, there is a maximum of one Single occupancy $ 70 child per room. Sleeping bags for children staying with Double occupancy $ 80 both parents will be permitted free of charge as long Triple occupancy $ 80 as the room is occupied to full bed capacity. Children Triple occupancy wfcot* $ 95 occupying a bed will be charged the full room and board Quadruple occupancy $ 80 rate; however, children under seven years of age will be Quadruple occupancy wfcot* $ 95 charged half-price for the meal portion of the package. Suites $150 (See section on Hotels above for alternate housing for * Number of cots is limited. families.) Full service hotel. Restaurants and lounge. Parking is Residence halls at Brown University have either three $6 per day for guests. Children 18 years and younger or four floors and no elevators or ramps. They are not are free in same room as parents. All major credit cards accessible to the handicapped. All single rooms have a accepted. single bed, bureau, one closet, one chair, one desk, linen, Holiday Inn a blanket and an overhead light. In addition to bed linen 21 Atwells Avenue pillow, and a blanket,. participants will receive two towels'' Providence, RI 02903 soap, and a disposal glass (exchangeable upon request at Telephone: 401-831-3900 the check-in desk). One towel can be exchanged daily at Single occupancy $ 65 a designated area. Participants are advised to bring their Double occupancy $ 65 own washcloths and hangers. Rooms will be prepared Triple occupancy $ 65 for occupancy in advance; however, there is no daily Triple occupancy wfcot* $71 maid service. There are no cots or cribs available. (See Quadruple occupancy $ 65 the section on Crib Rental.) In Wayland Hall, Harkness Quadruple occupancy wfcot* $ 71 Hall, Olney Hall, Diman Hall, and the Graduate Center Suites $135 there is one bathroom for each gender on each floor; all *Number of cots is limited. showers are open. Hegemann Hall is comprised of suites Full service hotel. Indoor swimming pool, jacuzzi, containing two or three bedrooms, a living room, and a free parking, restaurant, and lounge. Children 18 years bathroom with a private shower. Each hall is equipped and younger are free in same room as parents. All major with washers (50 cents) and dryers (50 cents). Vending credit cards accepted. machines are available in "Machine City," located in Keen~y Quad, for candy, popcorn, soda, juice, coffee, and cigarettes.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 841 Meetings

1 OMNI BILTMORE 6 R. R. STATION 2 HOUDAYINN 7 BUS STATION 3 PROVIDENCE PERFORMING ARTS CENTER (PPAC) 8 SI'ATE HOUSE 4 CHAMBER OF COMMERCE CENTER 9 MARRIOTI INN S AMERICAN MATHEMATICAL SOCIETY

842 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

Pets are not allowed in the residence halls. present their receipt which will enable them to receive There will be designated nonsmoking sleeping areas. two keys (one for the outside door and one for the The rooms are equipped with smoke alarms; the hallways room) and meal tickets at the Brown check-in desk. are equipped with smoke alarms and heat detectors. Those participants being assigned a room directly by Please note that none of the residence halls are air­ the Brown check-in desk will be required to fill out conditioned. a housing form, thus enabling them to receive keys. Spouses desiring a room key must follow this procedure Check-In Location and Times also. Please note that, although there is no deposit There will be one main check-in desk for Brown residence required for keys, a penalty of $2 will be imposed for each balls located at a central location (to be determined at a key lost or not returned. It is the responsibility of the later date) which will be staffed from 8:00 a.m. to 11:00 Mathematics Meetings Housing Bureau to collect this p.m. penalty. Therefore, it is requested that proper caution be Telephone numbers to call for assistance will be exercised to avoid this charge. At checkout, all keys must posted for those participants arriving after 11:00 p.m. be returned to the Brown check-in desk. Should the clerk when the check-in desk is closed. Parking stickers for not be present, please ensure that your name is left at nearby university lots may be purchased at the Brown the Brown check-in desk with the key. check-in desk at an estimated daily rate of $1.50. At the time of check-in, participants assigned rooms through the Mathematics Meetings Housing Bureau will

Brown University Room/Board Rates (per person)

Child 7 years Child under Children • under and older in 7 years in Adults 7 years in bed sleeping bag sleeping bag

8/5 $22.20 single $22.20 single No charge No charge $1 7. 70 double $17.70 double

8/6 $22.20 single $22.20 single No charge No charge $17.70 double $17.70 double

8/7 $27.00 single $24.60 single $4.80 single $2.40 single $22.50 double $20.10 double $4.80 double $2.40 double

8/8 $27.00 single $24.60 single $4.80 single $2.40 single $22.50 double $20.10 double $4.80 double $2.40 double

8/9 $27.00'single $24.60 single $4.80 single $2.40 single $22.50 double $20.10 double $4.80 double $2.40 double

8/10 $27.00 single $24.60 single $4.80 single $2.40 single $22.50 double $20.10 double $4.80 double $2.40 double

8/11 $27.00 single $24.60 single $4.80 single $2.40 single $22.50 double $20.10 double $4.80 double $2.40 double

8/12 $22.20 single $22.20 single No charge No charge $1 7. 70 double $17.70 double

• There is no charge for infants in arms.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 843 Meetings

1 HEGEMAN HALL 8 HALL B, GRADUATE CENTER 2 WAYLAND HALL 9 COMPUTER CENTER 1.25 inch= .Hi mi.le 3 OLNEY HALL 10 JOHN HAY LIBRARY SCALE OF MILES 4 DIMAN HALL 11 LIST ART CENTER 5 HARKNESS HALL 12 ROCKEFELLER CENTER R!SD AUDITORIUM 6 VENDING MACHINES 13 BARUS 8t HOLLEY * 7 SHARPE REFECTORY 14 FACULTY CENTER

844 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

Room and Board Rates Changes, Cancellations and Refunds Room and board rates for residence hall accommodations If written notice of cancellation is received by the Hous­ at Brown University can be found in a chart elsewhere ing Bureau by August 4, 1988, 50% of the preregistration in this announcement. Please note that there is no room fee and the cost of the tours, 90% of the residence taX applicable to these rates. hall package, 100% of the airport transfer and poster Should a family with two children request accommo­ prices will be refunded. ( 100% refunds can be made for dations, two rooms would be required and the double student, unemployed, and emeritus participants.) Mter rate (with appropriate adjustments for children under August 4, 90% minus one night's stay will be refunded on seven years of age) applies in each case. A family of residence hall packages, and 50% of the airport transfer three would. require one double room at the double rate and poster prices. It is regretted that no refunds of the plus a second room for the third occupant at the single preregistration fee or the cost of the tours can be made rate (with appropriate adjustments for children under after August 4. seven years of age). If written notice of cancellation is received by the Please note that after August 4 no adjustment can be Housing Bureau by July 25, 50% of the cost of the tickets made to the room and board package price should the for the MAA and TIME banquets and clambake will be participant arrive later than the original date given when refunded. Mter July 25, no refunds will be possible. reservation was made, or leave earlier than the original Please make all changes to or cancellations of ho­ departure date. Since no breakfast will be served on tel reservations and residence halls with the Hous­ August 6, 7 and 13, there is no charge for breakfast ing Bureau in Providence before July 15, 1988. The included in the room rate for the previous nights. The telephone number in Providence, Rhode Island, is cost of breakfast for children under seven years of age 401-272-9500 (extension 290), the E-mail address is is half-price; there is no charge for babies in arms. Meal PONY%"[email protected]", or Telex 797192. tickets are nonrefundable. After that date, changes should be made directly with Food Services the hotel and residence halls. Most hotels will refund the $50 deposit if notice of cancellation is received by The Sharpe Refectory located on the Brown University them 48 hours prior to arrival. All applicable refunds for campus offers a variety of choices for breakfast. Breakfast residence housing will be issued by the Housing Bureau. is served cafeteria-style and will not be available on a The Housing Bureau is unable to refund amounts less cash basis. Breakfast will be served between 6:30 a.m. than $1. and 9:00 a.m. A typical breakfast menu is: Mushroom or Mozzarella Cheese Omelette Scrambled or Fried Eggs Registration at the Meeting Boiled Eggs Meeting preregistration and registration fees only par­ Grilled Ham tially cover expenses of holding meetings. All mathemati­ Pancakes or French Toast cians who wish to attend sessions are expected to register. Lyonnaise Potatoes The fees for Centennial registration at the meeting listed Grits below are 30% more than the preregistration fees. Danish Pastry Choice of Beverage Centennial Celebration Choice of Cereal Member of AMS, AWM, CMS, MAA, Servings are generous; unlimited seconds are offered TIME, SIAM $ 89 on most. Emeritus Member of AMS, MAA, As mentioned above, breakfast at Sharpe Refectory is SIAM $ 30 included in the room and board package. Unfortunately, Nonmember $132 however, it will not be possible for participants to Student/Unemployed $ 30 purchase either lunch or dinner at the Refectory on a AMS Short Course cash basis. Student/Unemployed $ 15 There are several restaurants and specialty eat­ All Other Participants $ 45 ing establishments within the immediate vicinity of Brown University. They range from Italian, French, MAA Minicourses and Japanese full course meals to pizza, homemade ice (if openings available) cream, and unusual sandwiches. Minicourses # 2, 4, 5, 7 $ 30 Minicourses# 1, 3, 6, $ 50 Modes of payment which are acceptable, provided they are payable in U.S. dollars to the order of the

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 845 Meetings

American Mathematical Society, are U.S. Postal Money who has voluntarily resigned or retired from his or her Orders, certified U.S. bank checks, U.S. bank money latest position. orders, personal checks drawn on a U.S. bank, or credit Persons who qualify for emeritus membership in card (Visa or MasterCard only). either the Society or the Association or SIAM may An income tax deduction is allowed for education register at the emeritus member rate. The emeritus status expenses, including registration fees, cost of travel, meals refers to any person who has been a member of the AMS, and lodging incurred to (i) maintain or improve skills MAA, or SIAM for twenty years or more, and is retired in one's employment or trade or business or (ii) meet on account of age from his or her latest position. express requirements of an employer or a law imposed Nonmembers who preregister or register at the meet­ as a condition to retention of employment, job status, ing and pay the nonmember fee will receive mailings or rate of compensation. This is true even for education from AMS and MAA, after the meeting is over, contain­ that leads to a degree. However, the Tax Reform Act of ing information about a special membership offer. 1986 has introduced significant changes to this area. In general, the deduction for meals is limited to 80% of the Registration Dates, cost. Unreimbursed employee educational expenses are Times, and Locations subject to a 2% of adjusted gross income floor. However, there are exceptions to these rules. Therefore, one should AMS Short Course Outside Auditorium, contact one's tax advisor to determine the applicability Rhode Island School of Design of these provisions. Saturday, August 6 8:00 a.m. to 2:30 p.m. There is no extra charge for members of the families of registered participants, except that all professional Centennial Celebration mathematicians who wish to attend sessions must register Garden Room, Omni Biltmore independently. Sunday, August 7 3:00 p.m. to 7:00 p.m. All full-time students currently working toward a Monday, August 8 degree or diploma qualify for the student registration through 7:30 a.m. to 4:00 p.m. fees, regardless of income. Wednesday, August 10 The unemployed status refers to any person currently Thursday, August 11 7:30a.m. to 1:30 p.m. unemployed, actively seeking employment, and who is Friday, August 12 7:30 a.m. to 1:00 p.m. not a student. It is not intended to include any person MAA Minicourses (until filled) Petition Table Outside Room 167, Thomas J. Watson, Sr. Center for At the request of the AMS Committee on Human Rights Information Technology, Brown University of Mathematicians, a table will be made available in the Waterman Street (comer of Waterman and Brook) meeting registration area at which petitions on behalf of named individual mathematicians suffering from human Saturday, August 6 7:00 a.m. to 3:00 p.m. rights violations may be displayed and signed by meeting Sunday, August 7 7:00 a.m. to 3:00 p.m. participants acting in their individual capacities. Signs of moderate size may be displayed at the table, but must not represent that the case of the individual in Registration Desk Services question is backed by the Committee on Human Rights Assistance, Comments, and Complaints unless it has, in fact, so voted. Volunteers may be present A log for registering at the table to provide information on individual cases, participants' comments or com­ but notice must be sent at least seven (7) days in advance plaints about the meeting is kept at the Transparencies of the meeting to the Director of Meetings in Providence section of the registration desk. All participants are (telephone 401-272-9500). Since space is limited, it may encouraged to use this method of helping to improve also be necessary to limit the number of volunteers present future meetings. Comments on all phases of the meeting at the table at any one time. The Committee on Human are welcome. If a written reply is desired, participants Rights may delegate a person to be present at the table at should furnish their name and address. any or all times, taking precedence over other volunteers. Participants with problems of an immediate nature Any material which is not a petition (e.g., advertisements, requiring action at the meeting should see the Director resumes) will be removed by the staff. When registration of Meetings, who will try to assist them. closes, any material on the table will be discarded, so individuals placing petitions on the table should be sure to Audio-Visual Assistance remove them prior to the close of registration. A member of the AMS staff will be available to advise or consult with speakers on audio-visual usage.

846 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

Rooms where contributed paper sessions will be held Postmaster has agreed to continue this special offer for 60 are equipped with an overhead projector and screen. days after the Centennial. Interested parties should send Blackboards will not be available. their self-addressed material with proper postage to be cancelled to the Providence Post Office, 24 Corliss Street, Baggage and Coat Check Providence, Rhode Island 02904, Attention: Manager of Baggage and coats may be left in the Centennial reg­ Technical Sales and Service. istration area in Garden Room of the Omni Biltmore onlY during the hours that registration is open. The staff Personal and Telephone Messages cannot, however, take responsibility for lost or stolen Participants wishing to exchange messages during the articles. meeting should use the mailbox mentioned above. Mes­ sage pads and pencils are provided. It is regretted that Check Cashing such messages left in the box cannot be forwarded to The Centennial cashier will cash personal or travelers' participants after the meeting is over. checks up to $50, upon presentation of the official A telephone message center is located in the regis­ meeting registration badge, provided there is enough tration area to receive incoming calls for participants. cash on hand. Canadian checks must be marked for The center is open from August 7 through 12, during the payment in U.S. funds. It is advisable that participants hours that the Centennial registration desk is open only. bring travelers' checks with them. When funds are low Messages will be taken and the name of any individual the meetings cashier will not be able to cash checks and for whom a message has been received will be posted travelers' checks can be easily cashed at local banks, until the message has been picked up at the message restaurants, or hotels. center. Once the registration desk has closed for the day contacting participants other Daily Newsletter there is no mechanism for than calling them directly at their hotel or residence A newsletter containing program changes and informa­ hall. The telephone number of the message center is tion of general interest to participants will be available 401-331-9358. each morning, August 8- 12 at the registration desk. Participants should plan to pick up a copy every day. Information Table Local Information The information table at Joint Meetings of the AMS and MAA This section of the desk will provide information on is set up in the registration area for the dissemination of of a nonmathematical nature of possible interest local transportation, attractions, and events of interest. information to the members. The administration of the information table Lost and Found is in the hands of the AMS-MAA Joint Meetings Committee, as are all arrangements for such joint meetings. The following See the Centennial cashier. rules and procedures apply. Mail 1. Announcements submitted by participants should ordinarily be limited to a single sheet no more than S!"x 14". attending the meetings All mail and telegrams for persons 2. A copy of any announcement proposed for the table should be addressed as follows: Name of Participant, cfo is to be sent to the Director of Meetings, American Math­ Centennial Celebration, American Mathematical Society, ematical Society, Post Office Box 6248, Providence, Rhode P. 0. Box 6887, Providence, Rhode Island 02940. Mail Island 02940 to arrive at least one week before the first day and telegrams so addressed may be picked up at the of the scientific sessions. mailbox in the registration area during the hours the 3. The judgment on the suitability of an announcement registration desk is open. U.S. mail not picked up will be for display rests with the Joint Meetings Committee. It will forwarded after the meeting to the mailing address given make its judgments on a case by case basis to establish on the participant's registration record. precedents. 4. Announcements of events competing in time or place Special Postal Cancellation with the scheduled scientific program will not be accepted. The U.S. Postal Service has agreed to set up an official 5. Copies of an accepted announcement for the table are substation in the Centennial registration area Monday to be provided by the proponent. Announcements are not to be distributed in any other way at the meeting (for example, through Friday, August 8 to 12, from 11:00 a.m. to not by posting or personal distribution of handbills). 11, 3:00 p.m., with the exception of Thursday, August 6. It may be necessary to limit the number of events or when it will close at 1:30 p.m. Stamps may be purchased the quantity of announcements distributed at a meeting. and items mailed. A special cancellation will be used 7. At the close of registration, the table will be swept at this substation, offering participants an opportunity clean. A proponent who wishes the return of extra copies to obtain a pictorial cancellation for philatelic purposes should remove them. which will identify the AMS Centennial. The Providence

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 847 Meetings

Development, Transparencies 290 ' or the RI Department of Economic. Speakers wishing to prepare transparencies in advance of 401-277-2601, for more information. their talk will find the necessary materials and copying machines at this .section of the registration .desk. A Child Care member of the staff will assist and advise speakers on the best procedures and methods for preparation of their There are several registered day care facilities in Prov­ material. There is a modest charge for these materials. idence. Very few, however, have openings .during the week of the Centennial, and many also require that the Visual Index child be enrolled for a full week. The following have in­ An alphabetical list of registered participants, includ­ dicated that they are willing to accommodate Centennial ing local addresses and arrival and departure dates, is participants: maintained in the registration area. Carter Day Nursery, 239 Public Street, Providence, Rl, 401-751-9752. Contact Sister Mercain Hassett. Ages 3-5 years, certified kindergarten. Rates: $50fweek, no Miscellaneous Information daily or hourly rates available. Please call well m advance Athletic Facilities to make arrangements. · Brown Fox Point Day Care & Family Center, 150 The facilities at the YMCA (available to current Y Hope Street, Providence, Rl, 401-521-5460. Con~ct at 160 members; please check at membership window) Pam McGinn. Ages 3-5 years. Part-time five mornmgs an indoor pool; indoor track; Broad Street include or three afternoons per week, or full week. Rates: $35- handball and racquetball courts; universal basketball, $45/ part-time; full-time $45-$75. Please call two weeks weights; sauna; aerobics classes. in advance to make arrangements. public use are available at Nathan Tennis courts for Mt. Hope Day Care Center, Inc., 421 Hope Street, School on Sessions Street (off Elm- Bishop Junior High Providence, Rl, 401-521-7252. Contact Elizabeth Adam. grove), and at Hope High School on Hop~ Str~et. . Ages 3-6 years. Rates: $40 per week for half days, $55 athletic Any participant can use the Brown Uruverstty per week for full days. Please make reservations by the facilities at a cost of $2.75 per person per day or $7 end of June. per person per week. This includes the basketball.c~urt, Child Care Center Inc., B45 Blackstone Boulevard, (indoor courts are an additiOnal outdoor tennis courts Providence, 401-272-3959. Ages: 2 months to 6 years. charge), weight room, jogging track, squash, swimming YMCA Parent/Child Center, 438 Hope Street, Prov­ pool, sauna, and sun deck. idence 401-521-0155. Ages: 3 to 12 years. JCC of RI Preschool, 401 Elmgrove Avenue, Provi­ Book Stores dence, 401-861-8800. Ages: 3 months to 6 years. Providence Central YMCA, 160 Broad Street, Prov­ Street is The Brown University Bookstore at 244 Thayer idence, 401-456-0100. Ages: 1-1/2 to 6 years, approxi­ from 9:00 a.m. to 6:00 p.m. . open Monday-Saturday mately $60 per week Bookstore at 252 Thayer Street 1s The College Hill Federal Hill House Day Care Center, 9 Courtland from 9:00 a.m. to 11:00 p.m., open Monday-Thursday Street, Providence, 401-421-4722. Ages: 3 to 12, sliding on Friday & Saturday from 9:00 a.m. to midnight, and scale fees. on Sunday from 10:00 a.m. to 11:00 p.m. Smith Hill Day Care, 25 Danforth Street, Providence, 401-831-1720. Ages 2 to 6 years, approximately $2 per Camping hour. . The following commercial babysitting services are There are thirteen state or municipally owned camp­ available: variety of facilities. Some a~ept grounds, offering a We Sit Better of Rl, 1005 Fleet National Bank do not. There are many more pnvate reservations; others Building, Providence, RI. 401-421-1213. Referral service campgrounds, most of which encourage reservations. for babysitters. Members of the National Association of Because the Centennial takes place during a Rhode Is­ Referral Services. land holiday weekend (V-J Day is Monday), space may Both the Biltmore and the Holiday Inn have babysit- be at a premium. The closest campground is a forty­ referral services. . . minute drive from Providence. Overnight camping in ting There will be a list of local babysttters available at the on public hi~":"ays, in non-campi~g rest/picnic areas, Local Information Section of the Centennial Registration state/municipal parks, statefmuructpal beaches and their Desk during the meeting. Please contact the Mathe~at­ parking lots is prohibited. In addition, a Parent-Child Lounge will .be located ics Meetings Housing Bureau at 401-272-9500, extension adjacent to the Centennial registration area in the Grand

848 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

Ballroom of the Holiday Inn. It will be furnished with John Carter Brown Library (early Americana collec­ casual furniture, a crib, a television set, and VCR. tion), Monday-Friday, 9:00 a.m.-5:00p.m. Appropriate videotapes and cartoons will be available Providence Public Library, 150 Empire Street, Mon­ at the Telephone Message Center. Any child using this day, Tuesday, and Thursday, 9:30 a.m.- 9:00 p.m.; Fri­ lounge MUST be accompanied by a parent (not simply day and Saturday, 9:30 a.m.- 5:00 p.m. an adult) who must be responsible for supervision of Providence Athenaeum (rare books), 251 Benefit the child. This lounge will be unattended and parents Street, Monday-Friday, 8:30 a.m.- 5:00 p.m. assume all responsibility for their children. This lounge will be open only during the hours of registration and all Local Information persons must leave the lounge at the close of registration each day. In August, Providence is on Eastern Daylight Saving Time. Participants should be aware that Monday, August 8, Crib Rental (Victory over Japan Day) is celebrated as a legal holiday A very limited number of portable cribs are available for in Rhode Island. There is no delivery of mail by the U.S. rent from Rent It All, 738 N. Broadway, East Providence, Post Office, and all banks and state offices are closed, as RI 02914 (telephone 401-434-8479) for $4 per day or $20 are many businesses. per week (five days and over). There is a $18 charge each All sessions during the period August 8- 12 will be way for pickup and delivery. The cribs have mattresses, held in downtown facilities. Sessions scheduled to be held but linens are not provided. Participants renting these on Saturday and Sunday, August 6-7 will take place on cribs for use in the university residence halls should College Hill (also known as the East Side), where Brown notify the Brown check-in desk of the expected delivery. University is located. Both of these areas are very walk­ Payment may be made by VISA or MasterCard credit able, but the difficulty involved is in traveling between cards. them. College Hill is not a misnomer, and, although pic­ Participants not staying on campus will find portable turesque, it is fairly steep. Fortunately several buses run cribs available on a first-come, first-served basis from the regularly between Kennedy Plaza downtown and Thayer hotels. Street, the "main street" of College Hill. Taxis in Provi­ dence are not usually in evidence, except around hotels and bus and train stations, but it is easy to telephone for Handicapped them. A trip from the Holiday Inn to the Brown campus The Brown University housing facilities are not accessible would cost between $3 and $4. A small green trolley, to the handicapped; however, most (not all) classrooms called the Downtown Free Loop, circles around central used for the MAA Minicourses are. Participants with spe­ Providence approximately every twelve minutes, and will cial requests or questions regarding handicapped access stop just about anywhere on its route if the driver is at the university should contact Pat Henry, Conferences, given enough notice. Principal stops include the State 401-863-3500. House, First Baptist Church, Kennedy Plaza, and Davol All hotels are accessible to the handicapped. Partici­ Square. If this trolley is still in operation at the time pants with special requirements should keep this in mind of the Centennial, it is hoped that arrangements can be when requesting housing for the Centennial. made for this trolley to drop participants off at the AMS Participants with special' questions regarding handi­ headquarters office. In addition, the Society has arranged capped access in the city should contact The Governor's for a free shuttle service to and from Brown University Commission on the Handicapped, 401-277-3731. that will run daily during the meeting. Although this free shuttle is primarily intended to provide transportation for participants staying in the Brown residence halls, Libraries participants staying in the downtown hotels may also The following libraries are within walking distance of utilize this shuttle to visit the Brown University area, the hotels and residence halls: including the special mathematical exhibits described in Brown University Sciences Library, Comer of Thayer the section on Special Events. & Waterman Streets, Monday-Thursday 9:00a.m.-9:00 Other than the steep climb, the walk through Prov­ p.m., Friday 9:00 a.m.- 5:00 p.m., Saturday 10:00 a.m.- idence to Brown University is very short, and many of 5:00p.m. Providence's more interesting sights happen to be along Rockefeller Library, Brown University campus (same the way. The Art Deco Fleet National Bank, which faces hours as Sciences Library) Kennedy Plaza, is one such sight. It was once New John Hay Library (special collections), Brown Uni­ England's most prominent skyscraper, and it is still an versity campus, Monday-Friday, 9:00 a.m.- 5:00 p.m.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 849 Meetings integral part of Providence's skyline. The myth persists itself was built in 1775. The very fine Museum of Art that it is the original "Daily Planet" building of Super­ of the Rhode Island School of Design is also located man fame. On the other side of the bank (which can be on Benefit Street. It contains over 60,000 holdings in walked through) is the Arcade, America's firs~ shopping sculpture, painting, graphics, textiles, and the decorative mall, built in 1828. This exquisite, glass-roofed, Greek arts. Revival building contains three floors of specialty shops, Theatregoers will enjoy the summer comedies and food shops, and small restaurants. musicals presented by the Tony Award-winning Trinity Providence possesses an extraordinary number of Square Repertory Company in downtown Providence. eighteenth and nineteenth century buildings, both public If one is in search of greenery, Providence's parks and private, and College Hill is a pleasant place to enjoy include tiny Prospect Terrace on the East Side, which a remarkable collection of them. Benefit Street attracts features a statue of Rhode Island's founding father, Roger national attention as the longest street of original colonial Williams, and an excellent, breezy view of Providence, architecture in the country. Known as Providence's mile including the beautiful dome of the State House. The of history, it includes colonial, Federal, and Victorian visitor center of the Roger Williams National Memorial residences, meticulously restored. Although most are and a four-and-one-half acre park is located at Smith private homes, there are two house museums on Benefit Street and North Main Street. This is the site of the Street. One, the John Brown House, was referred to by founding of Providence by Roger Williams in 1636. John Quincy Adams as "the most magnificent and elegant Families may enjoy Roger Williams Park and Zoo, not mansion that I have ever seen on this continent." The far away on Elmwood Avenue. It is an elegant urban park more modest Stephen Hopkins House is a small, red, of approximately 450 acres, with a lake, paddleboats, and colonial once owned by Hopkins, a former Governor a merry-go-round. The zoo is newly renovated. There is of Rhode Island and a signer of the Declaration of a small Museum of Natural History on the grounds. Independence. Another building of special interest is Clusters of restaurants, most in close proximity to the Athenaeum, one of the oldest library companies in shopping areas, are located along Thayer Street, Wick­ America; here Edgar Allan Poe courted Sarah Helen enden Street, South and North Main Streets, in down­ Whitman, a resident of Benefit Street. The First Baptist town Providence, and on Federal Hill. There is a great Church, with its back to Benefit Street, is the home of diversity of cuisines and cultures to be found in Provi­ the first Baptist ;congregation in America, founded by dence's restaurants. One can find particularly good Italian Roger Williams and his followers in 1636. The church food in the Federal Hill (Atwells Avenue) section of the

Downtown Providence

850 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings city. There are several French-inspired restaurants; Mex­ There is a variety of parking garages and lots in ican restaurants exist, and the city's eastern offerings the downtown area. Several have "early bird" rates of include Indian, Chinese, and Japanese cuisines. Usually about $4.50 if you are in the lot by 9:00 a.m. and one can find lunch for under $5; dinner prices range out by 6:00 p.m. Otherwise, you should expect to pay from inexpensive to moderately expensive. this much for a two-and-one-half hour period. Rates are An advantage of the state's small size is that there fairly standard, with most charging a maximum of $7.50 is no place in Rhode Island more than than an hour's per day. The Parkade (associated with the Biltmore) is drive from Providence. With its more than 400 miles the most expensive with a maximum of $12.50 per day, of coastline, Rhode Island offers dozens of high quality but it is the only one to allow overnight parking (free public beaches on both sides of Narragansett Bay and to overnight guests who have their ticket validated at along the Atlantic as far south as Watch Hill. Lists of the hotel front desk). There is a lot near the bus station beaches and information on boating, fishing, and other that is more economical and allows overnight parking activities will be available at the Local Information for $7.50, but it is unprotected. The following lots fall Section of the Centennial Registration Desk. within the described rate structure: One of the oldest and most famous seaside resorts Meyers Park, across from the Performing Arts Center in the country, Newport is barely an hour away from Outlet Parking Garage, on Pine Street behind the Providence. The beautiful and opulent city was the Performing Arts Center location of the America's Cup Races until recent years. Majestic Parking Garage, on Fountain Street, around During the summer the Tall Ships can frequently be seen the comer from the Holiday Inn (also near the Provi­ in the harbor. Newport is also known for its sumptuous dence Public Library) summer "cottages," built around the tum of the century Parkade, behind the Omni Biltmore by wealthy industrialists and social leaders, among them Brown University has on-campus parking spaces the Vanderbilts and the Astors. available in overnight lots at an estimated cost of $1.50 per day. Parking permits will be sold at the check-in desk for the Brown residence halls. Medical Services Participants requiring medical assistance should refer to Smoking the following: Rhode Island Hospital, large multi-service teaching Please note that smoking is not allowed in any of the facility, open 24 hours, 593 Eddy Street, Emergency session rooms in the PPAC, Biltmore, Holiday Inn, or telephone: 277-4000 the Rhode Island School of Design. Women & Infants, offering obstetric and neonatal There are designated nonsmoking sleeping rooms in care, at Rhode Island Hospital the residence halls as well as hotels. Miriam Hospital, all services except psychiatric and obstetric, 164 Summit Avenue Special Events Veterans Administration Hospital, Davis Park, Prov­ idence AMS Open House Brown's Health Services, 13 Brown Street, open The Society invites all Centennial participants to visit Monday-Friday, 8:00 a.m.- 3:30 p.m. A physician is its headquarters office at 201 Charles Street. Guided on duty during these hours. The University police and tours will be given daily from 1:00 p.m. to 3:00 p.m. security people are Emergency Mec;lical Technicians and Tuesday through Thursday, August 9-11. Reservations may be called when health services are closed. They will must be made by signing up at the Tickets/Tours Section transport to a nearby hospital if necessary. of the Centennial Registration Desk. There is no charge for these tours which will be conducted by the various Parking department and division heads of the AMS staff. As previously mentioned, if the Downtown Free Loop Street parking is allowed in metered areas for 25 cents trolley is still operating in August, plans are to have it per hour. Meter time ranges from 30 minutes to 10 drop off participants at the AMS office. hours depending on location. Most meters have a two To get to the Society's headquarters from 195 North, hour limit. There is no overnight street parking. Meter take exit 23, State Offices. Tum left onto Orms Street at maids are frequent and plentiful. the end of the exit. Go left at the light at the bottom There is no charge for parking in the hotel's facility of the hill onto Charles Street. The Society is located for overnight guests staying at the Holiday Inn. There is in a one-story dark brown building about 200 yards on a $6 per day charge for guests at the Omni B.iltmore and the right. From 195 South, take exit 23, Charles Street. $12 per day for those not staying.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 851 Meetings

Bear right because traffic is one way at this point. Bear Clambake direction. to the left and around V.F. Liquors to reverse On Thursday afternoon, August II, a tr~ditiona~ New left around the Continue about 300 yards and bear England clambake will take place at Fran~1s Farm~~ Re­ Sunoco station. The Society's entrance is directly across hoboth Massachusetts. Transportation Will be provided. from the Sunoco station on Charles Street. The cl~mbake was originated by native Indians who covered the Opening Reception cooked their clams on large, hot stones and food with seaweed and hay. Francis Farm ha~ replaced All Centennial participants are invited to attend the the stones with heated iron ingots, since, unl~ke rock~, Opening Reception on Monday, August 8, at 7:00p.m. they can be reheated. The same family has contmued t?is the Rhode Island State House. Free transportation at tradition for 115 years. The "all-you-can-eat" menu m­ be provided. This predinner function features light will cludes chowder and crackers, clamcakes, clams hot from hors d'oeuvres and beverages. Music will be provided the bake with drawn butter, fresh fish, white and sweet Camera, a chamber music ensemble led by by Musica sausage, broWD: ~read, hard the Society's staff potatoes, onions, sweet corn, Ernest Nordman, recently retired from may after 25 years of service. butter, pickles, watermelon, and coffee. P~I~lp~nts substitute chicken for fish if advance notice IS gtven. State House was built in 1900 of white Georgia The Musical entertainment will be provided by the Old marble. It has the third largest unsupported marble dome Fiddlers' Club of Rhode Island, whose repertoire includes in the world, surpassed only by the Taj Mahal and St. all the old familiar songs. Peter's Cathedral in Rome. Among the many historical The farm has facilities and equipment fo: volleyb~, artifacts housed in this building is a full length portrait basketball, softball, and horseshoes. Entertam~e~t will of George Washington by Gilbert Stuart, a native Rhode Islander. be provided for the children, including .face-pamtmg. Tickets may be purchased at the Ttckets/T~urs s.ec­ Semicentennial in 1938, all participants gath­ At the tion of the registration desk until the close of regtstratton ered on the steps of the Low Memorial Library at on Tuesday, August 9. Tickets are $23 fo~ adults and Columbia University for a group photograph. In an at­ $15 for children ages 6 through 12. There ts. no ~h~ge tempt to duplicate this feat, all participants are requested for children under 6 years of age. However, tf bn~g~ng to gather on the front steps of the State House at 6:30 a child under 6 years of age, please inform the ttcket p.m., just prior to the Opening Reception. Copies of vendor when purchasing the ticket. Please note that a this photograph will be on sale later in the week at the 50% refund can be made on clambake tickets purchased Souvenirs Section of the Centennial Registration Desk. through preregistration until July 25. After July 25, no refunds are possible. RISD Museum The Rhode Island Shool of Design Museum of Art at 224 Benefit Street is one of the nation's finest sma~ler art museums. In recognition of the Society's Centennt~l, ~he directors of the museum are extending free admtss~on to Centennial participants wearing the official meetmg badge during the period August 8 - 12. Brochures abo~t the museum and containing its summer hours wtll be available at the Local Information section of the registration desk. The RISD museum's collections include art treasures from all over the world, from ancient to avant-gard. The museum is especially renowned for its French masterpieces and contemporary Amencan· pam· t' mg, and lists the gigantic Buddha from I Oth Century Japan as a "must see". The Pendleton House wing, patte':'Iled after an early 19th Century Providence house, c~ntams o~e of the foremost museum collections of Amencan furniture and decorative arts. Happy Hour Rhode Island State House On Tuesday and Wednesday, August 9-10, fro~ 5:30 p.m. to 7:30p.m., a no-host cash bar will operate m the

852 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

Cafe on the Terrace of the Omni Biltmore Hotel. Free preservation, turning it into one of New England's most dry snacks will be provided. Participants are encouraged beautiful residential areas. On this tour you will learn of to use this occasion to spend some time with old and Providence's rich history as a colonial capital, a bustling new friends. China Trade seaport, and an early center of industrial development as you view 18th century buildings on the city's waterfront. Included in this tour will be a visit to Souvenirs one of the private homes in this area that has undergone Photographs restoration to its original beauty. This tour is offered A copy of the group portrait of Semicentennial partici­ twice daily on Tuesday, August 9, and Thursday, August pants will be on display in the registration area. Orders ll, departing from the front of the Omni Biltmore at for reproductions of this photograph will be taken at 9:00 a.m. and ll :00 a.m. and returning at ll :00 a.m. and the Souvenirs Section of the Centennial Registration 1:00 p.m. Tickets are $12 for adults, $10 for children Desk. The price for these reproductions has not yet been age 12 or under. There is no charge for infants not determined, but should be modest. occupying a seat on the bus. Tickets may be purchased Copies of the group photograph taken on Monday, at the Tickets/Tours section of the registration desk as August 8, at the Opening Reception will also be on sale available. Please note that a 50% refund can be made at the Souvenirs Section of the Centennial Registration on tour tickets purchased through preregistration until Desk. August 4. After August 4, no refunds are possible. Commemorative Poster Living History Tour of Newport The Society has prepared a poster commemorating its A costumed character from Rhode Island history will Centennial featuring a photographic reproduction of serve as the guide for this bus tour of the colonial the sculpture Torus with Cross-Cap and Vector Field and Victorian city of Newport. Settled in the 18th by Helaman Rolfe Pratt Ferguson of Brigham Young century, Newport is home to the first synagogue and first University. The sculpture is a gift from the Mathematical Quaker meeting house in America. Newport is perhaps Association of America to the Society on the occasion best known as the home of the "summer cottages" of the of its Centennial. This striking poster is printed on country's wealthiest families, such as the Vanderbilts and museum quality glossy paper. The sculpture is white the Rockefellers. The highlight of the afternoon is a visit and photographed on a rich, blue background. These to Beechwood, the summer home of the Astor family. posters are on sale at the Souvenirs Section of the Centennial Registration Desk for $10 each. Those who purchased the poster through preregistration may pick up their copy at the Souvenirs Section. Please note that a I 00% refund can be made on posters purchased through preregistration until August 4. After August 4, only 50% can be refunded.

Tours Tour of Historic Providence Architectural historians regard the buildings of Provi­ dence's "East Side" as one of America's best examples of 18th century America. Homes from the colonial period to the contemporary cover every period of architectural importance in American Life. This area of Providence is not a recreated showplace; these homes have been lived in continuously since before the Revolutionary War. Benefit Street was created "for the common benefit of all" in the 1760s to relieve congestion on Providence's Main Street. Benefit Street followed a path along ex­ isting gardens, orchards, and family burial plots, and now has brick paved sidewalks lined with charmingly restored colonial, federal, Greek revival, and Victorian homes. Its rescue from blight was a triumph for historic John Brown House

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 853 Meetings

Here, instead of roped-off displays and security guards, by Meetings, Incentives, Conventions of America, Inc. one finds a lively recreation of life in the 1890s. Mrs. (MICA); their advertisement can be found elsewhere Astor's guests and servants are portrayed in Newport's in this meeting announcement. Although any travel only living history tour. This tour is offered on Monday, agent can obtain Supersaver or other such published August 8, Wednesday, August 10, and Friday, August promotional fares, only MICA can obtain the special 12, departing from the front of the Omni Biltmore at additional 5% discount over and above these fares, and 1:00 p.m. and returning at 5:00 p.m. Tickets are $15 the 35-40% off regular coach fare. The latter, of course, for adults, $12 for children age 12 or under. There is is financially beneficial only when one does not qualify no charge for infants not occupying a seat on the bus. for one of the promotional fares. Participants should pay Tickets may be purchased at the Tickets/Tours section particular attention to the cancellation policies stated in of the registration desk as available. Please note that the ad. a 50% refund can be made on tour tickets purchased through preregistration until August 4. After August 4, Airport Transfers no refunds are possible. Theodore Francis Greene Airport lies about nine miles south of Providence and is served by most major carriers. Cabs are metered, with a typical fare to the downtown Travel area averaging $15. By Air The Society has made arrangements with the Airport For some years now, the AMS-MAA Joint Meetings Limousine to provide ground transfers to and from the Committee has engaged a travel agent for the January Providence airport, the hotels, and the residence halls and August meetings in an effort to ensure that everyone at Brown University. The cost one way is $5.75 per attending these meetings is able to obtain the best person (no charge for infants on parent's lap) or $11.50 possible airfare. This service is presently being performed round trip. Participants wishing to purchase these trans­ fers through preregistration should have completed the AIRLINE INFORMATION

SPECIAL AIRFARES 1·800·888-MICA

MICA, Inc., the official travel management firm for the AMS Centennial Celebration to be held in Providence, August 8 - 12, 1988, has arranged for special discounts aboard American Airlines and USAir. Save 5% off published promotional fares, meeting all restrictions, or 35-45% off regular roundtrip coach fares, with a 7 day advance purchase, (American Airlines even provides 5% off the non-refundable fares). Only through MICA can you receive these substantial discounts on American Airlines and USAir. It may be possible to receive an even lower airfare depending upon your individual circumstances. The lowest promotional fares require a Saturday night stay, are subject to an airline change/ cancellation penalty and must usually be purchased at least 30 days prior to departure. Make your reservations today! For reservations on all airlines, call MICA directly on their nationwide toll-free number: 1-800-888-MICA. MICA reservationists will advise you of the most convenient flights and lowest airfares available. You may pay by credit card or ask to be invoiced. Your airfare is guaranteed when your ticket is written! MICA wishes to take this opportunity to congratulate the American Mathematical Society on the occasion of its Centennial, and to extend its best wishes for continued success during its next 1 00 years. Call Today: 1·800-888-M ICA And Savel Monday - Friday, 9:00 a.m. - 6:00 p.m. EST Meetings, Incentives, Conventions of America, Inc. (MICA, Inc.) AmericanAirlinH Suite 303, 195 Farmington Avenue, Farmington, CT 06032 (203) 678-1040

854 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

appropriate section of the. Prere~stration(Hou.sing Form By Auto and included payment With their preregiStration fee. It Interstate Route 95 (195) goes through the center of is mandatory that participants purchasing these trans­ Providence. If arriving from the South (as if from the fers provide accurate airline flight information. These airport) take 195 north to exit 21 (Broadway), at the transfers will be valid from Friday, August 5, through second set of lights, take a right, and the Holiday Inn Monday, August 15. Transfers will be mailed to each par­ will be on the left side. To get to the Omni Biltmore, ticipant with the acknowledgement of preregistration. A follow the previous directions but pass the Holiday Inn tOO% refund can be made for airport transfers cancelled and take a left at the next light. Follow the road as it by August 4. After August 4 only 50% can be refunded. curves to the right, but bear left onto Fountain Street. Return transfers should be reconfirmed at the Go through two sets of lights, and take a right at the Tickets/Tours Section of the Centennial Registration third set onto Dorrance Street. The hotel will be on your Desk during the meeting. Return transfers can also be right, across from Kennedy Plaza. purchased at this desk. If arriving from the north on 195 South, take exit 21, There will be a desk in the lobby of the airport for Atwells Avenue, and go left at the light at the end of the purpose of welcoming Centennial participants and the exit. The Holiday Inn will be on your left. For the dispensing information on transportation to and from Omni Biltmore, go by the Holiday Inn and take a left the airport. A comfortable area has been set aside for at the traffic light. Follow the directions in the previous those who must wait for the airport limousine or for paragraph, bearing left onto Fountain Street. outgoing flights. Brown University is most easily accessible from 1195 Participants staying at the Holiday Inn-Providence East (which merges with 195 North and South in the Downtown should be aware that the Inn has a courtesy center of Providence). Take the Wickenden Street exit van that runs between the airport and the hotel between (#2). Cross Wickenden Street at the end of the exit but the hours of 6:00 a.m. and 11 :00 p.m. When arriving in take an immediate left, following the signs to Benefit Providence, please use the green telephone in the baggage Street. Travel up Benefit Street about three-quarter miles claim area to contact the van. Although the van runs to George Street. Take a right on to George Street, which until 11:00 p.m., the last time it can be called is 10:30 leads to the residence halls. See campus map for exact p.m. location of these halls. There are several rental car companies located at the airport or directly across the street. Expect to pay By Train $30 per day for a compact car and $35 per day for an Amtrak provides regular train service from New York intermediate size car, which includes free mileage for the City and Washington, DC to the new Providence Train first 75 miles each day. Hertz rates are $10-$15 higher. Station. This Northeast Corridor route connects with It may be less expensive to contract for a week if you others nationwide. intend to use the car for five days. Weekly rates start at about $133 for a compact car (Dollar Rent-a-Car). All By Bus accept most major credit cards. The following are at the The Bonanza Bus terminal is also located downtown and airport or across the street (businesses with an asterisk has frequent service from New York City and Logan have toll free numbers to make reservations; consult your Airport in Boston. local yellow pages): *Avis, 738-5800 Videotapes Budget Car & Truck Rental, 739 .. 8900 {also has office at Omni Biltmore) · · The Society plans to videotape the eighteen lectures in the *Dollar Rent A Car, 739-8450 symposium Mathematics into the Twenty-First Century *Hertz, 738-7500 {also has office at Omni Biltmore) and the three AMS-MAA Joint Invited Addresses as a *National Car Rental, 737-4800 {also has office at record of the Centennial Celebration. It is anticipated Omni Biltmore) that the videotapes will be available for distribution later *Thrifty, 7J9-8660 in 1988.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 855 Meetings

Weather advised for warm, humid days. Average rainfall for the month of August is 3.9 inches; however, August can F. The The normal daytime high in the city is 80 degrees bring frequent and heavy thunderstorms where several low is 61 degrees F. Average daytime normal nighttime inches of rain can fall in a very short period of time. high tem­ humidity is 82%; nighttime is 54%. Record The highest rainfall for August on record is 7.92 inches. perature is 104 degrees F., while the low is 40 degrees F. Light sweaters or jackets are recommended for cool Everett Pitcher, Chairman evenings, while natural fiber clothing such as cotton is Centennial Committee

Proposed Amendments to the Bylaws of the American Mathematical Society

for Presentation at the Business Meeting of 8 August 1988 in Providence

The Council of 23 April 1988 recommended a change Article VIII in Article VIII, Section 4 of the bylaws. The change is Dues and Privileges of Members indicated by ~ 4MH words to be deleted and bold facing words to be inserted. In applying the amended Section 4. The minimum dues of an institutional version, it is intended to define scholarly activity member shall depend on the amettBt ef tnt~liskee in terms of size of faculty, number of graduate m&tea&l ereeitee te scholarly activity of that member students, and amount of published material reviewed Hi- eeFtlliB jettFB&ls Ek:tFiftg a StJeeiie ~- The for­ in Mathematical Reviews and to calculate dues by mula for computing these dues shall be established formula as a weighted average of measures of these from time to time by the Council, subject to approval categories. The amendment is offered at the Business by the Board of Trustees. Institutions may pay larger Meeting for the approval of the membership. dues than the computed minimum.

856 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Timetable (Eastern Standard Time)

The final version of the timetable and program, including room assignments, will be distributed at the meeting.

Saturday, . /,oan /ematic• /, August 6 /Ma:::atical Society / As~:~~~tion of America ~o;:~:zations

MORNING

7:00a.m. - 3:00 p.m. MINICOURSE REGISTRATION Outside Room 167, Thomas J. Watson, Sr. Center for Information Technology, Brown University

8:00 a.m. - 10:00 a.m. MINICOURSE #1 (Part A) EXP, EXPTest, and the creation of · testbanks Peter Frisk Room 167, Thomas J. Watson, Sr. Center for Information Technology, Brown University

8:00 a.m. - 10:00 a.m. MINICOURSE #2 (Part A) Contributions of algebraic coding theory to finite geometry E. F. Assmus, Jr. J.D. Key Room 168, Barus & Holley, Brown University

SHORT COURSE SERIES CHAOS AND FRACTALS: THE MATHEMATICS BEHIND THE COMPUTER GRAPHICS

8:00 a.m. - 2:30 p.m. SHORT COURSE REGISTRATION Outside Auditorium, Rhode Island School of Design

10:30 a.m. - 11 :45 a.m. SHORt COURSE 'LECTURE #1 Overview: Dynamics of simple maps Robert L. Devaney Auditorium, Rhode Island School of Design

10:30 a.m. - 12:30 p.m. MINICOURSE #3 (Part A) A survey of educational software David P. Kraines Vivian Kraines Room 167, Thomas J. Watson, Sr. Center for Information Technology, Brown University

JUL VIAUGUST 1988, VOLUME 35, NUMBER 6 857 TIMETABLE

Saturday, 6 (cont'd) ~~,m~o•ation of America zations August ~'~"matical Society ~' -

MORNING (cont'd) MINICOURSE #4 (Part A) 10:30 a.m.- 12:30 p.m. Coloring and path following algorithms for approximating roots and fixed points William F. Lucas Room 168, Barus & Holley, Brown University

AFTERNOON

2:00p.m.- 3:15p.m. SHORT COURSE LECTURE #2 The horseshoe map Philip J. Holmes Auditorium, Rhode Island School of Design

MINICOURSE #1 (Part B) 2:00p.m.- 4:00p.m. EXP, EXPTest, and the creation of testbanks Peter Frisk Room 167, Thomas J. Watson, Sr. Center for Information Technology, Brown University

MINICOURSE #2 (Part B) 2:00p.m.- 4:00p.m. Contributions of algebraic coding theory to finite geometry E. F. Assmus, Jr. J.D. Key Room 168, Barus & Holley, Brown University

3:30p.m.- 4:45p.m. SHORT COURSE LECTURE #3 Chaotic attractors James A. Yorke Auditorium, Rhode Island School of Design

MINICOURSE #3 (Part B) 4:30p.m.- 6:30p.m. A survey of educational software David P. Kraines Vivian Kraines Room 167, Thomas J. Watson, Sr. Center for Information Technology, Brown University

MINICOURSE #4 (Part B) 4:30p.m.- 6:30p.m. Coloring and path following algorithms for approximating roots and fixed points William F. Lucas Room 168, Barus & Holley, Brown University

858 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY TIMETABLE

sunday, ~ema~l August 7 ~;=matical Society ation of America izations ~~ ; - MORNING

7:00a.m.- 3:00p.m. MINICOURSE REGISTRATION Outside Room 167, Thomas J. Watson, Sr. Center for Information Technology, Brown University

8:00 a.m. - 10:00 a.m. MINICOURSE #5 (Part A) Teaching calculus with an HP-28 symbol manipulating calculator John W. Kenelly Room 168, Barus & Holley, Brown University

8:30a.m.- 3:30p.m. BOARD OF GOVERNORS' MEETING College & Sessions Rooms, Marriott

9:30 a.m. - 10:45 a.m. SHORT COURSE LECTURE #4 Julia sets Linda Keen Auditorium, Rhode Island School of Design

10:30 a.m.- 12:30 p.m. MINICOURSE #6 (Part A) An introduction to MA TLAB David R. Hill Room 167, Thomas J. Watson, Sr. Center for Information Technology, Brown University

10:30 a.m.- 12:30 p.m. MINICOURSE #7 (Part A) Groups, graphs, and computing Eugene M. Luks Room 168, Barus & Holley, Brown University

11:15 a.m.- 12:30 p.m. SHORT COURSE LECTURE #5 The Mandelbrot Set Bodll Branner Auditorium, Rhode Island School of Design

AFTERNOON

2:00p.m.- 3:15p.m. SHORT COURSE LECTURE #6 Introduction to fractals VIrginia Harrison Auditorium, Rhode Island School of Design

2:00p.m.- 4:00p.m. MINICOURSE #5 (Part B) Teaching calculus with an HP-28 symbol manipulating calculator John W. Kenelly Room 168, Barus & Holley, Brown University

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 859 TIMETABLE

Sunday, /rioan /homatlcaJ /., August 7 (cont'd) /Ma~::atical Society / As::~:tion of America ~zations

AFTERNOON .(cont'd)

3:00 p.m. - 7:00 p.m. REGISTRATION Garden Room, Omni Biltmore

3:45 p.m. - 5:00 p.m. SHORT COURSE LECTURE #7 Iterated function systems Michael F. Barnsley Auditorium, Rhode Island School of Design

4:30 p.m. - 6:30 p.m. MINICOURSE #6 (Part B) An introduction to MATLAB David R. Hill Room 167, Thomas J. Watson, Sr. Center for Information Technology, Brown University

4:30 p.m. - 6:30 p.m. MINICOURSE #7 (Part B) Groups, graphs, and computing Eugene M. Luks Room 168, Barus & Holley, Brown University

5:00 p.m. - 10:00 p.m. COUNCIL MEETING Bacchante Room, Omni Biltmore

EVENING

7:30p.m. SHORT COURSE FILMS Computer graphics and computer generated films Auditorium, Rhode Island School of Design

Monday, /rioan /hematical /., August 8 1 / Ma~::atical Society / As::~ ation of America L.or~~izations

MORNING

7:30 a.m. - 4:00 p.m. REGISTRATION Garden Room, Omni Biltmore

9:00a.m. -10:30 a.m. OPENING CEREMONIES Providence Performing Arts Center

860 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY TIMETABLE

Monday, ~ema~l August 8 (cont'd) ~"=matical Society ation of America ~~·zations - MORNING (cont'd)

11:15 a.m.- 12:15 p.m. KEYNOTE ADDRESS Renewing U.S. mathematics: An agenda to begin the second century Edward E. David, Jr. Providence Performing Arts Center

AFTERNOON

1:00 p.m.- 5:00p.m. EXHIBIT AND BOOK SALE BOOK SALE Grand Ballroom, Omni Biltmore Grand Ballroom, Omni Biltmore

1:00 p:m. - 5:00p.m. EXHIBITS Grand Ballroom, Omni Biltmore

2:00 p.m. - 3:00 p.m. INVITED ADDRESS Representations of finite groups as permutation groups Michael Aschbacher Providence Performing Arts Center

SESSIONS FOR CONTRIBUTED PAPERS

2:00p.m.- 5:25p.m. Differential equations State Suite A, Omni Biltmore

2:00p.m.- 5:40p.m. Number theory and algebra State Suite B, Omni Biltmore

2:00p.m.- 5:25p.m. Real and complex analysis State Suite C, Omni Biltmore

3:15p.m.- 4:15p.m. INVITED ADDRESS The geometry of solutions to nonlinear problems Luis A. Caffarelli Providence Performing Arts Center

4:30p.m.- 6:30p.m. SECTION OFFICERS' MEETING College & Canal Rooms, Marriott

SESSION FOR CONTRIBUTED PAPERS

5:30p.m.- 5:55p.m. History and biography State Suite C, Omni Biltmore

EVENING

7:00p.m. - 9:00p.m. OPENING RECEPTION Rhode Island State House

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 861 TIMETABLE

Tuesday, /,= /hematio• /., August 9 /Ma~::atical Society / As::~~tion of America / Or~~~zations

MORNING

7:30 a.m. - 4:00 p.m. REGISTRATION Garden Room, Omni Biltmore

8:30 a.m. - 9:30 a.m. INVITED ADDRESS Sufficiency as statistical symmetry Persi Diaconis Providence Performing Arts Center

9:00 a.m. - 5:00 p.m. EXHIBIT AND BOOK SALE BOOK SALE Grand Ballroom, Omni Biltmore Grand Ballroom, Omni Biltmore

9:00 a.m. - 5:00 p.m. EXHIBITS Grand Ballroom, Omni Biltmore

9:45 a.m. - 10:45 a.m. INVITED ADDRESS Problems from mathematical physics Charles L. Fetterman Providence Performing Arts Center

11 :00 a.m. - noon AMS-MAA INVITED ADDRESS The topological constraints on analysis RaouiH.Bott Providence Performing Arts Center

AFTERNOON

noon - 2:00 p.m. PI MU EPSILON COUNCIL LUNCHEON Conference Room, Chamber of Commerce Building 1 :00 p.m. - 3:00 p.m. OPEN HOUSE Headquarters Office

2:00 p.m. - 3:00 p.m. INVITED ADDRESS Working and playing with the two-dimensional disk Michael H. Freedman Providence Performing Arts Center

SESSIONS FOR CONTRIBUTED PAPERS

2:00 p.m. - 5:40 p.m. Foundations and general algebraic structures State Suite A, Omni Biltmore

2:00 p.m. - 5:40 p.m. Functional and abstract analysis State Suite C, Omni Biltmore

862 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY TIMETABLE

Tuesday, /hemati

3:15p.m.- 4:15p.m. INVITED ADDRESS The incompleteness phenomena Harvey M. Friedman Providence Performing Arts Center

AND 4:30 p.m. - 5:30 p.m. PRIZE SESSION BUSINESS MEETING Providence Performing Arts Center

HOUR 5:30 p.m. - 7:30 p.m. HAPPY Cafe on the Terrace, Omni Biltmore

EVENING

PME - RECEPTION 6:00 p.m. - 7:00 p.m. Omni Biltmore

PME - CONTRIBUTED PAPER 7:00p.m.- 10:00 p.m. SESSIONS State Suites A, B, & C, & Bacchante Room, Omni Biltmore

ASSOCIATION FOR WOMEN 7:30p.m. - 7:55p.m. IN MATHEMATICS MEMBERSHIP MEETING Providence Performing Arts Center

AWM - PANEL DISCUSSION 8:00 p.m. - 9:30 p.m. Centennial reflections on women in American mathematics Mabel S. Barnes Judy Green Jeanne LaDuke Vivienne Maione-Mayes Olga Taussky-Todd Providence Performing Arts Center

AWM - OPEN RECEPTION 9:30p.m.- 10:30 p.m. Lobby, Providence Performing Arts Center

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 863 TIMETABLE /, Wednesday, /,;,,. /hematloal August 10 /Ma~:::atical Society / As~:~~tion of America / Or~~~:zations

MORNING PME- DUTCH TREAT 6:30 a.m. - 8:00 a.m. BREAKFAST Refectory, Brown University

7:30 a.m. - 4:00 p.m. REGISTRATION Garden Room, Omni Biltmore

8:30 a.m. - 9:30 a.m. INVITED ADDRESS Modular forms and elliptic curves Benedict H. Gross Providence Performing Arts Center

SALE a.m. - 5:00 p.m. EXHIBIT AND BOOK SALE BOOK 9:00 Biltmore Grand Ballroom, Omni Biltmore Grand Ballroom, Omni

EXHIBITS 9:00 a.m. - 5:00 p.m. Grand Ballroom, Omni Biltmore

9:45 a.m.- 10:45 a.m. INVITED ADDRESS Developments in algebraic geometry Joseph Harris Providence Performing Arts Center

11 :00 a.m. - noon AMS-MAA INVITED ADDRESS Mathematics: Applied and pure Peter D. Lax Providence Performing Arts Center

AFTERNOON

1 :00 p.m. - 3:00 p.m. OPEN HOUSE Headquarters Office

2:00 p.m. - 3:00 p.m. INVITED ADDRESS A century of Lie theory Roger E. Howe Providence Performing Arts Center

SESSIONS FOR CONTRIBUTED PAPERS

2:00 p.m. - 4:25 p.m. Groups, rings, algebras and category theory State Suite A, Omni Biltmore

2:00 p.m. - 4:25 p.m. Applied mathematics State Suite C, Omni Biltmore

864 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY TIMETABLE

wednesday, /rioan / hematloal /., August 10 (cont'd) / Ma~::atical Society / As::~~tion of America /_ ~r~~:zations - AFTERNOON (cont'd)

3:15p.m.- 4:15p.m. INVITED ADDRESS A von Neumann algebra excursion: From quantum theory to knot theory and back Vaughan F. R. Jones Providence Performing Arts Center

4:30 p.m. - 5:30 p.m. INVITED ADDRESS Modular invariance in mathematics and physics VIctor G. Kac Providence Performing Arts Center

5:30p.m. - 7:30 p.m.

EVENING

PME- BANQUET 6:30p.m.- 8:15p.m. Conference Room, Chamber of Commerce Building

NATIONAL MEETING OF DEPARTMENT HEADS 7:00 p.m. - 8:00 p.m. MS2000 Panel on courses and curricula for mathematics master's degrees Samuel M. Rankin Ill State Suite B, Omni Biltmore

8:00 p.m. - 9:00 p.m. Academic employment of master's degree mathematicians in larger departments Alphonse H. Baartmans Alberto R. Galmarlno Donald F. Reynolds (moderator) Donald R. Whitaker State Suite C, Omni Biltmore

8:00 p.m. - 9:00 p.m. Academic employment of master's degree mathematicians in smaller departments David W. Ballew (moderator) Ronald M. Davis Karen L. Whitehead State Suite B, Omni Biltmore

mathematicians 8:00 p.m. - 9:00 p.m. Industrial employment of master's degree Tom Trotter (moderator) State Suite A, Omni Biltmore

7:00p.m.- 10:00 p.m. BANQUET FOR 25 YEAR MEMBERS Bacchante Room, Omni Biltmore -

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 865 TIMETABLE

Wednesday, /rioac /hematloat /, August 10 (cont'd) / Ma~::atical Society / As::~~tion of America L~~;zations - EVENING (cont'd)

8:30 p.m. - 9:30 p.m. PME- J. Sutherland Frame Lecture You, too, can tile the Conway way Doris W. Schattschneider Providence Performing Arts Center

Thursday, /homatioal /, August 11 / Ma~::atical/'''"" Society /_;:~~tion of America / Or~~~izations

MORNING

7:30 a.m. - 1 :30 p.m. REGISTRATION Garden Room, Omni Biltmore

8:30 a.m. - 9:30 a.m. INVITED ADDRESS Mathematical fluid dynamics: The interaction of nonlinear analysis and modern applied mathematics Andrew J. Majda Providence Performing Arts Center

9:00 a.m. - 1 :30 p.m. EXHIBIT AND BOOK SALE BOOK SALE Grand Ballroom, Omni Biltmore Grand Ballroom, Omni Biltmore

9:00 a.m. - 1 :30 p.m. EXHIBITS Grand Ballroom, Omni Biltmore

9:45a.m.- 10:45 a.m. INVITED ADDRESS Mathematics and computing in physiology and medicine: Examples from the past, present, and future Charles S. Peskin Providence Performing Arts Center

11 :00 a.m. - noon AMS-MAA INVITED ADDRESS Some major research departments of mathematics Saunders Mac Lane Providence Performing Arts Center

866 NOTICES OF THE AMERICAN MATHEMATICAL SOCIElY TIMETABLE

Thursday, ~~emaoc•ation of America zations August 11 (cont'd) ~~~"matical Society ~~' - AFTERNOON

1:00 p.m. - 3:00 p.m. OPEN HOUSE Headquarters Office

2:00 p.m. - 3:00 p.m. INVITED ADDRESS Progress on the renormalization conjectures in dynamical systems Dennis P. Sullivan Providence Performing Arts Center

3:15 p.m. - 8:00 p.m. CLAMBAKE Francis Farm

Friday, ~homatloal ~., August 12 ~·'~"matical Society ation of America nizations

MORNING

7:30 a.m. - 1 :00 p.m. REGISTRATION Garden Room, Omni Biltmore

8:30a.m.- 9:30a.m. INVITED ADDRESS Mathematics in computer science Robert E. Tarjan Providence Performing Arts Center

9:00 a.m. - noon EXHIBIT AND BOOK SALE BOOK SALE Grand Ballroom, Omni Biltmore Grand Ballroom, Omni Biltmore

9:00 a.m. - noon EXHIBITS Grand Ballroom, Omni Biltmore

9:45a.m.- 10:45 a.m. INVITED ADDRESS Three-dimensional geometry and topology William P. Thurston Providence Performing Arts Center

11 :00 a.m. - noon INVITED ADDRESS lnstantons and their relatives Karen K. Uhlenbeck Providence Performing Arts Center

-

JULVI AUGUST 1988, VOLUME 35, NUMBER 6 867 TIMETABLE

Friday, /,oan / hemat;ooJ / .. August 12 (cont'd) /Ma~::atical Society / As::~~tion of America / Or~~:zations

AFTERNOON

2:00 p.m. - 3:00 p.m. INVITED ADDRESS Quantum field theory and Donaldson polynomials Edward Witten Providence Performing Arts Center

SESSIONS FOR CONTRIBUTED PAPERS

2:00 p.m. - 5:40 p.m. Geometry and topology State Suite C, Omni Biltmore

2:00 p.m. - 5:25p.m. Probability, statistics and numerical analysis State Suite A, Omni Biltmore

3:15 p.m. - 4:15 p.m. STEELE PRIZE SESSION AND BUSINESS MEETING Providence Performing Arts Center

THEORIE DES VARiETES MINIMAI-ES ET APPLICATIONS (MINIMAL SUBMANIFOLDS) SEMINAIRE PALAISEAU (Asterisque, Number 154-155)

The study of minimal submanifolds is by now established Jr. at Ecole Polytechnique in Palaiseau, presents recent as one of the deep and esthetically appealing parts of contributions to the theory of minimal submanifolds in their mathematics. It combines in an exemplary fashion geometric diversity. It starts with an elementary approach to the subject, and analytical techniques both of a classical and of a more hence is appropriate as a source book for a graduate seminar. modern nature. In recent years it became a powerful tool to investigate the internal geometry of manifolds, a subject SOCIETE MATHEMATIQUE DE FRANCE, ASTERISQUE of interest today to both mathematicians and theoretical The AMS distributes Asterisque only in the U.S., Canada, and Mexico. physicists. Orders from other countries should be sent to the SMF, B.P. 126-05, 75226 Paris Cedex 05, France, or to OFFILIB, 48 rue Gay-Lussac, This volume, devoted to notes of a seminar held from October 75240 Paris Cedex 05, France. Individual members of either AMS or 1983 to June 1984 under the direction of H. B. Lawson SMF are entitled to the member price. (ISSN 0303-1179)

1st book 12. each add'l 11, 125 1980 Mathematics Subject Classifications: 57, 53, 58 Shipping/Handling: max. By air, 1st book 15. each add'l 13. 1100 max. 354 pages (softcover), 1988 Prepayment required. Order from AMS, P.O. Box Individual member $27, List price $39 1571, Annex Station, Providence, Rl 02901-9930, or To order, please specify AST/154/155NA call 800-556-7774 to use VISA or MasterCard.

868 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Program of the Sessions

Abstracts of papers presented by speakers in the Symposium on Mathematics into the Twenty-First Century, AMS-MAA Invited Speakers, and AMS Sessions for Contributed Papers will be found in the August 1988 issue of Abstracts of papers presented to the American Mathematical Society, which will be provided to registrants at the meeting. Abstracts for other sessions are not available. To maintain the schedule, beginning and ending times of presentations will be strictly enforced. For papers with more than one author, an asterisk follows the name of the author who plans to present the paper at the meeting. Where a presenter is visiting another institution, the permanent affiliation is given first, followed by the name of the institution being visited.

Saturday, August 6 MAA Minicourse #1: Part B 2:00 p.m.-4:00p.m. Room 167, Thomas J. Watson Sr. MAA Minicourse #1: Part A Center for Information Technology, Brown University 2:00 p.m. EXP, EXPTest, and the creation of testbanks. 8:00 a.m.-10:00 a.m. Room 167, Thomas J. Watson Sr. Peter Frisk, Rock Valley College Center for Information Technology, Brown University

8:00a.m. EXP, EXPTest, and the creation of testbanks. Peter Frisk, Rock Valley College MAA Minicourse #2: Part B

2:00 p.m.-4:00 p.m. Room 168, Barus & Holley, Brown MAA Minicourse #2: Part A University 2:00 p.m. Contributions of algebraic coding theory to finite 8:00 a.m.-10:00 a.m. Room 168, Barus & Holley, geometry. Brown University E. F. Assmus, Jr., Lehigh University and J. D. Key, Emmy Noether Lecturer at Bryn Mawr and on leave 8:00 a.m. Contributions of algebraic coding theory to finite from the University of Birmingham geometry. E. F. Assmus, Jr., Lehigh University and J.D. Key, Emmy Noether Lecturer at Bryn Mawr and on leave from the University of Birmingham MAA Minicourse #3: Part B

4:30 p.m.-6:30p.m. Room 167, Thomas J. Watson Sr. MAA Minicourse #3: Part A Center for Information Technology, Brown University 4:30 p.m. A survey of educational software. 10:30 a.m.-12:30 p.m. Room 167, Thomas J. Watson David P. Kraines, Duke University and VIvian Sr. Center for Information Technology, Brown University Kraines, Meredith College 10:30 a.m. A survey of educational software. David P. Kraines, Duke University and VIvian Kraines, Meredith College MAA Minicourse #4: Part B

4:30 p.m.-6:30 p.m. Room 168, Barus & Holley, Brown MAA Minicourse #4: Part A University 4:30 p.m. Coloring and path following algorithms for 10:30 a.m.-12:30 p.m. Room 168, Barus & Holley, approximating roots and fixed points. Brown University William F. Lucas, Claremont Graduate School 10:30 a.m. Cploring and path following algorithms for approximating roots and fixed points. William F. Lucas, Claremont Graduate School

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 869 Program of the Sessions

Sunday,August7 MAA Minicourse #7: Part B

MAA Minicourse #5: Part A 4:30 p.m.-6:30 p.m. Room 168, Barus & Holley, Brown University 8:00 a.m.-10:00 a.m. Room 168, Barus & Holley, 4:30 p.m. Groups, graphs, and computing. Brown University Eugene M. Luks, University of Oregon 8:00 a.m. Teaching calculus with an HP-28 symbol manipulating calculator. AMS Council John w. Kenelly, Clemson University 5:00 p.m.-10:00 p.m. Bacchante Room, Omni Biltmore

MAA Board of Governors

8:30 a.m.-3:30 p.m. College & Sessions Rooms, Monday, August 8 Marriott Keynote Address

MAA Minicourse #6: Part A 11:15 a.m.-12:15 p.m. Providence Performing Arts Center

10:30 a.m.-12:30 p.m. Room 167, Thomas J. Watson 11:15 a.m. Renewing U.S. mathematics: An agenda to begin the Sr. Center for Information Technology, Brown University (1) second century. Edward E. David, Jr., EED, Inc., Bedminster, New 10:30 a.m. An introduction to MATLAB. Jersey David A. Hill, Temple University

AMS Invited Address MAA Minicourse #7: Part A 2:00 p.m.-3:00 p.m. Providence Performing Arts Center 10:30 a.m.-12:30 p.m. Room 168, Barus & Holley, 2:00 p.m. Representations of finite groups as permutation Brown University (2) groups. Michael Aschbacher, California Institute of 10:30 a.m. Groups, graphs, and computing. Technology (844-20-51) Eugene M. Luks, University of Oregon

AMS Session on Differential Equations

MAA Minicourse #5: Part B 2:00 p.m.-5:25 p.m. State Suite A, Omni Biltmore

2:00 p.m.-4:00 p.m. Room 168, Barus & Holley, Brown 2:00 p.m. Some non-existence results for the abstract Cauchy University (3) problem. s. Zeidman, Universite de Montreal (844-34-15) 2:00 p.m. Teaching calculus with an HP-28 symbol manipulating 2:15p.m. A delayed-advanced two-body problem. calculator. (4) Jeffrey Hoag*, Providence College, and A. D. Driver, John W. Kenelly, Clemson University University of Rhode Island (844-34-68) 2:30 p.m. Oscillations of equations with piecewise constant (5) arguments. Geraslmos Ladas, University of Rhode Island MAA Minicourse #6: Part B (844-34-118) 2:45p.m. Asymptotic solutions of xt(t) = bx(t -1). (6). Bruce K. Driver, University of California at San Diego, 4:30 p.m.-6:30p.m. Room 167, Thomas J. Watson Sr. La Jolla, and A. D. Driver•, University of Rhode Island Center for Information Technology, Brown University (844-34-119) 4:30p.m. An introduction to MATLAB. David A. Hill, Temple University

870 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Program of the Sessions

3:00p.m. On the oscillations of the Goodwin oscillator. 3:00 p.m. Pisot numbers and binomial numbers. (7) K. Gopalsamy, Flinders University, Australia, and (21) Kuoduo J. Huang, California State University, Los M. R. Kulenovic* and G. Ladas, University of Rhode Angeles (844-11-128) Island (844-34-120) 3:15p.m. Samarth's conjecture in theory of numbers. 3:15p.m. Stability conditions for delay differential equations. (22) Samarth Verma, Congdon Park School, Duluth (8) Istvan Gyorl, University of Rhode Island (844-34-121) (844-11-136) (Sponsored by Robert L. McFarland) (Sponsored by Gerasimos Ladas) 3:30p.m. Kaninika's conjecture In theory of numbers. 3:30p.m. On oscillations of neutral equations with mixed (23) Kaninika Verma, Chester Park School, Duluth (9) arguments. (844-11-135) (Sponsored by Robert L. McFarland) Gerasimos Ladas, University of Rhode Island, and 3:45 p.m. Some aspects of theory of numbers. Stephen Schultz*, Providence College (844-34-125) (24) Krishnanand Verma, University of Minnesota, Duluth 3:45p.m. Semi-rigid partial differential operators and mlcrolocal (844-11-134) (Sponsored by Robert L. McFarland) (10) analytic hypoe/lipticity. 4:00 p.m. Algorithmic derivation of some units in algebraic A. Alexandrou Himonas, Princeton University (25) number fields. Preliminary report. (844-35-69) Malvina Baica, University of Wisconsin, Whitewater 4:00 p.m. Resonance and the 2nd BVP. Preliminary report. (844-12-33) (11) Victor L. Shapiro, University of California, Riverside 4:15p.m. A note on Indecomposable projective modules. (844-35-70) (26) Adil G. Naoum, University of Baghdad, Iraq 4:15p.m. Approximating solutions geometrically to a (844-13-16) (12) Monge-Ampare equation. 4:30 p.m. Algebraic integers in quadratic number field. Johanna Stenzel Schruben, University of Akron (27) Krishnanand Verma, University of Minnesota, Duluth, (844-35-107) and Ganesh Kumar*, University of Bihar, India 4:30 p.m. Wave propagation at computational domain (844-13-133) (Sponsored by Duane E. Anderson) (13) boundaries. Preliminary report. 4:45 p.m. Reflexive state of a Markov chain. Preliminary report. Henry A. Warchall, University of North Texas (28) James R. Weaver, University of West Florida (844-35-117) (844-15-36) 4:45 p.m. A characterization of Cauchy kernels. 5:00 p.m. Rational eigenvectors of singly stochastic matrices. (14) Konrad J. Heuvers, Michigan Technological (29) Russell Jay Hendel, North Adams State College University (844-39-103) (844-15-79) 5:00 p.m. Entropy of polynomial and rational maps. 5:15p.m. Greatest common divisor matrices. Preliminary report. (15) Shmuel Friedland, University of Illinois, Chicago (30) Scott Beslin and Steve Ligh*, University of (844-58-87) Southwestern Louisiana (844-15-137) 5:15p.m. Extinction of species in systems governed by 5:30p.m. Saslada's example of simple radical rings. Preliminary (16) difference equations of Lotka-Volterra type. (31) report. Preliminary report. Lung 0. Chung, North Carolina State University John E. Franke* and Abdui-Aziz Yakubu, North (844-16-139) Carolina State University (844-58-92)

AMS Session on Real and Complex Analysis AMS Session on Number Theory and Algebra 2:00 p.m.-5:25 p.m. State Suite C, Omni Biltmore 2:00 p.m.-5:40 p.m. State Suite B, Omni Biltmore 2:00 p.m. Fractal surfaces. 2:00 p.m. Favorable conditions for amicability. Preliminary (32) Peter R. Massopust, LaGrange College (844-26-85) (17) report. 2:15p.m. On norma/lattices and Wallman spaces. Mariano Garcia, Hostos Community College, City (33) George M. Eid, John Jay College of Criminal Justice, University of New York (844,11-30) City University of New York (844-28-35) 2:15p.m. Distances of k-free integers in arithmetic progressions 2:30p.m. Analysis of a class of probability preserving measure (18) In real quadratic number fields. (34) algebras on compact intervals. Werner G. H. Schaal, Fachbereich Mathematik der w. C. Connett and A. L. Schwartz*, University of Universitat Marburg, Federal Republic of Germany Missouri, St. Louis (844-28-1 06) (844-11-31) 2:45p.m. Weighted subspaces of Hardy spaces. 2:30 p.m. On the non-vanishing of Poincarfl series. (35) Hong Oh Kim*, Korea Advanced Institute of Science (19) C. J. Mozzochl, Institute for Advanced Study and Technology, Korea, and Em Geun Kwon, Andong (844-11-32) University, Korea (844-30-20) 2:45 p.m. Class numbers and units of several kinds of quadratic 3:00p.m. Function classes related to Ruscheweyh derivatives. (20) fields. (36) 0. P. Ahuja*, University of Papua, New Guinea, and Xianke Zhang, University of Maryland, College Park H. Silverman, College of Charleston (844-3Q-28) (844-11-123)

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 871 Program of the Sessions

Monday, August 8 (cont'd) AMS Session on History and Biography 5:30 p.m.-5:55 p.m. State Suite C, Omni Biltmore 5:30 p.m. A system more suitable to our meridian: Nicolas Pike's 3:15p.m. An inequality for real functions with applications to (47) "Arithmetic", the bicentennial. (37) function theory. Joe Albree, Auburn University at Montgomery Walter Bergweiler, Cornell University (844-30-34) (844-01-98) (Sponsored by Joe B. Hill) 3:30 p.m. The necessity of all of the Buchweitz tests. 5:45 p.m. Nikolai Luzin and "Idealism", (38) John Kasdan, Columbia University, School of Law (48) Charles E. Ford, Saint Louis University (844-01-108) (844-30-64) (Sponsored by Henre Jarquet) 3:45 p.m. Some extensions of Carlson's theorem. (39) Antoinette Trembinska, John Jay College of Criminal Justice, City University of New York (844-30-65) 4:00 p.m. Mapping class groups of quadratic rational maps and Tuesday,August9 (40) Aut2• Lisa R. Goldberg•, Institute for Advanced Study and Brooklyn College, City University of New York, and AMS Invited Address Linda Keen, Herbert H. Lehman College, City University of New York (844-30-86) 8:30 a.m.-9:30 a.m. Providence Performing Arts Center 4:15 p.m. Interpolation sequences for Orlicz spaces. report. (41} Maher M. H. Marzuq, Kuwait University (844-30-88) 8:30 a.m. Sufficiency as statistical symmetry. Preliminary (49) Persl Diaconls, Harvard University (844-62-41) 4:30 p.m. Alternate forms of multidimensional complex variables. (42) Preliminary report. E. Dale Martin, NASA Ames Research Center, Moffett Field, California (844-30-114) AMS Invited Address 4:45 p.m. On solutions of a cubic equation via hyperbolic (43) functions of a complex variable. Preliminary report. 9:45 a.m.-10:45 a.m. Providence Performing Arts Sadanand Verma, University of Nevada, Las Vegas Center (844-30-143) (Sponsored by L. J. Simonoff) 9:45 a.m. Problems from mathematical physics. 5:00 p.m. Higher order parturbations of the Chebyshev (50) Charles L. Fetterman, Princeton University (44) polynomials. Preliminary report. (844-81-53) Attlla Mate•, Brooklyn College, City University of New York, and Paul Naval, Ohio State University, Columbus (844-42-105) AMS-MAA Invited Address 5:15 p.m. Characterization of inverse-closed Carleman classes. (45) Jamll A. Siddiqi, Laval University (844-42-146) 11:00 a.m.-12:00 noon Providence Performing Arts Center AMS Invited Address 11 :00 a.m. The topological constraints on analysis. (51) Raoul H. Bott, Harvard University 3:15 p.m.-4:15p.m. Providence Performing Arts Center 3:15p.m. The geometry of solutions to nonlinear problems. (46) Luis A. Caffarelli, Institute for Advanced Study AMS Invited Address (844-35-47) 2:00 p.m.-3:00 p.m. Providence Performing Arts Center MAA Section Officers 2:00 p.m. Working and playing with the two-dimensional disk. (52) Michael H. Freedman, University of California at San Diego, La Jolla (844-57-42) 4:30 p.m.-6:30 p.m. College & Canal Rooms, Marriott

872 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Program of the Sessions

5:30 p.m. Tertiary decompositions of finite universal algebras AMS Session on Foundations and (67) omitting tame type 1. Preliminary report. General Algebraic Structures William H. Rowan, EMR, University of California, Berkeley (844-06-130) 2:00 p.m.-5:40 p.m. State Suite A, Omni Biltmore 2:00p.m. Some evaluations of C. S. Peirce's contributions to AMS Session on Functional (53) algebraic logic. Preliminary report. and Abstract Analysis Irving H. Anellis*, Iowa State University, and Nathan Houser, Indiana University-Purdue University, Indianapolis (844-03-11) 2:00 p.m.-5:40 p.m. State Suite C, Omni Biltmore 2:15p.m. Some properties of recursively enumerable sets 2:00 p.m. Multipliers of sequence spaces. Preliminary report. (54) uniform for equivalence relations. (68) George Brauer, University of Minnesota, Minneapolis Hong Ye, University of Connecticut, Storrs (844-40-19) (844-03-23) 2:15p.m. Best approximation in L(X, Y). 2:30 p.m. Fuzzy regressivity and retraceability. (69) W. Deeb and R. Khalil*, University of Kuwait, Kuwait (55) Leon Harkleroad, Cornell University and Bellarmine (844-41-13) College (844-03-39) 2:30 p.m. Lipschitz/an selections in approximation from 2:45 p.m. Topological applications of the Ultrapower theorem. (70) nonconvex sets of bounded functions. (56) Paul Bankston, Marquette University (844-03-11 0) Vasant A. Ubhaya, North Dakota State University, 3:00 p.m. Towards a structure theory for ideals on Fargo (844-41-132) (Sponsored by Warren E. (57) P-kappa-lambda (continued). Shreve) Donna M. Carr, State University of New York, College 2:45 p.m. An extended Poisson type sum formula for general at Plattsburgh, and Donald H. Pelletier*, York (71 ) integral transforms and aliasing error bound for the University (844-04-74) generalized sampling theorem. 3:15p.m. Unique Fibonacci formulas. Preliminary report. A. J. Jerri, The American University in Cairo, Egypt, (58) Joseph Arkin, Spring Valley, New York, David C. and Clarkson University (844-42-03) Arney, United States Military Academy, Gerald E. 3:00 p.m. Some Fourier-stleltjes coefficients revisited. Bergum, South Dakota State University, Stefan A. (72) Sylvester Reese, Queensborough Community Burr, City University of New York, and Bruce J. College, City University of New York (844-42-81) Porter*, United States Military Academy (844-05-05) 3:15p.m. A representation formula for the Dirichlet integral. 3:30 p.m. A perfect 4-dimensiona/ hypercube of order 7. (73) Preliminary report. (59) Joseph Arkin, David C. Arney*, and Bruce J. Porter, Kwang-Nan Chow* and David Protas, California United States Military Academy (844-05-06) State University, Northridge (844-44-97) 3:45 p.m. Tiling the ~Ch power of a power series. Preliminary 3:30 p.m. Boundary value problems for abstract Volterra (60) report. (7 4) functional-differential equations. Joseph Arkin*, Spring Valley, New York, David C. C. Corduneanu, University of Texas at Arlington Arney, United States Military Academy, Gerald E. (844-45-122) Bergum, South Dakota State University, Stefan A. 3:45p.m. Best approximation in LP(I, X), 0 < p < 1. Burr, City University of New York, and Bruce J. (75) W. Deeb* and R. Khalil, University of Kuwait, Kuwait Porter, United States Military Academy (844-05-07) (844-46-1 0) 4:00 p.m. Generalized binomial coefficients and uses. 4:00 p.m. Zeros and fixed points of mutt/functions with (61) Donald R. Snow, Brigham Young University (76) paracompact domains. (844-05-25) Joseph C. Bellenger, University of California, 4:15p.m. A conjecture on edge-cordial trees. Riverside (844-46-75) (62) Ho Kuen Ng* and Sin•Min Lee, San Jose State 4:15p.m. Some random best approximations and random fixed University (844-05-61) (77) point theorems. 4:30 p.m. Birigidity in the plane. V. M. Sehgal*, University of Wyoming, and S. P. (63) Brigitte Servatius, Worcester Polytechnic Institute Singh, Memorial University of Newfoundland (844-05-76) (844-47-89) 4:45 p.m. How to obtain an asymptotic expansion of a 4:30 p.m. Minimax inequalities, variational inequalities and fixed (64) combinatorial sequence from an analytic identity (78) point theorems. satisfied by its generating function. Preliminary report. Xie Ping Ding, Sichuan Normal University, People's Jacob Plotkin, Michigan State University, and John Republic of China (844-49-01) Rosenthal*, Ithaca College (844-05-94) 4:45 p.m. Hyperbolic aproperness. 5:00 p.m. Automorphism groups of unicyclic graphs. (79) Dan D. Pascali, Forest Hills, New York (844-47-142) (65) Donald McCarthy, St. John's University (844-05-126) 5:00 p.m. Convergence of reduced gradient method. 5:15p.m. A sum representation of meet-semilattice. (80) Abolghassem Alemzadeh, State University of New (66) S. M. Kim, Yonsei University, Korea (844-06-08) York, College at Cortland (844-49-141)

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 873 Program of the Sessions

Tuesday, August 9 (cont'd) AMS Invited Address 9:45 a.m.-10:45 a.m. Providence Performing Arts Center 5:15p.m. Algebraic and topological selections of multivalued (81) linear mappings. 9:45 a.m. Developments in algebraic geometry. Sung J. Lee, University of South Florida, and (85) Joseph Harris, Harvard University (844-14-52) M. Zuhalr Nashed*, University of Delaware (844-47-140) 5:30 p.m. Modular functions in lattice statistical mechanics. AMS-MAA Invited Address (82) Matthew P. Richey, St. Olaf College (844-82-145) 11:00 a.m.-12:00 noon Providence Performing Arts Center AMS Invited Address 11 :00 a.m. Mathematics: Applied and pure. (86) Peter D. Lax, Courant Institute of Mathematical 3:15 p.m.-4:15p.m. Providence Performing Arts Center Sciences, New York University 3:15p.m. The incompleteness phenomena. (83) Harvey M. Friedman, Ohio State University, Columbus (844-03-49) AMS Invited Address

2:00 p.m.-3:00 p.m. Providence Performing Arts Center MAA Prize Session and Business Meeting 2:00 p.m. A century of Lie theory. (87) Roger E. Howe, Yale University (844-22-48) 4:30 p.m.-5:30 p.m. Providence Performing Arts Center

PME Contributed Paper Sessions AMS Session on Groups, Rings, Algebras and Category Theory 7:00 p.m.-10:00 p.m. State Suites A, B, & C & Bacchante Room, Omni Biltmore 2:00 p.m.-4:25 p.m. State Suite A, Omni Biltmore 2:00 p.m. Invariant forms on central simple structurable (88) algebras. AWM Membership Meeting R. D. Schafer, Massachusetts Institute of Technology (844-17-27) 7:30 p.m.-7:55p.m. Providence Performing Arts Center 2:15p.m. A nonassociative powerassociative algebra of infinite (89) matrices. Alexander Ablan•, Iowa State University, and Paula AWM Panel Discussion A. Kemp, Southwest Missouri State University (844-17-77) 8:00 p.m.-9:30 p.m. Providence Performing Arts Center 2:30 p.m. On the monodromy group of everywhere tangent lines (90) to the octic surface in P3• Preliminary report. 8:00 p.m. Centennial reflections on women in American Harry D'Souza, University of Michigan, Flint mathematics. (844-17-78) 2:45 p.m. Nonassociative rings with a special kind of (91) endomorphism. Preliminary report. Tae-11 Suh, East Tennessee State University VVednesday,August10 (844-17-82) 3:00 p.m. Decomposition of quasi-injective objects over a (92) Grothendieck category. AMS Invited Address Jinzhong Xu*, University of Toronto, and Ho Kuen Ng, San Jose State University (844-18-40) 8:30 a.m.-9:30 a.m. Providence Performing Arts Center 3:15p.m. Gerstenhaber-schack cohomology and Hattori's theory (93) for mapping cones of squares. 8:30 a.m. Modular forms and elliptic curves. Klaus Wemer Wiegmann, University of Duisburg, (84) Benedict H. Gross, Harvard University (844-14-59) Federal Republic of Germany and University of Rochester (844-18-60)

874 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Program of the Sessions

3:30p.m. Involutions in the genera/linear group GL(n, F). 4:15p.m. Evaluating the hazard function using time dependent (94) Jo-Ann Cohen and Kwangll Koh*, North Carolina (107) probabilities from pharamacoklnetlc models. State University (844-20-73) Preliminary report. 3:45 p.m. Weight spaces of Invariants of certain unipotent group Dennis W. Quinn, Air Force Institute of Technology, (95) actions. Wright-Patterson Air Force Base, Ohio (844-92-124) Jozset Horvath, West Chester University of Pennsylvania (844-20-1 04) 4:00 p.m. Harmonic analysis, geometric quantization, and chaos. AMS Invited Address (96) Mohamed W. I. Sesay, University of the District of Columbia (844-81-144) 3:15 p.m.-4:15p.m. Providence Performing Arts Center 4:15p.m. Moufang loops with a unique nontrivial square are 3:15 p.m. A von Neumann algebra excursion: From quantum (97) code loops. (1 08) theory to knot theory and back. Orin Chein*, Temple University, and Edgar G. Vaughn F. R. Jones, University of California, Berkeley Goodaire, Memorial University of Newfoundland (844-57-54) (844-20-67)

AMS Session on Applied Mathematics AMS Invited Address

4:30 p.m.-5:30 p.m. Providence Performing Arts Center 2:00 p.m.-4:25 p.m. State Suite C, Omni Biltmore 4:30 p.m. Modular invarlance In mathematics and physics. 2:00 p.m. On unsteady hydromagnetlc flows of a dusty fluid (109) Victor G. Kac, Massachusetts Institute of Technology (98) batwean two oscillating plates. (844-17-46) Lokenath Debnath, University of Central Florida (844-76-62) 2:15p.m. The Thomas rotation formalism underlying the Lorentz Joint Policy Board for Mathematics: (99) group of the special theory of relativity. Abreham Ungar, North Dakota State University, National Meeting of Department Heads Fargo (844-83-63) 2:30 p.m. Holors in accelerated coordinate transformations. 7:00 p.m.-9:00 p.m. State Suite B, Omni Biltmore (100) Domina Eberle Spencer*, University of Connecticut, Storrs, and Shama Y. Uma, Bridgewater State College (844-83-1 00) PME J. Sutherland Frame Lecture 2:45 p.m. The Holor transformation of the Doppler shift In (1 01) accelerated coordinate systems. · 8:30 p.m.-9:30 p.m. )~rovidence Performing Arts Center Shama Y. Uma•, Bridgewater State College, and Domina Eberle Spencer, University of Connecticut, 8:30 p.m. You, too, can file the Conway way. Storrs (844-83-1 01) (11 0) Doris W. SC:hattschneider, Moravian College 3:00 p.m. Conditions for the existence of tangential forces on (1 02) current elements. Parry Moon, Massachusetts Institute of Technology, , and Domina Eberle Spencer, University of Thursday,August11 Connecticut, Storrs, Shama Y. Uma, Bridgewater State College, and Philip Jay Mann•, University of Connecticut, Storrs (844-83-102) AMS Invited Address 3:15p.m. Conditions on utility guaranteairig existence of optimal (103) extraction paths in unbounded horizon problems. 8:30 a.m.-9:30a.m. Providence Performing Arts Center Jennett Highfill and Mi~hael Mc~y*, Bradley University (844-90-93) 8:30 a.m. Mathematical fluid dynamics: The Interaction of 3:30 p.m. Three-dimensional controlled roundings do not always (111) nonlinear analysis and modern applied mathematics. (104) exist. Preliminary report. Andrew J. Ma)da, Princeton University (844-76-44) Lawrence R. Ernst, Bureau of the Census, Washington, D.C. (844-90-109) 3:45 p.m. Bounds for the coefficient of additivity of the (105) characteristics function of a game. Miguel Paredes, Pan American University (844-90-112) 4:00 p.m. On a simple model for the spread of a sexually (106) transmitted disease. Preliminary report. Alejandro Necochea, Pan American University (844-92-111)

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 875 Program of the Sessions

Thursday, August 11 (cont'd) AMS Invited Address 11:00 a.m.-12:00 noon Providence Performing Arts AMS Invited Address Center 11 :00 a.m. lnstantons and their relatives. 9:45 a.m.-10:45 a.m. Providence Performing Arts (117) Karen K. Uhlenbeck, University of Texas at Austin Center (844-53-50)

9:45 a.m. Mathematics and computing in physiology and ( 112) medicine: Examples from the past, present, and future. AMS Invited Address Charles S. Peskin, Courant Institute of Mathematical Sciences, New York University (844-92-55) 2:00 p.m.-3:00 p.m. Providence Performing Arts Center 2:00 p.m. Quantum field theory and Donaldson polynomials. AMS-MAA Invited Address (118) Edward Witten, Institute for Advanced Study (844-57-43) 11:00 a.m.-12:00 noon Providence Performing Arts Center AMS Session on Geometry and Topology 11 :00 a.m. Some major research departments of mathematics. (113) Saunders Mac Lane, University of Chicago 2:00 p.m.-5:40 p.m. State Suite C, Omni Biltmore 2:00 p.m. A geometric measure of the non-rectangularity of a AMS Invited Address (119) hyperbola. Mangho Ahuja, Southeast Missouri State University (844-51-24) 2:00 p.m.-3:00 p.m. Providence Performing Arts Center 2:15p.m. Lattice polygons In space. 2:00 p.m. Progress on the renormalization conjectures in (120) Stanley Rabinowitz, Alliant Computer Systems (114) dynamical systems. Corporation, Littleton, Massachusetts (844-51-95) Dennis P. Sullivan, Graduate School and University 2:30 p.m. Parallel plane curves: relations, properties, limits, and Center, City University of New York (844-58-45) (121) beyond. Rodney T. Hood, Franklin College of Indiana (844-51-116) 2:45 p.m. Pasch's postulate and metrization. Preliminary report. Friday, August 12 (122) James W. Petticrew, Pan American University (844-51-129) 3:00 p.m. Covering curves with restricted curvature. Preliminary AMS Invited Address (123) report. Mostafa Ghandeharl, Santa Clara University (844-52-09) 8:30 a.m.-9:30 a.m. Providence Performing Arts Center 3:15p.m. Automorphism groups of almost r-paracontact 8:30 a.m. Mathematics in computer science. (124) manifolds of P-sasaklan type. (115) Robert E. Tarjan, Princeton University (844-68-56) Andrzej Bucki, Lycoming College (844-53-72) 3:30 p.m. Some results on the topological Markov chains. (125) Zuo Ling Zhou, Zhongshan University, People's AMS Invited Address Republic of China (844-54-04) 3:45 p.m. A conjecture concerning X x I. Preliminary report. 9:45 a.m.-10:45 a.m. Providence Performing Arts (126) Ernest P. Lane, Appalachian State University Center (844-54-29) 4:00 p.m. Semi-reasonable topologies. Preliminary report. 9:45 a.m. Three-dimensional geometry and topology. (127) Kathryn F. Porter, Ball State University (844-54-71) Princeton University (844-55-57) (116) William P. Thurston, 4:15p.m. Construction of vector field on space models of Torus (128) and Klein Bottle. Preliminary report. Margaret M. LaSalle, University of Southwestern Louisiana (844-54-91) 4:30 p.m. Extending continuous functions to connected (129) functions. Preliminary report. B. D. Garrett, Tennessee State University (844-54-99)

878 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Program of the Sessions

4:45 p.m. Sufficient conditions for existence of counterexample 3:15p.m. Confounding and the Hadamard matrix problem. (130) to 3-dimensional Poincare conjecture. Preliminary (139) Preliminary report. report. Jay H. Beder, University of Wisconsin, Milwaukee Francis D. Lonergan, Webster, Massachusetts (844-62-113) (844-55-14) 3:30 p.m. Computation principles of mathematical physics. 5:00 p.m. A spectral sequence for classifying rational homotopy (140) Chenggui Huang, Tianjin Normal University, People's (131) type. Republic of China (844-65-17) Ronald N. Umble, Millersville University of 3:45 p.m. Interpolation by division. Pennsylvania (844-55-66) (141) Herbert E. Salzer, Brooklyn, New York (844-65-22) 5:15p.m. On the unitary bordlsm ring U.(Zp) ® Zp. 4:00 p.m. The solution of Poisson Boltzmann equation between (132) Chlng-Mu Wu, Tamkang University, People's (142) two spheres-a modified iterative method. Republic of China (844-57-18) Abdul J. Jerri, Clarkson University and The American 5:30 p.m. Topological structures in computer and systems University in Cairo, Egypt, and Russ L. Herman*, (133) sciences. Clarkson University (844-65-26) Efim Khalimsky, Borough of Manhattan Community 4:15p.m. PFIS, a nonlinear system solver. College, City University of New York (844-99-147) (143) S. K. Dey, Eastern Illinois University (844-65-83) 4:30 p.m. Improved error and C1 derivative approximations for (144) continuous Runge-Kutta formulas. Curtis Outlaw, Leroy Derr and Dlran Sarafyan*, University of New Orleans (844-65-96) AMS Session on Probability, 4:45 p.m. An algorithm for the inversion of generalized Statistics and Numerical Analysis (145) Vandermonde matrices. Luis Verde-Star, Universidad Aut6noma Metropolitana, Mexico (844-65-115) 2:00 p.m.-5:25 p.m. State Suite A, Omni Biltmore 5:00 p.m. Computational electromagnetics. (146) Brian McCartin, United Technologies Research 2:00 p.m. Open toss problem. Preliminary report. Center, East Hartford, Connecticut (844-65-127) (134) Prem N. Bajaj, Wichita State University (844-60-37) 5:15p.m. The shape of clusters created from percolation of 2:15p.m. Sufficient statistics for some pairwise independent (147) coalescing random walks. Preliminary report. (135) stationary stochastic processes. Preliminary report. Bao G. Nguyen, University of Mississippi and Illinois James B. Robertson, University of California, Santa Institute of Technology (844-82-138) Barbara (844-60-38) 2:30 p.m. Oriented percolation processes with interaction In (136) population genetics. Preliminary report. AMS Steele Prize Session R. B. Campbell, University of Northern Iowa and Business Meeting (844-60-80) 2:45 p.m. Markov chains generated by endomorphlsms, Ill. 3:15 p.m.-4:15 p.m. Providence Performing Arts Center (137) Preliminary report. John R. Durbin, University of Texas at Austin Felix E. Browder (844-60-84) Chairman, Centennial Program Committee New Brunswick, New Jersey 3:00 p.m. Simultaneous confidence bounds in multivariate (138) analysis. Frank T. Blrtel Jack Tomsky, Lockheed Missiles and Space Associate Secretary Company, Palo Alto, California (844-62-02) New Orleans, Louisiana

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 877 Presenters of Papers

Numbers following the names indicate the speakers' position(s) on the program. • AMS Invited Lecturer t Keynote Speaker o AMS-MAA Invited Lecturer 1C PME J. Sutherland Frame Lecturer

Abian, A., 89 Franke, J. E., 16 Ligh, s., 30 Schruben, J. S., 12 Ahuja, M., 119 • Freedman, M. H., 52 Lonergan, F. D., 130 Schultz, S., 9 Ahuja, 0. P., 36 Friedland, S., 15 <>Mac Lane, S., 113 Schwartz, A. L., 34 Albree, J., 47 • Friedman, H. M., 83 • Majda, A. J., 111 Sehgal, V. M., 77 Alemzadeh, A., 80 Garcia, M., 17 Mann, P. J., 102 Servatius, B., 63 Anellis, I. H., 53 Garrett, B. D., 129 Martin, E. D., 42 Sesay, M. W. I., 96 Arkin, J., 60 Ghandehari, M., 123 Marzuq, M. M. H., 41 Shapiro, V. L., 11 Arney, D. C., 59 Goldberg, L. R., 40 Massopust, P. R., 32 Siddiqi, J. A., 45 • Aschbacher, M., 2 • Gross, B. H., 84 Mate, A., 44 Snow, D. R., 61 Baica, M., 25 Gyori, I., 8 McAsey, M., 103 Spencer, D. E., 100 Bajaj, P. N., 134 Harkleroad, L., 55 McCarthy, D., 65 Suh, T.-I., 91 Bankston, P., 56 • Harris, J., 85 McCartin, B., 146 • Sullivan, D. P., 114 Beder, J. H., 139 Hendel, R. J., 29 Mozzochi, C. J., 19 • Tarjan, R. E., 115 Bellenger, J. C., 76 Herman, R. L., 142 Naoum, A. G., 26 •Thurston, W. P., 116 Bergweiler, W., 37 Heuvers, K. J., 14 Nashed, M. Z., 81 Tomsky, J., 138 o Bott, R. H., 51 Himonas, A. A., 10 Necochea, A., 106 Trembinska, A., 39 Brauer, G., 68 Hoag, J., 4 Ng, H. K., 62 Ubhaya, V. A., 70 Bucki, A., 124 Hood, R. T., 121 Nguyen, B. G., 147 • Uhlenbeck, K. K., 11 7 • Caffarelli, L. A., 46 Horvath, J., 95 Paredes, M., 105 Uma, S. Y., 101 Campbell, R. B., 136 • Howe, R. E., 87 Pascali, D. D., 79 Umble, R.N., 131 Chein, 0., 97 Huang, C., 140 Pelletier, D. H., 57 Ungar, A., 99 Chow, K.-N., 73 Huang, K. J., 21 •Peskin, C. S., 112 Verde-Star, L., 145 Chung, L. 0., 31 Jerri, A. J., 71 Petticrew, J. W., 122 Verma, K., 23 Corduneanu, C., 74 • Jones, V. F. R., 108 Porter, B. J., 58 Verma, K., 24 t David, E. E., Jr., 1 •Kac, V. G., 109 Porter, K. F., 127 Verma, S., 43 Debnath, L., 98 Kasdan, J., 38 Quinn, D. W., 107 Verma, S., 22 Deeb, W., 75 Khalil, R., 69 Rabinowitz, S., 120 Warchall, H. A., 13 Dey, S. K., 143 Khalimsky, E., 133 Reese, S., 72 Weaver, J. R., 28 • Diaconis, P., 49 Kim, H. 0., 35 Richey, M. P., 82 Wiegmann, K. W., 93 Ding, X. P., 78 Kim, S. M., 66 Robertson, J. B., 135 • Witten, E., 118 Driver, R. D., 6 Koh, K., 94 Rosenthal, J., 64 Wu, C.-M., 132 D'Souza, H., 90 Kulenovic, M. R., 7 Rowan, W. H., 67 Xu, J., 92 Durbin, J. R., 137 Kumar, G., 27 Salzer, H. E., 141 Ye, H., 54 Eid, G. M., 33 Ladas, G., 5 Sarafyan, D., 144 Zaidman, S., 3 Ernst, L. R., 104 Lane, E. P., 126 Schaal, W. G. H., 18 Zhang, X., 20 • Fefferman, C. L., 50 LaSalle, M. M., 128 Schafer, R. D., 88 Zhou, Z. L., 125 Ford, C. E., 48 <>Lax, P. D., 86 n Schattschneider, D. W., 110

878 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY American Mathematical Society Short Course Series Introductory Survey Lectures on Chaos and Fractals: The Mathematics Behind the Computer Graphics Providence, Rhode Island, August 6- 7, 1988

The American Mathematical Society, in conjunction with its graphics images associated with these terms have heightened Centennial Celebration, will present a two-day Short Course interest among scientists in these ideas. The purpose of this titled Chaos and Fractals: The Mathematics Behind the Com­ Short Course is to explain the rich mathematical concepts be­ puter Graphics on Saturday and Sunday, August 6-7, 1988, hind these terms. Basic ideas from dynamical systems theory, at the Rhode Island School of Design. The program is under including chaos, strange attractors, Julia sets, and the Man­ the direction of RoBERT L. DEVANEY of Boston University delbrot set, will be introduced and basic concepts from frac­ and LINDA KEEN of Lehman College, CUNY. tal geometry, especially fractal dimension and self-similarity, The terms "chaos" and "fractal" have received widespread will be discussed. Applications in engineering, meteorology attention in the media in recent years. The alluring computer and data compression will be described.

Saturday, August 6: Overview: Dynamics ofSimple Maps, RoBERT L. DEVANEY, Boston University. The Horseshoe Map and Applications in Engineering, PHILIP J. HoLMES, Cornell University. Chaotic Attractors, JAMEs A. YoRKE, University of Maryland. Sunday, August 7: Julia Sets, LINDA KEEN, Lehman College, CUNY. The Mandelbrot Set, BoDIL BRANNER, The Technical University of Denmark, Copenhagen. Introduction to Fractals, VIRGINIA HARRISON, University of California, Berkeley. Iterated Function Systems, MicHAEL F. BARNSLEY, Georgia Institute of Technology. Evening Session: Computer graphics and computer generated films. The lectures will assume no familiarity with chaotic dynamics or fractal geometry; however, those who wish to get the most benefit from the course should consult the first chapter of Introduction to Chaotic Dynamical Systems, R. L. Devaney, Addison-Wesley, 1985, or the introductory chapter of Iterated Maps of the Interval as Dynamical Systems, P. Collet and J.-P. Eckmann, Birkhauser, 1980.

Synopses of the talks and accompanying· reading lists, which The Short Course was recommended by the AMS-MAA provide a variety of sources for study prior to the course, Committee on Employment and Educational Policy (CEEP), appeared in the April issue of Notices. Complete lecture notes whose members are Morton Brown, Stefan A. Burr, Edward will be mailed to those who preregister for the course, and A. Connors (chair), Philip C. Curtis, Jr., Don 0. Loftsgaar­ will be available at the Short Course Registration Desk for den, David J. Lutzer, and Audrey A. Terras. The Short those registering on site. Course series is under the direction of the CEEP Short Course All who wish to participate in the Short Course may do Subcommittee, whose members are Stefan A. Burr (chair), so upon payment of a $35 advance registration fee ($45 on Lisl Novak Gaal, Robert P. Kurshan, Barbara L. Osofsky, site). There are reduced fees for students and unemployed Marjorie L. Stein, and James J. Tattersall. individuals. Please refer to the sections titled Housing and Registration at the Meetings in this issue of Notices for de­ tails.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 879 Lawrence, Kansas University of Kansas October 28-29

First Announcement

The eight-hundred-and-forty-fifth meeting of the Ameri­ Operator theory and applications to geometry, STEVEN can Mathematical Society will be held at the University E. HURDER, University of Illinois at Chicago and of Kansas in Lawrence, Kansas on Friday, October 28, NOBERTO SALINAS, University of Kansas and Saturday, October 29, 1988. Commutative algebra, DANIEL KATz and JEFFERY LANG, University of Kansas Potential theory and partial differential equations Addresses Invited in nonsmooth domains, JILL PIPHER, University of By invitation of the Committee to Select Hour Speakers Chicago and GREGORY VECHOTA, University of Illinois for Central Sectional Meetings, there will be four invited at Chicago one-hour addresses. The speakers, their affiliations, and 3-manifolds, PETER SCOTT, University of Michigan titles, when available, are: Most of the papers to be presented at these special BJORN DAHLBERG, Washington University, Elliptic sessions will be by invitation. However, anyone submit­ boundary value problems in non-smooth domains. ting an abstract for the meeting who feels that his or STEVEN E. HuRDER, University of Illinois at Chicago, her paper would be particularly appropriate for one of Geometry and the index theory offoliations. these special sessions should indicate this clearly on the PETER SCOTT, University of Michigan, Ann Arbor, abstract form and submit it by August 2, 1988, three Least area surfaces in 3-manifolds. weeks before the deadline for contributed papers, in SIDNEY M. WEBSTER, University of Minnesota, Min­ order that it may be considered for inclusion. Partici­ neapolis, title to be announced. pants are reminded that a charge of $16 is imposed for retyping abstracts that are not in camera-ready form. Special Sessions Contributed Papers By invitation of the same committee, there will be twelve special sessions of selected twenty-minute papers. Topics There will also be sessions for contributed ten-minute and the names and affiliations of the organizers follow. papers. Abstracts should be prepared on the standard Partial differential equations- Geometric equations, AMS form available from the AMS office in Providence ANDREW ACKER, Wichita State University or in Departments of Mathematics. Abstracts should be Geometry and mathematical physics, JOHN K.. BEEM, sent to the Editorial Department, American Mathemat­ University of Missouri and PHILLIP E. PARKER, Wichita ical Society, Post Office Box 6248, Providence, Rhode State University Island 02940, so as to anive before the August 24, Numerical linear algebra, RALPH BYERS, University 1988 abstract deadline. Participants are reminded that of Kansas a charge of $16 is imposed for retyping abstracts that Algebraic geometry, BRUCE CRAUDER and SHELDON are not in camera-ready form. Late papers will not be KATz, Oklahoma State University accommodated. Control theory, TYRONE DUNCAN, University of Kansas Registration Applications of set theory, WILLIAM FLEISSNER, Uni­ versity of Kansas The registration desk will be open on Friday, October Real analysis, JAMES FoRAN, University of Missouri 28, and Saturday, October 29 at times and a location at Kansas City to be announced in the September issue of Notices. The Flat bundles and geometric structures, WILLIAM MARK registration fees are $30 for members of the AMS, $45 GOLDMAN, University of Maryland

880 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

for nonmembers, and $10 for students or unemployed Local Information mathematicians. Information on travel, accommodations, food service, and other pertinent local information will be available in Petition Table the September issue of Notices. A petition table will be set up in the registration area. Andy Roy Magid Additional information about petition tables can be Associate Secretary found in a box in the Atlanta meeting announcement on Norman, Oklahoma page 68 of the January issue of Notices.

CATEGORIES OF HIGHEST WEIGHT MODULES: MOMENTS IN MATHEMATICS APPLICATIONS TO CLASSICAL Henry J. Landau, Editor HERMITIAN SYMMETRIC PAIRS (Proceedings of Symposia in Applied Mathematics, Volume 37) Function theory, spectral decomposition of Thomas J. Enright and Brad Shelton operators, probability, approximation, electrical (Memoirs of the AMS, Number 367) and mechanical inverse problems, prediction of stochastic processes, the design of algorithms for The category of highest weight representations is of signal-processing VLSI chips-these are among a special interest within the full set of representations host of important theoretical and applied topics of a real semisimple Lie group. This book describes illuminated by the classical moment problem. To the structure of the generalized Verma modules as survey some of these ramifications and the research well as the Kazhdan-Lusztig data for the simple which derives from them, the AMS sponsored the modules in this category for the classical groups. Short Course Moments in Mathematics at the In particular, the authors give explicit formulas for Joint Mathematics Meetings, held in San Antonio, composition factors of generalized Verma modules Texas, in January 1987. This volume contains the and Kazhdan-Lusztig polynomials. six lectures presented during that course. Contents Contents H. J. Landau, Classical background of the Categories of highest weight modules moment problem Reduction of singularities J. H. B. Kemperman, Geometry of the moment The Zuckerman derived functors problem An equivalence of categories Donald Sarason, Moment problems and operators A second equivalence of categories in Hilbert space Highest weight modules for Hermitian symmetric Thomas Kailath, Signal processing applications pairs of some moment problems Statement of the main results Christian Berg, The multidimensional moment Wall shifting problem and semigroups Induction from lower rank Persi Diaconis, Application of the method of Projective resolutions and Ext moments in probability and statistics Kazhdan-Lusztig polynqmials . 1980 Mathematics Subject Classifications: Decompostions of U( u-)-free self-dual 44A60,60E15, 47A20. 47A40. 60F05 g-modules ISBN 0-8218-0114-7, LC 87-19384 ISSN 0160-7634 1980 Mathematics Subject Classification: 22 168 pages. October 1987 ISBN 0-8218-2429-5. LC 87-1446 Hardcover: Individual member 118. list price ISSN 0065-9266 $30, Institutional member $24 100 pages (softcover), May 1987 Softcover: Individual member 115, list price Individual member 18, list price $13. $25. Institutional member $20 Institutional member $10 To order, please specify PSAPM/37NA To order, please specify MEM0/367NA (hardcover), PSAPMS/37NA (softcover)

Shipping/Handling: 1st book $2, each add'l $1, $25 max. By air, 1st book $5, each add'l $3, Shipping/Handling: 1st book $2, each add'l $1, $25 1100 max. max. By air, 1st book $5, each add'l $3, $100 max. Prepayment required. Order from AMS, P.O. Box Prepayment required. Order from AMS, P.O. Box 1571. Annex Station, Providence, Rl 02901-9930, 1571, Annex Station, Providence, Rl 02901-9930. or or call 800-556-7774 to use VISA or MasterCard. call 800-556-7774 to use VISA or MasterCard.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 881 Claremont, California Claremont McKenna College November 12 -13

First Announcement

The eight-hundred-and-forty-sixth meeting of the Amer­ abstract form and submit it by August 2, 1988, three ican Mathematical Society will be held at Claremont weeks before the deadline for contributed papers, in McKenna College, Claremont, California, on Saturday order that it may be considered for inclusion. Partici­ and Sunday, November 12 and 13, 1988. pants are reminded that a charge of $16 is imposed for retyping abstracts that are not in camera-ready form. Invited Addresses Contributed Papers By invitation of the Committee to Select Hour Speak­ ers for Far Western Sectional Meetings, there will be There will also be sessions for contributed ten-minute three invited one-hour addresses. The speakers, their papers. Abstracts should be prepared on the standard affiliations, and some of the titles follow: AMS form available from the AMS office in Provi­ WILLIAM JACOB, Oregon State University, Galois co­ dence or in the Departments of Mathematics. Abstracts homology and K-theory: Applications to division algebras should be sent to the Editorial Department, American and quadratic forms. Mathematical Society, Post Office Box 6248, Providence, ROBERT BROOKS, University of Southern California, Rhode Island 02940, so as to arrive before the August 23, title to be announced. 1988 abstract deadline. Participants are reminded that FRANCIS BONAHON, University of Southern Califor­ a charge of $16 is imposed for retyping abstracts that nia, Riemann surfaces and measured laminations. are not in camera-ready form. Late papers will not be accommodated. Special Sessions Activities of Other Organizations By invitation of the same committee, there will be five special sessions of selected twenty-minute papers. The Mathematical Association of America will meet on The topics, names and affiliations of the organizers and Saturday, November 12. LEONARD GILLMAN, University speakers are: of Texas and President of MAA will give an AMS-MAA Low dimensional geometry, FRANCIS BoNAHON and address. Information about other sessions of the MAA DAVID GABAI, California Institute of Technology. will be announced later. Computers and software in mathematical research, ROBERT BORRELLI, Harvey Mudd College, and COURT­ Registration NEY S. COLEMAN, Harvey Mudd College. The spectrum of the Laplacian, RoBERT BRooKs and The meeting registration desk will be located in the lobby S.-Y. CHENG, University of California, Los Angeles. of Bauer Center. The desk will be open from 8:30a.m. Differential and difference equations, STAVROS N. to 2:00 p.m. on both Saturday and Sunday, November BUSENBERG, Harvey Mudd College, and MARIO 12 and 13. The registration fees are $30 for both days MARTELLI, California State University, Fullerton. for members of the AMS, $45 for nonmembers, and $10 Division algebras, WILLIAM JACOB and ADRIAN for students and unemployed mathematicians. There is a WADSWORTH, University of California, San Diego. special one-day fee for MAA members on Saturday only Most of the papers to be presented at these special of $15. sessions will be by invitation. However, anyone submit­ ting an abstract for the meeting who feels that his or her paper would be particularly appropriate for one of these special sessions should indicate this clearly on the

882 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

Petition Table Ramada Inn (2.5 miles) 840 South Indian A petition table will be set up in the registration area. Hill Boulevard Oaremont, CA 91 711 Additional information about petition tables can be (Next to San found in a box in the Atlanta meeting announcement on Bernardino Freeway-Interstate 10) page 68 of the January issue of Notices. Telephone: 800-228-2828 or 714-621-4831 Rooms (1-2 guests): $49 plus tax Accommodations Additional person: $6 per night Rooms have been blocked at the following hotels and motels. Participants should make their own reservations Food Service directly with the hotel of their choice, identifying them­ The hotels listed above have their own restaurants. In selves as attending the American Mathematical Society's addition, there are many good restaurants in Claremont meeting at Claremont McKenna College. Rates quoted and the surrounding area. A list will be provided at the include applicable tax and are subject to change. meeting. Shuttle service from and to Ontario International Airport is provided free of charge. The driving time is 15 to 20 minutes. Luncheon Griswold's Inn (walking distance) There will be a joint luncheon for MAA and AMS 555 West Foothill Boulevard participants on Saturday, November 12 at noon in the Claremont, CA 91711 Athenaeum on the college campus. (Comer of Indian Hill Boulevard) Telephone: 800-854-5733 (except California), 800- Travel 821-0341 (in California) or 714-626-2411 Claremont is located 35 miles east of Los Angeles. Rooms (1-4 persons): $50 plus tax Most major airlines serve Ontario International Air­ port (California). The drive from LAX is slow on Friday Rooms must be reserved before October 21. afternoons, but not as slow on Saturdays and Sundays. Lance W. Small Associate Secretary La Jolla, California

METHODS AND APPLICATIONS OF MATHEMATICAL LOGIC Walter A. Carnielli and Luiz Paulo de Alcantara, Editors (Contemporary Mathematics, Volume 69)

This volume constitutes the proceedings of the Seventh in logic has become important because various areas-such Latin American Symposium on Mathematical Logic, held as software engineering, mathematical biology, physics, and July 29-August 2, 1985, at the University of Campinas in linguistics-now appear to need mathematical methods of the Brazil. Striking a balance between breadth of scope and kind studied in logic. depth of results, the papers in this collection range over a 1980 Mathematics Subject Classifications: 03, 06, 01, 04, 08, 1 0, 52, variety of topics in classical and non-classical logics. The 60, 68, 81 and others book proyides readers with an introduction to the active ISBN 0-8218-5076-8, LC 87-33651 lines of research in mathematical logic and particularly ISSN 0271-4132 emphasizes the connections to other fields, especially 256 pages (softcover), March 1988 Individual member philosophy, $17, List price $28, computer science, and probability theory. The Institutional member $22 potential applicability of the mathematical methods studied To order, please specify CONM/69NA

Shipping/Handling: 1st book $2, each additional Sl, maximum $25; by air, 1st book $5, each additional S3, maximum SlOO Prepayment required. Order from American Mathematical Society, P.O. Box 1571, Annex Station Providence, Rl 02901-9930, or call toll free 800-556-7774 to charge with VIsa or MasterCard

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 883 Invited Speakers and Special Sessions

Invited Speakers Ralph Byers, Numerical linear algebra at AMS Meetings Bruce Crauder and Sheldon Katz, Algebraic geometry Tyrone Duncan, Control theory The individuals listed below have accepted invitations William Fleissner, Applications of set theory to address the Society at the times and places in­ James Foran, Real analysis dicated. For some meetings, the list of speakers is William Mark Goldman, Flat bundles and geometric incomplete. structures Steven E. Hurder and Noberto Salinas, Operatory Lawrence, October 1988 theory and applications to geometry Bj0m Dahlberg Peter Scott Daniel Katz and Jeffery Lang, Commutative algebra Steven E. Hurder Sidney M. Webster Jill Pipher and Gregory Vechota, Potential theory and partial differential equations in nonsmooth domains Claremont, November 1988 Peter Scott, 3-manifolds William Jacob Francis Bonahon Robert Brooks November 1988 Meeting in Claremont Far Western Section Phoenix, January 1989 Associate Secretary: Lance W. Small John B. Conway Cathleen S. Morawetz Deadline for organizers: Expired Percy Alec Deift (AMS-MAA) Deadline for consideration: August 2, 1988 David Fried Steve Smale Francis Bonahon and David Gabai, Low dimensional R. L. Graham (AMS-MAA) geometry (AMS-MAA) Luc Tartar Robert Borrelli and Courtney S. Coleman, Computers Peter Landweber and software in mathematical research and Topics Robert Brooks and S. -Y. Cheng, The spectrum of the Organizers Laplacian of Special Sessions Stavros N. Busenberg, Differential and difference equa­ The list below contains all the information about tions Special Sessions at meetings of the Society available William Jacob and Adrian Wadsworth, Division alge­ at the time this issue of Notices went to the printer. bras The section below entitled Information for Organizers describes the timetable for announcing the existence January 1989 Meeting in Phoenix of Special Sessions. Associate Secretary: Lance W. Small Deadline for organizers: Expired consideration: September 21, 1988 October 1988 Meeting in Lawrence Deadline for Central Section Melvyn S. Berger, Mathematics of nonlinear science Associate Secretary: Andy Roy Magid John B. Conway, Harry Gonshor, and Martin Kruskal, Deadline for organizers: Expired Surreal numbers Deadline for consideration: August 2, 1988 Percy Deift,.Integrable systems Andrew Acker, Partial differential equations-Geometric David Eisenbud and Craig Huneke, Commutative equations algebra and algebraic geometry John K. Beem and Phillip E. Parker, Geometry and mathematical physics

884 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Invited Speakers and Special Sessions

Larry C. Grove and M. F. Newman, Computational mittee from a list of proposed Special Sessions in essentially group theory the same manner as Invited Speakers are selected. The William A. Harris, Singular perturbation theory number of Special Sessions at a Summer or Annual Meeting Victor C. Katz and Florence Fasanelli, History of is limited. The algorithm that determines the number of Mathematics Special Sessions allowed at a given meeting, while simple, is not repeated here, but may be found in "Rules for Special Albert Mardin and Burton Rodin, Computational Sessions" which can be found on page 614 in the Aprill988 aspects of complex analysis issue of Notices. Sidney Port, Stochastic processes Each Invited Speaker is invited to generate a Special Marc A. Rieffell, Operator algebras and geometry Session, either by personally organizing one or by having a Hal L. Smith, Mathematics in population biology Special Session organized by others. Proposals to organize a Special Session are sometimes requested either by the April1989 Meeting in Worcester Program Committee or by the Associate Secretary. Other Eastern Section proposals to organize a Special Session may be submitted Associate Secretary: W. Wistar Comfort to the Associate Secretary in charge of that meeting (who is Deadline for organizers: July 15, 1988 an ex-officio member of the committee and whose address Deadline for consideration: January 4, 1989 may be found below). These proposals must be in the hands of the Program Committee well in advance of the May 1989 Meeting in Chicago meeting and, in any case, at least nine (9) months prior Central Section to the meeting at which the Special Session is to be held Associate Secretary: Andy Roy Magid in order that the committee may consider all the proposals Deadline for organizers: August 15, 1988 for Special Sessions simultaneously. Proposals that are sent Deadline for consideration: February 8, 1989 to the Providence office of the Society, to the Notices, or Jeffery Bergen, Noncommutative ring theory directed to anyone other than the Associate Secretary will Jonathan Cohen, Numerical methods in harmonic have to be forwarded and may not be received in time to be analysis considered for acceptance. Stephen Doty, Algebraic groups and related topics It should be noticed that Special Sessions must be Christine Haught, Recursion theory announced in the Notices in such a timely fashion that Cary Huffman and Neal Brand, Codes and designs any member of the Society who so wishes may submit an abstract for consideration for presentation in the Special August 1989 Meeting in Boulder Session before the deadline for such consideration. This Associate Secretary: Andy Roy Magid deadline is usually three (3) weeks before the Deadline for Deadline for organizers: November 15, 1988 Abstracts for the meeting in question. Deadline for consideration: April 25, 1989 Special Sessions are very effective at Sectional Meetings and can usually be accommodated. They are selected by the October 1989 Meeting in Hoboken Committee to Select Hour Speakers for the Section. The Eastern Section processing of proposals for Special Sessions for Sectional Associate Secretary: W. Wistar Comfort Meetings is handled by the Associate Secretary for the Deadline for organizers: January 15, 1989 Section, who then forwards the proposals to the Committee Deadline for consideration: August 9, 1989 to Select which makes the final selection of the proposals. Each Invited Speaker at a Sectional Meeting is invited to October 1989 Meeting in Muncie organize a Special Session. Just as for national meetings, Central Section · no Special Session at a Sectional Meeting may be approved Associate Secretary: Andy Roy Magid so late that its announcement appears past the deadline Deadline for organizers: January 15, 1989 after which members can no longer send abstracts for Deadline for consideration: August 9, 1989 consideration for presentation in that Special Session. The Society reserves the right of first refusal for the publication of proceedings of any Special Session. These Information for Organizers proceedings appear in the book series Contemporary Math­ Special Sessions at Annual and Summer Meetings are held ematics. under the supervision of the Program Committee for Na­ More precise details concerning proposals for and orga­ tional Meetings. They are administered by the Associate nizing of Special Sessions may be found in the "Rules for Secretary in charge of that meeting with staff assistance Special Sessions" or may be obtained from any Associate from the Meetings and Editorial Departments in the Society Secretary. office in Providence. According to the "Rules for Special Sessions" of the Society, Special Sessions are selected by the Program Com-

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 885 Invited Speakers and Special Sessions

Send Proposals for Special Sessions to the Information for Speakers Associate Secretaries A great many of the papers presented in Special Sessions by the Associate The programs of sectional meetings are arranged but any Secretary for the section in question: at meetings of the Society are invited papers, member of the Society who wishes to do so may submit Far Western Section (Pacific and Mountain) an abstract for consideration for presentation in a Special Lance W. Small, Associate Secretary Session, provided it is received in Providence prior to Department of Mathematics the special early deadline announced above and in the University of California, San Diego La Jolla, CA 92093 announcements of the meeting at which the Special Session (Telephone 619-534- 3590) has been scheduled. Contributors should know that there Central Section is a limitation in size of a single special session, so that it is sometimes true that all places are filled by invitation. Andy Roy Magid, Associate Secretary Department of Mathematics Papers not accepted for a Special Session are considered as University of Oklahoma ten-minute contributed papers. 601 Elm PHSC 423 Abstracts of papers submitted for consideration for pre­ Norman, OK 73019 sentation at a Special Session must be received by the (Telephone 405-325-2052) Providence office (Editorial Department, American Mathe­ Eastern Section matical Society, P. 0. Box 6248, Providence, Rl 02940) by W. Wistar Comfort, Associate Secretary the special deadline for Special Sessions, which is usually Department of Mathematics three weeks earlier than the deadline for contributed papers Wesleyan University for the same meeting. The Council has decreed that no Middletown, CT 06457 paper, whether invited or contributed, may be listed in the (Telephone 203- 34 7-9411) program of a meeting of the Society unless an abstract of the Southeastern Section paper has been received in Providence prior to the deadline. Frank T. Birtel, Associate Secretary Department of Mathematics Tulane University New Orleans, LA 70118 (Telephone 504-865-5646) As a general rule, members who anticipate organizing Special Sessions at AMS meetings are advised to seek approval at least nine months prior to the scheduled date of the meeting. No Special Sessions can be approved too late to provide adequate advance notice to members who wish to participate.

ASYMPTOTIC BEHAVIOR OF DISSIPATIVE Hausdorff dimension, and the stability of the flow on the SYSTEMS global attractor under perturbation. The remainder of the book deals with particular equations occurring in applications Jack K. Hale and especially emphasizes delay equations, reaction-diffusion (Mathematical Surveys and Monographs, Volume 25) equations, and the damped wave equations. In each of the examples presented, the author shows how to verify the This book is directed at researchers in nonlinear ordinary existence of a global attractor, and, for several examples, he and partial differential equations and at those who apply discusses some properties of the flow on the global attractor. these topics to other fields of science. About one third of the book focuses on the existence and properties of the flow on 1980 Mathematics Subject Classifications: 34, 35, 58 the global attractor for a discrete or continuous dynamical ISBN 0-8218-1527-X, LC 87-33495 system. The author presents a detailed discussion of abstract ISSN 0076-5376 200 pages (hardcover), March 1988 smooth maps and properties and examples of asymptotically Individual member $32, List price $54, semigroups. He also covers some of the continuity properties Institutional member $43 of the global attractor under perturbation, its capacity and To order, please specify SURV/25NA

Shipping/Handling: 1st book $2, each additional Sl, maximum $25; by air, 1st book SS, each additional $3. maximum $100 Prepayment required. Order from American Mathematical Society, P.O. Box 1571, Annex Station Providence, Rl 02901-9930, or call toll free 800-556-7774 to charge with VIsa or MasterCard

886 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Call For Topics For 1990 Conferences

Suggestions are invited from mathematicians, either Sciences, a one-, two-, or three-week conference may singly or in groups, for topics of the various confer­ be proposed. ences that will be organized by the Society in 1990. The deadlines for receipt of these suggestions, as well as some relevant information about each of the con­ 1990 AMS Symposium in Pure Mathematics ferences, are outlined below. An application form to This symposium in pure mathematics has traditionally be used when submitting suggested topic(s) for any been conducted in the spring of even numbered of these conferences (except the Short Course Series) years in conjunction with a sectional meeting. The may be obtained by writing to the Meetings Depart­ symposium can be held independently of a sectional ment, American Mathematical Society, P.O. Box 6248, meeting and serves to honor great accomplishments Providence, RI 02940, or telephoning 401-272-9500. in mathematics. Proceedings are normally published Individuals willing to serve as organizers should by the Society as volumes in the series Proceedings of be aware that the professional meeting staff in the Symposia in Pure Mathematics. Society's Providence office will provide full support Topics in recent years have been: and assistance before, during, and after each of these 1982-Several complex variables, organized by YuM­ conferences. Organizers should also note that for all ToNG Sm of Stanford University conferences, except Summer Research Conferences, it 1984-Pseudodifferential operators and Fourier inte­ is required that the proceedings be published by the gral operators with applications to partial differential Society, and that proceedings of Summer Research equations, organized by FRANCOIS TREVEs of Rutgers Conferences are frequently published. A member of University the Organizing Committee must be willing to serve as 1987-The mathematical heritage of Herman Weyl, editor of the proceedings. organized by R. 0. WELLS, JR. of Rice University. All suggestions must include ( 1) the names and Deadline For Suggestions: September 1, 1988 affiliations of proposed members and chairman of the Organizing Committee; (2) a two- or three-page de­ tailed outline of the subject(s) to be covered, including the importance, timeliness of the topic, and estimated 1990 AMS Summer Institute attendance; (3) a list of the recent conferences in the same or closely related areas; (4) a tentative list Summer institutes are intended to provide an un­ of names and affiliations of the proposed principal derstandable presentation of the state of the art in speakers; ( 5) a list of likely candidates who would be an active field of research in pure mathematics and invited to participate and their current affiliations; and usually extend over a three-week period. Dates for (6) any other observations which may affect the size of a summer institute must not overlap those of the the conference and the amount of support required. Society's summer meeting, which at the time of this Any suggestions as to sites and dates should be made printing have not yet been determined. There should as early as possible in order to allow adequate time for be a period of at least one week between them. Pro­ planning. By action of the AMS Board of Trustees, the ceedings are published by the Society as volumes in the Meetings Department of the Society is responsible for series Proceedings ofSymposia in Pure Mathematics. the final selection of the site for each conference and Topics in recent years have been: for all negotiations with the host institution. Individ­ 1986-Representations of finite groups and related uals submitting suggestions for the conferences listed topics, organized by JONATHAN L. ALPERIN of the below are requested to recommend sites or geographic University of Chicago. areas which would assist the Meetings Department in 1987-Theta functions, organized by LEON EHREN­ their search for an appropriate site. In the case of Joint PREIS of Temple University and ROBERT GUNNING of Summer Research Conferences in the Mathematical Princeton University.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 ..... Call for Topics

1988-Operator theory/Operator algebras and applica­ 1988-Computational solution of nonlinear systems tions, organized by WILLIAM B. ARVESON of University of equations, organized by EUGENE ALLGOWER of of California, Berkeley, and RONALD G. DOUGLAS of Colorado State University. State University of New York at Stony Brook. 1989-The mathematics of random media, organized 1989-Several complex variables and geometry, orga­ by WERNER KOHLER of Virginia Polytechnic Institute nized by STEVEN G. KRANTz of Washington Univer­ and BENJAMIN WHITE of Exxon Research & Engineer­ sity. ing Company. Deadline For Suggestions: September 1, 1988 Deadline For Suggestions: September 1, 1988

1990 AMS-SIAM-SMB Symposium 1990 AMS-IMS-SIAM Summer Some Mathematical Questions in Biology Research Conferences in the This one-day symposium now under joint sponsorship Mathematical Sciences with the Society for Mathematical Biology is usually These conferences are similar in structure to those held in conjunction with the annual meeting of a held at Oberwolfach and represent diverse areas of biological society closely associated with the topic. mathematical activity, with emphasis on areas cur­ Papers from the symposia are published by the Society rently especially active. Careful attention is paid to as volumes in the series Lectures on Mathematics in subjects in which there is important interdisciplinary the Life Sciences. activity at present. Topics for the sixth series of Topics in recent years have been: one-week conferences, being held in 1988, are The 1985-Plant biology, organized by ROBERT M. MIURA mathematics and physics of order and disorder, Spa­ of the University of British Columbia. tial statistics and imaging, Mathematical developments 1986-Modeling circadian rhythms, organized by GAIL arising from linear programming, Geometric Problems A. CARPENTER of Northeastern University. in Fourier Analysis, Computational number theory, 1987-Models in population biology, organized by Current progress in hyperbolic systems: Riemann prob­ ALAN HASTINGS of the University of California, Davis. lems and computations, Mathematical problems posed 1988-Dynamics of excitable media, organized by by anisotropic materials, Geometric and topological HANs G. 0THMER of the University of Utah. invariants of elliptic operators, Elliptic genera and 1989-Sex allocation and sex change: Experiments and elliptic cohomology, Control theory & multibody sys­ models, organized by MARC MANGEL of the University tems. If proceedings are published by the Society, they of California, Davis. will appear as volumes in the series Contemporary Deadline For Suggestions: September 1, 1988 Mathematics. Deadline For Suggestions: February 1, 1989

1990 AMS-SIAM Summer Seminar 1990 AMS Short Course Series The goal of the summer seminar is to provide an environment and program in applied mathematics The AMS Short Courses consist of a series of in­ troductory survey lectures and discussions ordinarily in which experts can exchange the latest ideas and extending over a period of one and one-half days newcomers can learn about the field. Proceedings are published by the Society as volumes in the series starting immediately prior to the Joint Mathematics Meetings held in January and August each year. Each Lectures in Applied Mathematics. Topics in recent years have been: theme is a specific area of applied mathematics or mathematics used in the study of a specific subject or 1985-Reacting flows: Combustion and chemical reac­ tors, organized by G.S.S. LUDFORD of Cornell Univer­ collection of problems in one of the physical, biologi­ sity. cal, or social sciences, technology, or business. Topics 1987-Computational Aspects of VLSI Design with in recent years have been Computation Complex­ an Emphasis on Semiconductor Device Simulation, ity Theory (January 1988), Moments in Mathemat­ organized by RANDOLPH BANK of the University of ics (January 1987), Approximation Theory (January California, San Diego. 1986), Actuarial Mathematics (August 1985), Fair Al­ location (January 1985). Proceedings are published by the Society as volumes in the series Proceedings of

888 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Call for Topics

Symposia in Applied Mathematics, with the approval Submit suggestions to: Professor Stefan A. Burr, Chair­ of the Editorial Committee. man, Short Course Subcommittee, Department of Deadline for Suggestions: Suggestions for the January Computer Science, CUNY, City College, New York, 1990 course should have been submitted by July 1, NY 10031. 1988; suggestions for the August 1990 course should be submitted by December 1, 1988.

MATHEMATICAL INTUITIONISM: INTRODUCTION TO PROOF THEORY A. G. Dragalin (Translations of Mathematical Monographs, Volume 67)

In the area of mathematical logic, a great deal of attention is now being devoted to the study of nonclassical logics. Nonclassical logics are used in the theory of computations, 1980 Mathematics Subject Classification: 03 in information theory, ISBN 0-8218-4520-9, LC 87-32766 and for the description of systems ISSN 0065-9282 of heuristic programming. lntuitionistic logic is a particularly 224 pages (hardcover), March 1988 important nonclassical logic. The aim of this book is to Individual member $45, List price $75, present the most important methods of proof theory in Institutional member $60 intuitionistic logic and to acquaint the reader with the principal To order, please specify MMON0/67 NA axiomatic theories based on intuitionistic logic. The exposition, accessible to a wide audience, requires only an introductory course in classical mathematical logic.

Shipping/Handling: 1st book $2, each additional $1, maximum $25; by air, 1st book $5, each additional $3, maximum $100 Prepayment required. Order from American Mathematical Society, P.O. Box 1571, Annex Station Providence, Rl 02901-9930, or call toll free 800-556-7774 to charge with VIsa or MasterCard

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 889 Mathematical Sciences Conferences July 1988 Meetings and 20-31. Ninth Latin American School of Mathematics, Santiago, Chile. (Novem­ ber 1987, p. 1140) 22-28. Twelfth IMACS World Congress on Scientific Computation, Paris, France. (April 1988, p. 635) 24-30. International Conference on Rad­ icals - Theory and Applications, Sendai, Japan. (May/June 1988, p. 727) 24-August 5. Constructive Methods in Computing Science, Marktoberdorf, Ger­ THIS SECTION contains announcements of meetings and conferences of interest to many. (May/June 1988, p. 727) segment of the mathematical public, including ad hoc, local, or regional meetings, some First International Conference on and meetings or symposia devoted to specialized topics, as well as announcements 25-28. of regularly scheduled meetings of national or international mathematical organizations. Optimal Design and Analysis of Experi­ (Information on meetings of the Society, and on meetings sponsored by the Society, will ments, Neuchatel, Switzerland. (February be found inside the front cover.) 1988, p. 311) AN ANNOUNCEMENT will be published in Notices if it contains a call for papers, 25-28. Third KIT Mathematics Work­ and specifies the place, date, subject (when applicable), and the speakers; a second shop, Taejon, Korea. (May/June 1988, full announcement will be published only if there are changes or necessary additional p. 727) information. Once an announcement has appeared, the event will be briefly noted in each issue until it has been held and a reference will be given in parentheses to the month, 25-29. Third International Conference year, and page of the issue in which the complete information appeared. Asterisks (*) on Fibonacci Numbers and Their Appli­ mark those announcements containing new or revised information. cations, Pisa, Italy. (April 1988, p. 635) IN GENERAL, announcements of meetings and conferences held in North America carry 25-30. Third International Congress on only date, title of meeting, place of meeting, names of speakers (or sometimes a general Computational and Applied Mathematics, statement on the program), deadlines for abstracts or contributed papers, and source University of Leuven, Belgium. (August more of further information. Meetings held outside the North American area may carry 1987, p. 814) detailed information. In any case, if there is any application deadline with respect to participation in the meeting, this fact should be noted. All communications on meetings 25-30. International Symposium on En­ and conferences in the mathematical sciences should be sent to the Editor of Notices, gineering Mathematics, Beijing, China. care of the American Mathematical Society in Providence. (April 1988, p. 635) DEADLINES for entries in this section are listed on the inside front cover of each issue. In 27-August 3. Sixth International Con­ order to allow participants to arrange their travel plans, organizers of meetings are urged gress on Mathematical Education, Bu­ to appear in more than to submit information for these listings early enough to allow them dapest, Hungary. (March 1988, p. 463) one issue of Notices prior to the meeting in question. To achieve this, listings should be received in Providence SIX MONTHS prior to the scheduled date of the meeting. * 28-30. Algorithmic Aspects of Geometry and Algebra, Cornell University, Ithaca, New York. 1987-1988. Academic Year Devoted to April 17-December 24, 1988. INFORMATION: Cornell University, Differential Geometry, University of Mathematisches Forschungsinstitut Ober­ Mathematical Sciences Institute, 294 North Carolina, Chapel Hill, North Car­ wolfach (Weekly Conferences), Federal Caldwell Hall, Ithaca, New York 14853- olina. (November 1987, p. 1137) Republic of Germany. (October 1987, 2602, 607-255-8005. 198 7-1988. Academic Year Devoted to p. 995) 6. Nonstandard Analysis, Several Complex Variables, Mittag-Leffler October 1988-August 1989. Symposium 31-August University of Massachusetts, Amherst, Institute, Djursholm, Sweden. (January on Singularity Theory and Its Applica­ and Smith College, North­ 1987, p. 131) tions, University of Warwick, Coven­ Massachusetts, (January 1988, try, United Kingdom. (May/June 1988, ampton, Massachusetts. 19 8 8-198 9. Academic Year Devoted to p. 159) Operator Algebras, Mittag-Leffler Insti­ p. 724) tute, Djursholm, Sweden. (February 1988, January 1-December 23, 1989. Math­ p. 307) ematisches Forschungsinstitut Oberwol­ August 1988 fach (Weekly Conferences), Federal Re­ March 6-December 24, 1988. Math­ Annual Conference and public of Germany. (April 1988, p. 629) 1-5. Fifteenth ematisches Forschungsinstitut Oberwol­ Exhibition on Computer Graphics and fach (Weekly Conferences), Federal Re­ Interactive Techniques (SIGGRAPH '88), public of Germany. (March 1988, pp. Georgia World Congress Center, Atlanta, 457-458) Georgia. (October 1987, p. 999)

890 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings and Conferences

1-5. Thirty-eighth Gordon Research Con­ 9-12. Henstock Real Analysis Sympo­ Applications, Colorado State University, ference on Statistics in Chemistry and sium, University of Ulster, Coleraine, Fort Collins, Colorado. (October 1987, Chemical Engineering, New Hampton, Northern Ireland. (March 1988, p. 463) p. 999) New Hampshire. (April 1988, p. 636) 9-13. First International Symposium on 16-September 9. Summer School on 1-6. Group Actions and Invariant The­ Algebraic Structures and Number Theory, Dynamical Systems, International Centre ory, McGill University, Montreal, Que­ Hong Kong. (November 1987, p. 1140) for Theoretical Physics, Trieste (Italy). bec, Canada. (May/June 1988, p. 727) * 11-13. Mathematical Theory of Queueing (April1988, p. 636) 1-7. International Symposium on Num­ Systems, Cornell University, Ithaca, New 17-24. Functional Analysis/Optimisation, ber Theory and Analysis, Tsing Hua York. Australian National University, Canberra. University, Beijing, People's Republic of (April 1988, p. 636) INFORMATION: Cornell University, China. (February 1988, p. 311) Mathematical Sciences Institute, 294 19-24. Georgia Topology Conference, 1-12. Conference on Computational Tools Caldwell Hall, Ithaca, New York 14853- University of Georgia, Athens, Georgia. of Numerical Approximation and Con­ 2602, 607-255-8005. (February 1988, p. 312) trol, Montana State University, Bozeman, 20-26. Groups, Pusan, Republic of Ko­ Montana. (May/June 1988, p. 728) 12-14. International Conference on Math­ rea. (June 1987, p. 686) *2-5. Software Tools for Parallel Su­ ematical Modelling in Sciences and Tech­ 21-25. Crypto 88 Conference, Santa Bar­ percomputing, Cornell University, Ithaca, nology, Madras, India. (Note date change, bara, California. February 1988, p. 312) October 1987, p. 999) New York. 21-27. Seventeenth International Con­ 13-17. Kaehler Geometry and Several INFORMATION: Cornell University, gress of Theoretical and Applied Mechan­ Complex Variables, Eastern University, Mathematical Sciences Institute, 294 ics, Grenoble, France. (January 1987, Charleston, Illinois. (April 1988, p. 636) Caldwell Hall, Ithaca, New York 14853- p. 135) 2602, 607-255-8005. 13-17. NSF-CBMS Regional Confer­ 21-27. International Conference on Gen­ ence: Rigidity in Several Complex Vari­ eral Algebra, KremsfDonau, Austria. 4-11. Algebraic Logic Conference, Bu­ ables, Eastern Illinois University, Charles­ (November 1987, p. 1140) dapest, Hungary. (October 1987, p. 999) ton, Illinois. (May /June 1988, p. 728) 21-September 7. Eighteenth Ecole d'ete 5-8. Second Boston Workshop for Math­ 14-18. Institute of Mathematical Statis­ de Calcul des Probabilities, Saint-Flour, ematics Faculty, Wellesley College, tics Annual Meeting, Fort Collins, Col­ Cantal. (March 1988, p. 463) Wellesley, Massachusetts. (January 1988, orado. (October 1987, p. 999) * 22. Third Annual AI and Simulation p. 159) 14-27. Harmonic Analysis on Reduc­ Workshop, Minneapolis, Minnesota. 6-7. AMS Short Course on Chaos and tive Groups, Bowdoin College, Bruns­ Fractals: The Mathematics Behind the wick, Maine. (This conference has been SPONSORS: AAAI/SCS/IMACS. Computer Graphics, Providence, Rhode postponed. For further details, see the INFORMATION: P. Fishwick, Depart­ Island. (April 1988, p. 597) announcement for July 30-August 12, ment of Computer and Information 1989, in this section of Mathematical Science, University of Florida, Build­ INFORMATION: M. Foulkes, Ameri­ Sciences Meetings and Conferences) ing CSE, Gainesville, Florida 32611. can Mathematical Society, Post Office Box 6248, Providence, Rhode Island 15-19. New Directions in Dynamical 22-25. 1988 Joint Statistical Meetings, 02940. Systems, Brown University, Providence, New Orleans, Louisiana. (June 1987, Rhode Island. (October 1987, p. 999) p. 686) 8-12. AMS Centennial Celebration, Prov­ 15-19. International Symposium on Ap­ 22-25. Fifth International Conference on idence, Rhode Island. (April1987, p. 553) proximation, Optimization and Comput­ the New Quality Philosophy in Statistical INFORMATION: For further details, see ing, University of Regina, Saskatchewan, Research and Statistical Education (Satel­ the Meetings section of the April 1988 Canada. (February 1988, p. 312) lite Conference), New Orleans, Louisiana. Notices. 15-19. Advances in the Theory of Frechet (March 1988, p. 464) Spaces, Istanbul, Turkey. (May/June 1988, 22-26. Conference on Categorical Topol­ Fifth Marcel Grossman Meeting,. 8-12. p. 728) ogy and its Relations to Algebra, Analysis Perth, Western Australia. (April 1988, * 15-19. Joint Logic Programming Meet­ and Combinatorics, Prague, Czechoslo­ p. 636) ing, University of Washington, Seattle, vakia. (October 1987, p. 1000) of SeH-Homotopy Equiv­ 8-13. Spaces Washington. 22-26. International Conference on Op­ Centre de recherches mathema­ alences, erator Theory: Advances and Applications, Universite de Montreal, Montreal, INFORMATION: K. A. Bowen, 313 Link tiques, University of Calgary, Alberta, Canada. Quebec. (February 1988, p. 311) Hall, Syracuse University, Syracuse, New York 13210, 315-423-2466 or (November 1987, p. 1140) 8-14. Conference on Algebraic Logic, 2467. 22-26. 20th Nordic Congress of Mathe­ Hungary. (May/June 1988, Budapest, maticians, Trondheim, Norway. (Novem­ p. 728) 15-20. Universal Algebra, Karlovy Vary, ber 1987, p. 1140) 9-12. International Symposium in Real Czechoslovakia. (March 1988, p. 463) 22-26. Conference in Dift'erential Geom­ Analysis, University of Ulster, Coleraine, 16-19. Institute of Mathematical Statis­ etry In Honor of M. Perdigio Do Carmo, Northern Ireland. (February 1987,p. 364) tics Symposium on Probability and its

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 891 Meetings and Conferences

Rio de Janeiro, Brazil. (March 1988, Biostatistics, Innsbruck, Austria. (April INFORMATION: A. Eisinberg, Systems p. 464) 1988, p. 637) Department, University of Calabria, 22-27. Conference on Numerical Meth­ Province de Cosenza, Italy. ods and Applications, Sofia, Bulgaria. September 1988 18-24. DMV-Jahrestagung 1988, Re­ (March 1988, p. 464) gensburg, Federal Republic of Germany. * 5-23. Workshop on Dynamical Systems, 22-September 16. Summer School: Jor­ (March 1988, p. 464) dan Algebras and Jordan Triples: Geome­ Trieste, Italy. 19-23. Sixth International ColloquiUID try and Analysis, Poitiers, France. (March INFORMATION: International Centre on Differential Geometry, Universidad 1988, p. 464) for Theoretical Physics, Trieste, Italy, de Santiago de Compostela, Santiago 23-30. 1988 Association for Symbolic P. 0. B. 586, Miramare, StradaCostiera de Compostela, Spain. (February 1988, Logic European Summer Meeting, Padova, 11. Telephone: 2240-1. p. 312) Italy. (January 1988, p. 160) 6-10. International Neural Network So­ 19-23. Histoire de Ia Relativite Generale, 27-31. First Islamic Countries Confer­ Marseille, France. (March 1988, p. 464) ence on Statistical Sciences, Lahore, Pak­ ciety 1988 Annual Meeting, Boston, Mas­ 23-25. Fall Foliage Combinatorial Group istan. (March 1988, p. 464) sachusetts. (February 1988, p. 312) * 10-11. Fourth Seminar on Algebra, Theory Conference, Shaker Village, New 28-31. European Conference for Math­ Hampshire. (May/June 1988, p. 729) ematics in Industry, Glasgow, Scotland. Mashad University, Mashad-Iran. 24-25. Last Fall Foliage Topology Sem­ (March 1988, p. 464) PROGRAM: The seminar is devoted to inar, Shaker Village, New Hampshire. * 29-31. IEEE Workshop on Languages group theory, ring theory, algebrated (MayfJune 1988, p. 729) for Automation, University of Maryland, geometry and related areas. 25-30. International Symposium in Honor College Park, Maryland. INFORMATION: M. R. R. Moghaddam, Department of Mathematics, Statis­ of Rene Thom, Paris, France. (February INFORMATION: P. A. Ligomenides, tics and Computer Science, Post Of­ 1988, p. 312) Cybernetics Research Laboratory, Elec­ fice Box 1159, Mashad University, 26-30. Computational Intelligence '88, trical Engineering Department, Uni­ Mashad-Iran. University of Milano, Milano, Italy. versity of Maryland, College Park, (March 1988, p. 465) Maryland 20742, 301-454-6842. 12-16. Theorie des Nombres, Marseille, 26-0ctober 1. Fifth International Confer­ France. (March 1988, p. 464) 29-September 1. Fourteenth Interna­ ence on Complex Analysis, Halle, German tional Conference on Very Large Databases, 12-16. Eurographics '88: Research, Prac­ Democratic Republic. (October 1987, Hyatt Regency Hotel, Long Beach, Cali­ tice and Experience, Nice, France. (March p. 1000) fornia. (March 1988, p. 464) 1988, p. 464) * 28-0ctober 9. Sixth International Sum­ 29-September 2. ICO Topical Meeting * 12-16. Third International Symposium mer School on Probability Theory and on Optical Computing, Orsay, France. on Systems Analysis and Simulation, Mathematical Statistics, Varna, Bulgaria. Berlin, German Democratic Republic. (October 1987, p. 1000) INFORMATION: Department of Prob­ 29-September 2. Orbit Method in Repre­ INFORMATION: A. Sydow, ZK.I der ability and Statistics, Post Office Box sentation Theory, Copenhagen, Denmark. ADW der DDR, Kurstrasse 33, DDR- 373, 1090 Sofia, Bulgaria. (October 1987, p. 1000) 1086 Berlin, German Democratic Re­ 30-0ctober 1. Sixteenth Annual Math­ 29-September 2. Workshop on Sym­ public. plectic Topology, Mathematical Sciences ematics and Statistics Conference, Miami 13-16. Workshop on Arboreal Group University, Oxford, Ohio. (April 1988, Research Institute, Berkeley, California. Theory, Mathematical Sciences Research (February 1988, p. 321) p. 637) Institute, Berkeley, California. (January 29-September 2. COMPSTAT '88 Eighth 1988, p. 160) Symposium on Computational Statistics, 13-23. Heyting Summer School and Con­ October 1988 Copenhagen, Denmark. (March 1988, ference on Mathematical Logic, Chaika 1. Algebra Day, Carleton University, p. 464) * near Varna, Bulgaria. (February 1988, Ottawa, Canada. 29-September 2. Harmonic Analysis p. 312) in Lie Groups, Copenhagen, Denmark. ORGANIZER: Ottawa-Carleton Insti­ 14-16. Fifth IMA International Con­ (April 1988, p. 637) tute for Graduate Studies and Re­ ference on Control Theory, Strathclyde, search in Mathematics and Statistics, 29-September 2. International Confer­ Scotland. (April 1988, p. 637) ence on Mathematical Statistics, Olsztyn, University of Ottawa-Carleton Uni­ 14-17. Meeting of Topology, Centro Poland. (April 1988, p. 637) versity. Congressi Cocumella, Sorrento (Na­ INVITED SPEAKERS: M. Lorenz, North­ 29-September 2. Thirteenth Interna­ ples), Italy. (March 1988, p. 464) em Illinois University; D. J. S. Robin­ tional Symposium on Mathematical Pro­ * 18-21. IMACS International Sympo­ son, University of Illinois at Urbana­ gramming, Tokyo, Japan. (April 1988, sium on System Modelling and Simula­ Champaign; A. R. Wadsworth, Uni­ p. 637) tion, University of Calabria, Province de versity of California, San Diego; T. 30-September 2. Ninth Annual Meeting Cosenza, Italy. Wakamatsu, Jobu, Japan. of the International Society of Clinical

892 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings and Conferences

INFORMATION: L. Ribes, Department 21-22. Tenth Midwest Probability Collo­ write to Department of Mathemat­ of Mathematics and Statistics, Car­ quium, Northwestern University, Evanston, ics, University of Notre Dame, Notre leton University, Ottawa, Ontario K1S Illinois. (MayjJune 1988, p. 730) Dame, Indiana 46556. 5B6, Canada. Telephone: 613-564- Twenty-ninth Foundations of * 24-26. 13-1 7. French-Israeli Conference on 7131. Computer Science, White Plains, New Combinatorics and Algorithms, Israel. York. 3-5. Knowledge-Based Robot Control, (February 1988, p. 313) Bonas, France. (MayjJune 1988, p. 729) INFORMATION: C. Papadimitrou, De­ 18-19. Quantitative Approaches to Di­ 4-6. Colloque Ergoia 88 Ergonomie et partment CSE, University of Califor­ abetes, Sydney, Australia. (April 1988, Intelligence Artificelle, Biarritz, France. nia/San Diego, La Jolla, California p. 638) (April 1988, p. 637) 92043, 619-534-2086. 20-25. Eighth Annual Conference: To­ *5-7. IFAC/IMACS/IFIP Symposium on 28-29. Seventeenth Midwest Differential wards a Whole Decade for Scientific and Robot Control SYROCO '88, Karlsruhe, Equations Conference, Iowa State Uni­ Technological Cooperation for Interna­ Federal Republic of Germany. versity, Ames, Iowa. (MayjJune 1988, tional Development, Jamaica, West In­ dies. (March 1988, p. 465) INFORMATION: G. Hirzinger, DFVLR p. 730) Oberpfaffenhofen, D-8031 Wessling, 28-30. Central Sectional Meeting, Law­ 21-December 16. College on Global Methods in Oberbayern, Federal Republic of Ger­ rence, Kansas. Geometric and Topological 1988, many. Analysis, Trieste, Italy. (March INFORMATION: J. Balletto, American p. 465) 9-13. International Conference on Com­ Mathematical Society, Post Office Box puter Languages, Castle Premier, Miami 6248, Providence, Rhode Island 02940. December 1988 Beach, Florida. (March 1988, p. 465) *3D-November 4. Ninth International on Expert Sys­ 10-12. IEEE Workshop on Visual Lan­ Conference on Computer Communications, *5-8. IMACS Conference guages, University of Pennsylvania, Pitts­ Tel Aviv, Israel. tems for Numerical Computing, Purdue burgh, Pennsylvania. (March 1988, p. 465) University, West Lafayette, Indiana. * I 0-12. Frontiers '88: Second Symposium SPONSORS: The International Council for Computer Communication; the PURPOSE: This international confer­ on the Frontiers of Massively Parallel ence is designed to aid nonspecialists Computation, George Mason University, Institute of Electrical and Electronic Informa­ in the use of computers for scientific Fairfax, Virginia. Engineers (Israeli Section); tion Processing Association (Israel). applications. It is intended to bring INFORMATION: J. R. Fischer, Code INFORMATION: The Secretariat, Ninth together active researchers, to allow viewpoints, 635, NASA Goddard Space Flight International Conference on Com­ for an exchange of ideas, Center, Greenbelt, Maryland 20771, puter Communications - ICCC 88, and techniques. 301-286-3464. Post Office Box 50006, Tel Aviv PAPERs: The conference will include contributed papers. Au­ 61500, Israel. Telephone: 972/3/ invited and Pro­ pa­ 10-14. Workshop on Mathematical 654571. thors who wish to contribute gramming, Catholic University of Rio De pers should submit 3 copies of an Janeiro, Brazil. (MayjJune 1988, p. 729) 31-November 18. Workshop in Math­ extended abstract (2-3 pages, typed 12-14. Sensor-Based Robots: Algorithms ematical Ecology, Trieste, Italy. (March single-spaced on 8 by 11 inch pa­ and Architectures, Bonas, France. 1988, p. 465) per) no later than July 31, 1988 to (May /June 1988, p. 729) the address given below. All papers submitted will be reviewed. Authors * 12-15. Discrete Mathematics and Com­ November 1988 puter Science, University of Montreal, will be notified regarding acceptance Montreal, Quebec, Canada. 4-5. Southeast Differential Equations of their contribution no later than Conference, Athens, Georgia. (March 1988, September 15, 1988. INVITED SPEAKERs: G. Brassard; M. p. 465) INFORMATION: Abstracts and requests Cosnard; J. Della Dora; J. Fonlupt; should be addressed 12-13. 846th Meeting of the AMS, for information P. Frankl; P. L. Hammer; P. Hell; F .. Houstis, Department of Com­ Claremont, California. to E. Jaeger; M. Nivat; F. Robert; N. Sauer; puter Science, Purdue University, West D. Therien; W. Tutte; M. Watkins. INFORMATION: J. Balletto, American Lafayette, Indiana 4 7907, 317-494- ORGANIZERS: A. Achache; G. Hahn; Mathematical Society, Post Office Box 6003. M. Pouzet; I. G. Rosenberg. 6248, Providence, Rhode Island 02940. INFORMATION: G. Hahn, Departement 6-8. First International Conference on d'I. R. 0., Universite de Montreal, C. * 12-13. Twenty-fourth Midwest Partial Matter Elements Analysis, Guangzhou, P. 6128, Succ. A, Montreal, Quebec Differntial Equations Seminar, University Guangdong, People's Republic of China. H3C3J7 Canada. Telephone: 514-343- of Notre Dame, Notre Dame, Indiana. (October 1987, p. 1000) 6806. INFORMATION: N. K. Stanton, 219- 12-15. International Conference on Com­ 239-7436; Mei-Chi Shaw, 219-239- puter Vision, Tarpon Springs, Florida. 6537; G. Roberts, 219-239-5402; or (May /June 1988, p. 730)

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 893 Meetings and Conferences

12-17. International Course on Com­ 9-13. Sixth Texas International Sympo­ 15-17. 1989 New Zealand Mathemat. putational Geometry, Dipartimento di sium on Approximation Theory, College ics Colloquium, Palmerston North, New Matematica, Universita, Catania, Italy. Station, Texas. (April 1988, p. 638) Zealand. (May/June 1988, p. 731) (May/June 1988, p. 730) 9-14. K-Theory and Dynamics, Uni­ 19-20. Central Section Meeting, Loyola 13-15. IMA/SIAM International Con­ versity of Florida, Gainesville, Florida. University, Chicago, Illinois. ference on Mathematics of Signal Pro­ (April 1988, p. 638) INFORMATION: J. Balletto, American cessing, Warwick, England. (April 1988, 11-14. Joint Mathematics Meetings, Mathematical Society, Post Office Box p. 638) Phoenix, Arizona. (Aprill987, p. 553) 6248, Providence, Rhode Island 02940. * 13-17. KAC-Moody Lie Algebras and INFORMATION: H. Daly, American Physics Conference, North Carolina State 23-27. International Conference on Com­ Mathematical Society, Meetings De­ University, Raleigh, North Carolina. puting and Information, Toronto, On­ partment, Post Office Box 6248, Prov­ tario, Canada. (April 1988, p. 638) SPEAKERS: V. G. Kac; R. V. Moody; idence, Rhode Island 02940. D. Olive. 28-June l 0. AMS-SIAM Summer Sem­ INFORMATION: Conference Organiz­ 15-20. American Association for the Ad­ inar on the Mathematics of Random Me­ ing Committee, Department of Math­ vancement of Science Annual Meeting, dia, Virginia Polytechnic Institute and ematics, North Carolina State Univer­ San Francisco, California. (May/June State University, Blacksburg, Virginia. sity, Raleigh, North Carolina 27695- 1988, p. 731) (May/June 1988, p. 731) 8205, 919-737-3968 or 737-2370. 23-27. Workshop on Algorithms, Word 29-June l. Third International Confer­ Problems, and Classification in Combina­ ence in Mathematics: Fractional Calculus * 14-16. Raj Chandra Bose Memorial torial Group Theory, Mathematical Sci­ and Its Applications, Nihon University, Conference on Combinatorial Mathemat­ ences Research Institute, Berkeley, Cali­ Tokyo, Japan. (May/June 1988, p. 731) ics and Applications, Calcutta, India. fornia. (February 1988, p. 313) ORGANIZERS: Indian Statistical Insti­ June 1989 tute in collaboration with the Calcutta February 1989 Statistical Association. 5-16. Workshop on the Geometry of INFORMATION: All correspondence and 21-23. Seventeenth Annual Computer Hamiltonian Systems, Mathematical Sci­ inquiries should be addressed to A. R. Science Conference, Commonwealth Con­ ences Research Institute, Berkeley, Cali­ Rao, Secretary, Bose Conference, Di­ vention Center, Louisville, Kentucky. fornia. (April 1988, p. 638) vision of Theoretical Statistics and (MayfJune 1988, p. 731) * 12-16. Computers and Mathematics, Mathematics, Indian Statistical Insti­ Massachusetts Institute of Technology, tute, 203 B. T. Road, Calcutta 700 Cambridge, Massachusetts. March 1989 035, India. CONFERENCE THEMES: This confer­ 19-22. ENAR Spring Meeting, Lexing­ ence will focus on the use of the 27-31. Holiday Symposium on Fermat's ton, Kentucky. (March 1988, p. 465) Last Theorem, New Mexico State Univer­ computer as a research tool in the sity, Las Cruces, New Mexico. (May /June mathematical sciences. Emphasis will 1988, p. 730) April1989 be placed on the current and potential use of computer algebra and computer 3-6. IEEE International Conference on graphics as a research tool. The pro­ January 1989 Control and Applications, Jerusalem, Is­ gram consists of invited addresses, rael. (April 1988, p. 638) 2-5. International CoUoquium in Ring contributed papers and minicourses, l 0-13. IEEE Artificial Neural Networks Theory, Bar-Ilan University, Ramat-Gan, mathematical software tutorials, and Conference, Sheraton International Con­ Israel. (May /June 1988, p. 730) a special computer graphics event. ference Center, Reston, Virginia. (Note CoNTRIBUTED PAPERS: Drafts are due 4-6. American Statistical Association date change, March 1988, p. 465) by November 30, 1988, to E. Kaltofen, Winter Conference: Statistics in Soci­ 15-16. Eastern Section Meeting, Col­ Department of Computer Science, ety, San Diego, California. (March 1988, Holy Cross, Worcester, Mas­ RPI, Troy, New York 12180-3590. p. 465) lege of the sachusetts. (May/June 1988, p. 731) CONTRIBUTED MINICOURSES: Propos­ 6-12. Nonlinear Wave Equations, George als for 2- to 3-hour courses are due Mason University, Fairfax, Virginia. by October 10, 1988, to P. Gianni, (April 1988, p. 638) May 1989 IBM Research, Post Office Box 218, 8-10. Symposium in Honor of the Seven­ 8-10. Forty-Third Annual Quality Con­ Yorktown Heights, New York 10598. tieth Birthday of Ted Harris, Los Angeles, gress, Toronto, Canada. (March 1988, INFORMA noN: H. Schmidt, Confer­ California. (May/June 1988, p. 730) p. 465) ence Secretary, 62 Eastview, Pleas­ 8-ll. First Caribbean Conference on 8-12. Workshop on Arithmetic Groups antville, New York 10570, 914-769- Fluid Dynamics, Saint Augustine, Trini­ and Buildings, Mathematical Sciences 2725. dad, West Indies. (June 1987, p. 686) Research Institute, Berkeley, California. (March 1988, p. 465)

894 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings and Conferences

20-September 6. Nineteenth Ecole d'ete de Calcul des Probabilities, Saint-Flour, July 1989 May 1990 Cantal. (March 1988, p. 466) 3-7. Computational Ordinary Differen­ 28-September 1. IFIP 89: Eleventh 25-31. Tenth International Conference tial Equations, London, England. (April World Computer Conference, San Fran­ on Pattern Recognition, Resorts Hotel, 1988, p. 638) cisco, California. (April 1988, p. 639) Atlantic City, New Jersey. (March 1988, 10-12. International Conference on Com­ 29-September 6. Forty-seventh Session p. 466) putational Techniques and Applications, of the International Statistical Institute, Griffith University, Brisbane. (April1988, Paris, France. (April 1988, p. 639) August1990 p. 639) 10-21. Microprogram on Noncommu­ 6-9. 1990 Joint Statistical Meetings, tative Rings, Mathematical Sciences Re­ September 1989 Anaheim, California. (March 1988, p. 466) search Institute, Berkeley, California. 16-0ctober 20. Sixth World Congress (May /June 1988, p. 731) on Medical Information, Beijing, China. January 1991 30-August 4. Sixteenth Annual Confer­ (April 1988, p. 639) 16-19. 97th Annual Meeting, San Fran­ ence and Exhibition on Computer Graph­ 25-29. Third International Conference cisco, California. ics and Interactive Techniques (SIG­ on the Theory of Groups and Related GRAPH '89), Hynes Auditorium, Boston, Topics, Australian National University, INFORMATION: H. Daly, American Massachusetts. (March 1988, p. 466) Canberra. (May /June 1988, p. 732) Mathematical Society, Post Office Box 30-August 12. Harmonic Analysis on Re­ 6248, Providence, Rhode Island 02940. ductive Groups, Bowdoin College, Bruns­ 1989 wick, Maine. (January 1988, p. 160) October August 1991 21-22. Eastern Section Meeting, Stevens Institute of Technology, Hoboken, New 17-21. American Sociological Associa­ August 1989 Jersey. (May/June 1988, p. 732) tion Annual Meeting, Cleveland, Ohio. 1-9. Eighteenth International Congress 27-28. Central Section Meeting, Ball (May/June 1988, p. 732) of the History of Science, Hamburg, 1- State University, Muncie, Indiana. 19-22. 1991 Joint Statistical Meetings, 5; Munich, 6-9, Federal Republic of (May /June 1988, p. 732) Atlanta, Georgia. (March 1988, p. 466) Germany. (April 1988, p. 639) 6-10. 1989 Joint Statistical Meetings, 1990 Washington, District of Columbia. (March January 1988, p. 466) 17-20. Joint Mathematics Meetings, 13-19. Fourth Conference on Differen­ Louisville, Kentucky. (April1987, p. 553) tial Equations and Applications, Rousse, INFORMATION: H. Daly, American Bulgaria. (May/June 1988, p. 731) Mathematical Society, Meetings De­ partment, Post Office Box 6248, Prov­ idence, Rhode Island 02940.

applications in many statistical settings, including selection FOUR SETS OF TABLES. among normal means using either the indifference-zone or PERCENTAGE POINTS OF MULTIVARIATE the subset approach and in multiple comparisons involving STUDENT t DISTRIBUTIONS contrasts among means. These and other applications are Robert E. Bechhofer and Charles W. Dunnett described in detail, and examples of the uses of the tables are (Selected Tables in Mathematical Statistics, Volume 11) given. In addition, the volume contains interpolation methods which extend the usefulness of the tables. This volume presents tables dealing With the central multivariate student t distribution in which there is a common 1980 Mathematics Subject Classification: 62 variance estimate in the denominators of the variates and the ISBN 0-8218-1911-9, LC 7 4-6283 numerators are equicorrelated. The tables contain one-sided ISSN 0094-8837 384 pages (hardcover), February 1988 and two-sided upper equicoordinate percentage points for Individual member $28, List price $46, this distribution. In addition, the volume provides tables based Institutional member $37 on the assumption that the variates have a certain block To order, please specify TABLES/11 NA correlation structure. The entries have been computed to an accuracy of 5 decimal places. Shipping/Handling: 1st book $2, each add'l $1. $25 max. By air. 1st book $5, each add'l $3, $100 max. These tables, prepared under the aegis of the Institute for Prepayment required. Order from AMS. P.O. Box 1571. Annex Mathematical Statistics, are considerably more comprehensive Station. Providence. Rl 02901-9930, or call 800-556-777 4 to than previously published tables of this type. They have use VISA or MasterCard.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 895 New AMS Publications

1980 Mathematics Subject Classifications: 57, 54, 26 ISBN 0-8218-0117-1, LC 88-14445 THE COLLECTED PAPERS 1680 pages, 2 volumes (hardcover), July 1988 OF R. H. BING Individual member $93, List price $155, Sukhjit Singh, Steve Armentrout, Institutional member $124 and Robert J. Daverman, Editors To order, please specify COLBING/N

A powerful mathematician and a great problem solver, R. H. Bing laid the foundation for a number of areas of topology. THE MATHEMATICAL HERITAGE Many of his papers have continued to serve as a source of OF HERMANN WEYL major theoretical developments and concrete applications R. 0. Wells, Jr., Editor in recent years. One outstanding example was Michael H. (Proceedings of Symposia in Pure Mathematics, Volume 48) Freedman's use of Bing's Shrinking Criterion to solve the four-dimensional Poincare Conjecture. Hermann Weyl was one of the most influential mathematicians This two-volume set brings together over one hundred of of the twentieth century. Viewing mathematics as an organic Bing's research, expository, and miscellaneous papers. whole rather than a collection of separate subjects, Weyl These works range over a great variety of topics in topology, made profound contributions to a wide range of areas, including the topology of manifolds, decomposition spaces, including analysis, geometry, number theory, Lie groups, and continua, metrization, general topology, and geometric mathematical physics, as well as the philosophy of science topology. In addition, there are a number of papers in the and of mathematics. The topics he chose to study, the areas of convex functions, linearity, and conformal varieties. lines of thought he initiated, and his general perspective on The introductory section in the first volume provides historical mathematics have proved remarkably fruitful and have formed background on Bing's life and achievements. the basis for some of the best of modern mathematical research. This collection will appeal to mathematicians in all areas, and especially those in topology, as well as students, historians, This volume contains the proceedings of the AMS Symposium and educators in the mathematical sciences, for it provides a on the Mathematical Heritage of Hermann Weyl, held in May complete historical summary of the mathematical events in 1987 at Duke University. In addition to honoring Weyl's great the life of the man and the mathematician, R. H. Bing. accomplishments in mathematics, the symposium also sought Contents to stimulate the younger generation of mathematicians by highlighting the cohesive nature of modern mathematics as I. R. H. Bing: An introduction seen from Weyl's ideas. The symposium assembled a brilliant An editorial preface array of speakers and covered a wide range of topics. All of R. H. Bing: A study of his life, by S. Singh the papers are expository and will appeal to a broad audience A chronology of R. H. Bing of mathematicians, theoretical physicists, and other scientists. Ph.D. students of R. H. Bing Contents R. H. Bing: October 20, 1914-April 28, 1986, Raoul Bott, On induced representations by R. D. Anderson and C. E. Burgess Dennis Sullivan, Differentiable structures on fractal-like sets, Abstracts by R. H. Bing determined by intrinsic scaling functions on dual Cantor sets II. Papers of R. H. Bing R. P. Langlands, Representation. theory and arithmetic Ill. Classifications of works of R. H. Bing David A. Vogan, Jr., Noncommutative algebras and unitary representations Publications of R. H. Bing: Classified by the year Roger Howe, The oscillator semigroup Publications of R. H. Bing: Classified by subject matter Roger Howe, The Classical Groups and invariants of binary Works not included in these volumes forms Permissions

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898 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New AMS Publications

James Arthur, Characters, harmonic analysis, and an are provided with each review. The fifth volume of this set L2-Lefschetz formula contains author and key indexes, making it very easy to locate J, Lepowsky, Perspectives on vertex operators and the Monster items written by a specific author or to get information about 1. M. Singer, Some problems in the quantization of gauge collections or conference proceedings dealing with global theories and string theories analysis. L. Nirenberg, Fully nonlinear elliptic equations Contents Robert L. Bryant, Surfaces in conformal geometry H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Volume 1 Algebraic cycles, Bott periodicity and the Chern characteristic Global analysis, analysis on manifolds General theory of differentiable manifolds map s.-T. Yau, Uniformization of geometric structures Infinite-dimensional manifolds R. G. Douglas, Elliptic invariants for differential operators Calculus on manifolds; nonlinear operators Michael Atiyah, New invariants of 3- and 4-dimensional Spaces and manifolds of mappings manifolds Volume 2 Clifford Henry Taubes, Moduli spaces and homotopy theory Variational problems in infinite-dimensional spaces R. Penrose, Fundamental asymmetry in physical laws Ordinary differential equations on manifolds; dynamical Edward Witten, Free fermions on an algebraic curve systems 1980 Mathematics Subject Classifications: 00, 01 Volume 3 ISBN 0-8218-1482-6 ISSN 0082-0717 Ordinary differential equations on manifolds; dynamical 352 pages (hardcover), July 1988 systems Individual member $28, List price $47, Volume 4 Institutional member $38 To order, please specify PSPUM/48N Partial differential equations on manifolds; differential operators Pseudogroups and general structures on manifolds ~----:· __ ._,,;;?p;ijjj,,llllllli~='G---·_.,.,~~~~~~~•rru w REVIEWS IN GLOBAL ANALYSIS, 1980-86 Volume 5 Introduction by Anthony J. Tromba Series contents Author index Key index The term "global analysis" refers to the general area of analysis on manifolds, in which the methods of modern 1980 Mathematics Subject Classification: 58 algebra, analysis, geometry, and topology are blended. ISBN 0-8218-0104-X, LC 88-10565 Although the beginnings of these ideas can be traced to 4060 pages, 5 volumes (softcover), August 1988 Individual member $177, List price $295, the 17th century, major contributions in this direction were Institutional member $236, Reviewer $148 made by Lie, Riemann, and Poincare toward the end of To order, please specify REVGL0/86N the last century, followed by the work of G. D. Birkhoff, E. Cartan, and Morse in the early part of this century. However, it is only in recent years that the subject has attained its present central position in mathematics. The subject has many rich applications to fields outside mathematics-such as mechanics, quantum physics, and general relativity-as New Series well as within mathematics itself. Today, this vital and active field is undergoing a virtual I History of Mathematics explosion of new and important results. Reviews in Global Analysis makes information about the most recent This series begins in 1988 with books about American contributions to this rapidly groWing field ·accessible both to mathematics during the past century. The titles in this series specialists working in global analysis, and to those in other will present historical perspectives on individuals who have areas of pure and applied mathematics. profoundly influenced the development of mathematics, as These five volumes contain the more than 18,000 reviews that well as those who have made great contributions to the appeared in Mathematical Reviews from 1980 through 1986 mathematical community, or will trace the development of and have a primary or a secondary classification in Global special areas of research. The following book is volume one Analysis (classification number 58). Relevant cross-references in this new series.

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JULY/AUGUST 1988, VOLUME 35, NUMBER 6 897 New AMS Publications

Standing orders are accepted for any book series published Mathematicians, historians of science, and students alike will by the Society. Proforma invoices are sent to standing order find this book illuminating and rewarding. That the lessons of customers prior to the publication of each new volume. the past can guide the resolution of present problems makes Shipment is made upon receipt of payment and publication. this book important reading for all who are concerned with the To begin a standing order for this new series or. for any development of mathematics. It will also make a fine addition other AMS series, please contact the Memt:>ership and Sales to any library collection. Department. Contents J. L. Synge, For the 100th birthday of the American Mathematical Society ~~~iili;i' 1:.,::te~:,;._,,.1 ,~1·"~1 George E. Andrews, J. J. Sylvester, Johns Hopkins, and A CENTURY OF MATHEMATICS Partitions IN AMERICA Solomon Lefschetz, Luther Pfaler Eisenhart D. V. Widder, Some mathematical reminiscences Peter L. Duren, Editor Stephen C. Kleene, The role of logical investigations in with the assistance of Richard A. Askey mathematics since 1930 and Uta C. Merzbach R. P. Boas, Memories of bygone meetings Hassler Whitney, Moscow 1935: Topology moving toward In the 100 years since the founding of the AMS, the America American mathematical community has grown from a small D. Montgomery, Oswald Veblen group heavily dependent on European mathematicians to P. R. Halmos, Some books or Auld Lang Syne a large and influential group that in many areas sets the Nathan Reingold, Refugee mathematicians in the United States standard for the rest of the world. By the 1930s, there was of America, 1933-1941: Reception and reaction a flourishing mathematical community to welcome the influx Solomon Lefschetz, Reminiscences of a mathematical of mathematicians fleeing Europe. These refugees supplied immigrant in the U.S. additional strength and new vigor to a field that increased Ivan Niven, The threadbare thirties dramatically as a result of World War II and the postwar Lipman Bers, The European mathematicians' migration to recognition of mathematics. America · Irving Kaplansky, Abraham Adrian Albert This volume, the first in the new History of Mathematics D. H. Lehmer, A half century of reviewing series, brings together a variety of perspectives on the G. Baley Price, American mathematicians in WWI political, social, and mathematical forces that have shaped Mina Rees, The mathematical sciences in WW/1 the American mathematical community in the past century. Peter Hilton, Reminiscences of Bletch/ey Park, 1942-45 Humorous, edifying, and poignant, this book presents the J. Barkely Rosser, Mathematics and mathematicians In WW/1 personal recollections of a number of mathematicians who Herman H. Goldstine, A brief history of the computer have influenced the development of mathematics in this Saunders Mac Lane, Concepts and categories in perspective country. Olga Taussky, Some non-commutativity methods in ·algebraic One of the highlights of the volume is Lipman Bars's number theory paper which was presented as an AMS-MAA Joint Invited M. Hall, Mathematical biography Address in Atlanta in January 1988 and which gives a moving S.-S. Chern, American differential geometry-some personal account of the reception that he and other European refugee notes mathematicians received in this country. Described here G. Baley Price, The mathematical scene, 1940-1965 are some of the success stories of this century-such as W. S. Massey, Reminiscences or forty years as a mathematician classification of finite simple groups, delineated by Daniel C. Davis, The purge Gorenstein-as well as some of the problems-such as R. W. Hamming, The use of mathematics the McCarthy period, chronicled by Chandler Davis. Paul R. D. Knuth, Algorithmic themes Halmos, one of the most influential textbook writers, tells of D. Gorenstein, The classification of the finite simple groups, a the textbooks he used when he was a student and young persona/journey: The early years professor and how they influenced him. Among the papers reprinted here are some that have appeared in journals not ordinarily read by mathematicians, such as the article by science historian Nathan Reingold, which appeared in The Annals of Science.

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898 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New AMS Publications

1980 Mathematics Subject Classification: 01 V. Conferences ISBN 0-8218-0124-4 VI. Supporting Journals ISSN 0899-2428 VII. Books and Book Series 400 pages (hardcover), August 1988 VIII. Translation Individual member $34, Ust price $57, Institutional member $46 IX. Publication Problems To order, please specify HMATH/1N X. International Congresses XI. Membership and Dues AMS Centennial Publications ______XII. The Council The following three books are being published in commemoration XIII. Presidents of the AMS's Centennial Celebration. The Semicentennial volumes _are XIV. Officers reprints of books originally produced in 1938 for the Semicentenmal XV. Executive Directors of the Society. XVI. Trustees XVII. Planning and Organization [____ ~--·.;'i,_'iiit~>";;n,.}M1<*1J XVIII. Policy Committees AMERICAN MATHEMATICAL XIX. Other Committees SOCIETY CENTENNIAL XX. Fellowships PUBLICATIONS, Volume 1: XXI. Political and Social Questions A History of the Second Fifty Years, XXII. Finances XXIII. Headquarters American Mathematical Society 1939-1988 XXIV. Archives Everett Pitcher 1980 Mathematics Subject Classification: 01 ISBN 0-8218-0125-2 This is volume one of a two-volume set which is being 300 pages (hardcover), August 1988 published to commemorate the AMS Centennial. (Volume Individual member $24, List price $40, 2 will contain the Proceedings of the AMS Centennial and Institutional member $32 will be published at a later date.) Professor Everett Pitcher To order, please specify HMPITCHER/N served as an AMS Associate Secretary for 8 years and as the Society Secretary for the past 22 years. His long association ~~I!IIJI$,~:~0~Aillll!lllkfi with the Society, his detailed knowledge of its workings, AMERICAN MATHEMATICAL SOCIETY and his historical perspective on the American mathematical community make him the ideal author for such a work. SEMICENTENNIAL PUBLICATIONS, Volume 1: A Semicentennial History of the Professor Pitcher chronicles the Society's activities over the American Mathematical Society 1888-1938 past fifty years, as it grew in membership, in volume and Raymond Clare Archibald diversity of its publications, in the number of meetings and conferences it organizes, and in the range of services it '7he preparation of the manuscript for this History during the provides to the mathematical community. The book presents past two years has been the most enjoyable piece of literary a picture of the AMS in 1938 and delineates the political and work which I have ever undertaken. It has seemed as if contacts social influences that shaped its subsequent development. with past enthusiasms, visions, largeness of outlook, and charm Some of the key personalities in the Society's history, notably of personality, could still be established. " the Presidents, are also described. This book is the crowning achievement in Professor Pitcher's years of dedication and So begins the preface of this delightful and fascinating work, service to the Society. in which R. C. Archibald, librarian of the Society from 1920 to 1941, outlines the history of the AMS in its first fifty years. This book complements the. history of the Society's first fifty Drawing upon his own and others' recollections and upon years, written in 1938, the Society's semicentennial year, original Society records, Archibald has assembled a portrait by Raymond Clare Archibald,. who was. the AMS librarian of the early years of the AMS and of the development of at that time. Archibald's history is volume one of American mathematics in this country. Mathematical Society Semicentennial. Publications. Contents The story begins at Columbia University, where Thomas I. Introduction Scott Fiske founded the New York Mathematical Society, II. Primary Journals later to be called the American Mathematical Society. Among Ill. Meetings other subjects, Archibald touches on the development of IV. Mathematical Reviews the various AMS publications, the growth of the important

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JULY/AUGUST 1988, VOLUME 35, NUMBER 6 899 New AMS Publications

Chicago Section, the influence of the National Research Norbert Wiener, The historical background of harmonic Council (established just after World War 1), and the meetings analysis of the Society. E. J. McShane, Recent developments in the calculus of variations One of the most interesting features of the book is the section T. Y. Thomas, Recent trends in geometry describing the past AMS Presidents up to 1938. Among them R. L. Wilder, The sphere in topology are such distinguished mathematicians as E. H. Moore, W. F. G. C. Evans, Dirichelet problem Osgood, Maxima BOcher, E. B. Van Vleck, Frank Morley, J. L. Synge, Hydrodynamical stability Henry B. Fine, G. D. Birkhoff, Oswald Veblen, and Solomon G. D. Birkhotf, Fifty years of American mathematics Lefschetz. Instilled with profound respect, admiration, and affection, these short sketches contain many personal touches 1980 Mathematics Subject Classifications: 00, 01 and anecdotes. ISBN 0-8218-0119-8 316 pages (hardcover), July 1988 Contents Individual member $23, List price $39, Introductory. The Society, 1888-1920 Institutional member $31 The Society, 1921-1938 To order, please specify PROCSEMI/N Financial affairs. Special funds Combination offer. AMS Semicentennial Publications Set. The Bulletin Volume 1: A Semicentennial History of the AMS 1888-1938 The Transactions and Volume II: Semicentennial Addresses of the AMS. Colloquium Lectures and Colloquium Publications Set price: Individual member $44, List $73, Institutional member $58 The Chicago group and the Chicago section To order, please specify SEMISET/N Annual meetings Summer meetings t~l'liii:Btf}Mf'lil'''~"*1t~J Josiah Willard Gibbs lectureship The library OPERATOR THEORY AND The Council and Board of Trustees ARITHMETIC IN Hoo The Secretaries Hari Bercovici The Vice-Presidents (Mathematical Surveys and Monographs, Volume 26) The Presidents 1980 Mathematics Subject Classifications: 00, 01 Jordan's classification theorem for linear transformations on ISBN 0-8218-0118-X a finite-dimensional vector space is a natural highlight of the 262 pages (hardcover), July 1988 deep relationship between linear algebra and the arithmetical Individual member $25, List price $42, properties of polynomial rings. Because the methods and Institutional member $34 To order, please specify HMREPRINT/N results of finite-dimensional linear algebra seldom extend to or have analogs in infinite-dimensional operator theory, it is therefore remarkable to have a class of operators which has a classification theorem analogous to Jordan's classical result AMERICAN MATHEMATICAL SOCIETY and has properties closely related to the arithmetic of the ring SEMICENTENNIAL PUBLICATIONS, H00 of bounded analytic functions in the unit disk. Co is such Volume II: Semicentennial Addresses a class and is the central object of study in this book. of the American Mathematical Society A contraction operator belongs to Co if and only if the associated functional calculus on H00 has a nontrivial kernel. The second Semicentennial volume contains brief treatises Co was discovered by Bela Sz.-Nagy and Ciprian Foiafi in their on eight mathematical areas and a historical summary of work on canonical models for contraction operators on Hilbert American contributions to mathematics during the Society's space. Besides their intrinsic interest and direct applications, first fifty years. Some of the treatises are chronological or operators of class Co are very helpful in constructing contain many references to early contributors; others are examples and counterexamples in other branches of operator concerned chiefly with presenting a view of a particular theory. In addition, Co arises in certain problems of control subject as it stood at that time. and realization theory. Contents In this survey work, the author provides a unified and concise E. T. Bell, Fifty years of algebra In America, 1888-1938 presentation of a subject that was covered in many articles. J. F. Ritt, Algebraic aspects of the theory of differential The book describes the classification theory of Co and equations relates this class to other subjects such as general dilation

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900 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New AMS Publications

theory, stochastic realization, representations of convolution The starting point is a simple function of several variables algebras, and Fredholm theory. satisfying a number of q-difference equations. The author presents an elementary method for using these equations This book should be of interest to operator theorists as well as to obtain transformations of the original function. A bilateral theoretical engineers interested in the applications of operator series, formed from this function, is summed as an infinite theory. In an effort to make the book as self-contained as product, thereby providing an elegant and fruitful result which possible, the author gives an introduction to the theory of goes back to Ramanujan. By exploiting a special case, the dilations and functional models for contraction operators. author is able to evaluate the coefficients of several classes prerequisites for this book are a course in functional analysis of infinite products in terms of divisor sums. He also touches and an acquaintance with the theory of Hardy spaces in on general transformation theory for basic series in many the unit disk. In addition, knowledge of the trace class of variables and the basic multinomial, which is a generalization operators is necessary in the chapter on weak contractions. of a finite sum. Contents These developments lead naturally to the arithmetic domains An introduction to dilation theory of partition theory, theorems of Liouville type, and sums of The class Co squares. Contact is also made with the mock theta-functions of Ramanujan, which are linked to the rank of partitions. The Classification theory author gives a number of examples of modular functions Applications of Jordan models with multiplicative coefficients, along with the beginnings of an elementary constructive approach to the field of modular Characteristic functions and the class Co equations. Weak contractions Requiring only an undergraduate background in mathematics, Fredholm theory this book provides a rapid entry into the field. Students Miscellaneous applications of partitions, basic series, theta-functions, and modular equations, as well as research mathematicians interested in 1980 Mathematics Subject Classifications: 47; 46 an elementary approach to these areas, will find this book ISBN 0-8218-1528-8, LC 88-1 0344 useful and enlightening. Because of the simplicity of its ISSN 0076-5376 280 pages (hardcover), July 1988 approach and its accessibility, this work may prove useful as Individual member $40, List price $67, a textbook. Institutional member $54 To order, please specify SURV /26N BIOGRAPHY------­ Nathan J. Fine received his PhD in 1946 from the University of Pennsylvania and taught there from 1947 to 1963. He was an NSF Postdoctoral Fellow (1953-1954), a Guggenheim Fellow (1958-1959), BASIC HYPERGEOMETRIC SERIES and E. S. Hedrick Memorial Lecturer (1966). In 1963, he moved to Pennsylvania State University, where he remained until his retirement AND APPLICATIONS in 1978. Author of An Introduction to Modern Mathematics and a Nathan J. Fine co-author of Rings of Quotients of Rings of Functions, he has also (Mathematical Surveys and Monographs, Volume 27} written numerous papers In analysis, special functions, topology, number theory, combinatorics, and other topics. The theory of partitions, founded by Euler, has led in a natural Contents way to the idea of basic hypergeometric series, also known as Fundamental properties of basic hypergeometric series Eulerian series. These series were first studied systematically by Heine, but many early results are attributed to Euler, Partitions Gauss, and Jacobi. Today, research in q-hypergeometric Mock theta-functions and the functions L(N}, J(N} series is very active, and there are now major interactions with Lie algebras, combinatorics, special functions, and number Other applications theory. Modular equations However, the theory has been developed to such an extent 1980 Mathematics Subject Classifications: 05, 11 , 33 and with such a profusion of powerful and general results that ISBN 0-8218-1524-5, LC 88-6235 the subject can appear quite formidable to the uninitiated. By ISSN 0076-5376 144 pages (hardcover), July 1988 providing a simple approach to basic hypergeometric series, Individual member $23, List price $39, this book provides an excellent elementary introduction to the Institutional member $31 subject. · To order, please specify SURV /27N

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JULY/AUGUST 1988, VOLUME 35, NUMBER 6 901 New AMS Publications f· ,,'i,.;;;;p,:;c'~l'},,,:~- _c.-~;] Contents DIRECT AND INVERSE SCATTERING Part I. The Forward Problem ON THE LINE 1. Distinguished Solutions Richard Beals, Percy Deift, and Carlos Tomei 2. Fundamental Matrices (Mathematical Surveys and Monographs, Volume 28) 3. · Fundamental Tensors 4. Behavior of Fundamental Tensors as lxl __. oo; This book deals with the theory of linear ordinary differential the Functions Ak operators of arbitrary order. Unlike treatments that focus 5. Behavior of Fundamental Tensors as z __. oo on spectral theory, this work centers on the construction of 6. Behavior of Fundamental Tensors as z __. 0 special eigenfunctions (generalized Jost solutions) and on the 7. Construction of Fundamental Matrices inverse problem: the problem of reconstructing the operator 8. Global Properties of Fundamental Matrices; from minimal data associated to the special eigenfunctions. In the Transition Matrix o the second order case this program includes spectral theory 9. Symmetries of Fundamental Matrices and is equivalent to quantum mechanical scattering theory; the 10. The Green's Function for L essential analysis involves only the bounded eigenfunctions. 11. Generic Operators and Scattering Data For higher order operators, bounded eigenfunctions are again 12. Algebraic Properties of Scattering Data sufficient for spectral theory and quantum scattering theory, 13. Analytic Properties of Scattering Data but they are far from sufficient for a successful inverse theory. 14. Scattering Data for m; Determination of v from v 15. Scattering Data for L" The authors give a complete and self-contained theory of 16. Generic Selfadjoint Operators and Scattering Data the inverse problem for an ordinary differential operator 17. The Green's Function Revisited of any order. The theory provides a linearization for the 18. Genericity at z = 0 associated nonlinear evolution equations, including KdV and 19. Genericity at z 'I 0 Boussinesq. The authors also discuss Darboux-Backlund 20. Summary of Properties of Scattering Data transformations, related first-order systems and their Part II. The Inverse Problem evolutions, and applications to spectral theory and quantum 21. Normalized Eigenfunctions for Odd Order Inverse Data mechanical scattering theory. 22. The Vanishing Lemma Among the book's most significant contributions are a new 23. The Cauchy Operator construction of normalized eigenfunctions and the first 24. Equations for the Inverse Problem complete treatment of the self-adjoint inverse problem in 25. Factorization near z = 0 and Property (20.6) order greater than two. In addition, the authors present the 26. Reduction to a Fredholm Equation first analytic treatment of the corresponding flows, including a 27. Existence of h# detailed description of the phase space for Boussinesq and 28. Properties of h# other equations. 29. Properties of 1-'#{x,z) and 1-'{x,z) as z __. oo and as x __. -oo and The book is intended for mathematicians, physicists, 30. Proof of the Basic Inverse Theorem engineers in the area of soliton equations, as well as those 31. The Scalar Factorization Problem for o or interested in the analytical aspects of inverse scattering 32. The Inverse Problem at x = +oo and the bijectivity in the general theory of linear ordinary differential operators. of the map L ~--+ S(L) = (Z(L), v(L)) single work covering the analytical Because there is no other 33. The Even Order Case theory of inverse scattering for operators of degree greater 34. The Second Order Problem than two, this book is likely to be a valuable resource to many. Part Ill. Applications Required background consists of a basic knowledge of 35. Flows complex variable theory, the theory of ordinary differential 36. Eigenfunction Expansions and Classical equations, linear algebra, and functional analysis. The authors Scattering Theory have attempted to make the book sufficiently complete and 37. Inserting and Removing Poles self-contained to make it accessible to a graduate student 38. Matrix Factorization and First Order Systems having no prior knowlege of scattering or inverse scattering Appendix A. Rational Appro~imation theory. The book may therefore be suitable for a graduate textbook or as background reading in a seminar. Appendix B. Some Formulas

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1g80 Mathematics Subject Classifications: 34, 35, 58 ISBN 0-8218-1530-X, LC 88-14487 ISSN 0076-5376 224 pages (hardcover), July 1988 IndiVidual member $32, List price $53, Institutional member $42 To order, please specify SURV f28N

AMENABILITY Alan L. T. Paterson (Mathematical Surveys and Monographs, Volume 29)

The subject of amenability has its roots in the work of Lebesgue at the turn of the century. In the 1940s, the subject began to shift from finitely additive measures to means. This shift is of fundamental importance, for it makes the substantial resources of functional analysis and abstract harmonic analysis available to the study of amenability. The ubiquity of amenability ideas and the depth of the mathematics involved points to the fundamental importance of the subject. BIOGRAPHY------­ This book presents a comprehensive and coherent account of Alan L. T. Paterson obtained his PhD at the University of Edinburgh amenability as it has been developed in the large and varied in 1969. His initial research interests lay in Banach algebra theory, but he later became literature during this century. The book has a broad appeal, interested in the phenomenon of amenability in its various forms. In 1984-1985, he was visiting professor at the for it presents an account of the subject based on harmonic University of Western Ontario and the University of British Columbia. and functional analysis. In addition, the analytic techniques He will be taking a position at the University of Mississippi, Oxford, should be of considerable interest to analysts in all areas. MS in August 1988. In addition, the book contains applications of amenability to Contents a number of areas: combinatorial group theory, semigroup theory, statistics, differential geometry, Lie groups, ergodic Introduction-Basic concepts and problems of amenability theory, cohomology, and operator algebras. Amenable locally compact groups and amenable semigroups The main objectives of the book are to provide an introduction The algebra of invariant means to the subject as a whole and to go into many of its topics in Free groups some depth. The book begins with an informal, nontechnical and the amenability of Lie groups account of amenability from its origins in the work of F0lner conditions Lebesgue. The initial chapters establish the basic theory of Ergodic theorems for amenable locally compact groups amenability and provide a detailed treatment of invariant, finitely additive measures (i.e., invariant means) on locally Locally compact groups of polynomial growth compact groups. The author then discusses amenability for Sizes of sets of invariant means Lie groups, ··almost invarianr· properties of certain subsets of an amenable group, amenability and ergodic theorems, 1980 Mathematics Subject Classifications: 43, 47, 22, 03 polynomial growth, and invariant mean cardinalities. Also ISBN 0-8218-1529-6 ISSN 0076-5376 included are detailed discussions of the two most important 416 pages (hardcover), August 1988 achievements in amenability in the 1980s: the solutions to von Individual member $54, List price $90, Neumann's conjecture and the Banach-Ruziewicz Problem. Institutional member $72 To order, please specify SURV /29N The main prerequisites for this book are a sound understanding of undergraduate-level mathematics and a knowledge of abstract harmonic analysis and functional analysis. The book is suitable for use in graduate courses, and the lists of problems in each chapter may be useful as student exercises;

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JULY/AUGUST 1988, VOLUME 35, NUMBER 6 903 New AMS Publications

Wilfrid Kendall, Martingales on manifolds and harmonic maps GEOMETRY OF RANDOM MOTION Yuri Kifer, Harmonic functions on Riemannian manifolds Rick Durrett and Mark A. Pinsky, Editors Remi Leandre, Quantitative and geometric applications of the (Contemporary Mathematics, Volume 73) Malliavin calculus Ming Liao, An independence property of Brownian motion Ming Liao and Mark Pinsky, Stochastic parallel translation for In July 1987, an AMS-IMS-SIAM Joint Summer Research Riemannian Brownian motion conditioned to hit a fixed point of Conference on Geometry of Random Motion was held at a sphere Cornell University. The initial impetus for the meeting came Peter March, Probabilistic interpretation of Hadamard's from the desire to further explore the now-classical connection variational formula between diffusion processes and second-order (hypo)elliptic Carl Mueller, A counterexample for Brownian motion on differential operators. To accomplish this goal, the conference manifolds brought together leading researchers with varied backgrounds Berndt Oksendal, Using Brownian motion to study quasi-regular and interests: probabilists who have proved results in functions geometry, geometers who have used probabilistic methods, E. J. Pauwels and L. C. G. Rogers, Skew-product and probabilists who have studied diffusion processes. decompositions of Brownian motions Focusing on the interplay between probability and differential Mark Pinsky, Local stochastic differential geometry geometry, this volume examines diffusion processes on Ross Pinsky, Transience and recurrence for multi-dimensional various geometric structures, such as Riemannian manifolds, diffusions: a survey and a recent result Lie groups, and symmetric spaces. Some of the articles Steven Rosenberg, Semigroup domination and vanishing specifically address analysis on manifolds, while others center theorems on (nongeometric) stochastic analysis. The majority of the John C. Taylor, The lwasa decomposition and the limiting articles deal simultaneously with probabilistic and geometric behavior of Brownian motion on a symmetric space of techniques. noncompact type N. Th. Varopoulos, Green's functions and harmonic functions Requiring a knowledge of the modern theory of diffusion on manifolds processes, this book will appeal to mathematicians, 1980 Mathematics mathematical physicists, Subject Classifications: 58, 53, 60 and other researchers interested ISBN 0-8218-5081-4, LC 88-13964 in Brownian motion, diffusion processes, Laplace-Beltrami ISSN 0271-4132 operators, and the geometric applications of these concepts. 352 pages (softcover), June 1988 The book provides a detailed view of the leading edge of Individual member $19, List price $32, research in this rapidly moving field. Institutional member $26 To order, please specify CONM/73N Contents Isaac Chavel, Edgar Feldman, and Jay Rosen, Fluctuations of the Wiener sausage for surfaces Michael Cranston and Carl Mueller, A review of recent and GEOMETRY OF GROUP REPRESENTATIONS older results on the absolute continuity of harmonic measure William M. Goldman and Andy R. Magid, R. W. R. Darling, Constructing stochastic flows: some examples Editors Josef Dodziuk and Leon Karp, Spectral and function theory (Contemporary Mathematics, Volume 74) for combinatorial laplacians Peter G. Doyle, On deciding whether a surface is parabolic or The representations of a finitely generated group in a hyperbolic topological group G form a topological space which is an Tyrone E. Duncan, A solvable stochastic control problem in analytic variety if G is a Lie group, or an algebraic variety if G spheres is an algebraic group. The study of this area draws from and K. David Elworthy, Brownian motion and the ends of a manifold contributes to a wide range of mathematical subjects: algebra, Masatoshi Fukushima, On holomorphic diffusions and analysis, topology, differential geometry, representation plurisubharmonic functions theory, and even mathematical physics. In some cases, the Peter B. Gilkey, Leading terms in the asymptotic expansion of space of representations is the object of the study, in others the heat equation it is a tool in a program of investigation, and, in many cases, it Joseph Glover, Probability theory and differential equations is both. Pei Hsu, Brownian motion and Riemannian geometry Leon Karp and Mark Pinsky, First-order asymptotics of the Most of the papers in this volume are based on talks delivered principal eigenvalue of tubular neighborhoods at the AMS-IMS-SIAM Summer Research Conference on the

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904 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New AMS Publications

Geometry of Group Representations, held at the University of Colorado in Boulder in July 1987. The conference was THE FINITE CALCULUS ASSOCIATED designed to bring together researchers from the diverse areas WITH BESSEL FUNCTIONS of mathematics involving spaces of group representations. In keeping with the spirit of the conference, the papers are Frank M. Cholewinski directed at nonspecialists, but contain technical developments (Contemporary Mathematics, Volume 75) to bring the subject to the current research frontier. Some of the papers include entirely new results. Readers will gain Although Bessel functions are among the most widely used an understanding of the present state of research in the functions in applied mathematics, this book is essentially geometry of group representations and their applications. the first to present a calculus associated with this class of Contents functions. The author obtains a generalized umbral calculus associated with the Euler operator and its associated William Abikoff, Kleinian groups-geometrically finite and Bessel eigenfunctions geometrically perverse for each positive value of an index parameter. For one particular value of this parameter, the functions G. W. Brumfiel, The real spectrum compactification of and operators Teichmuller space can be associated with the radial parts of n-dimensional Euclidean space objects. Some G. W. Brumfiel, A semi-algebraic Brower fixed point theorem of the results of this book are tor real affine space in part extensions of the work of Rota and his co-workers on the ordinary umbral calculus G. W. Brumfiel, The tree of a non-archimedean hyperbolic and binomial plane enumeration. The author also introduces a wide variety of new polynomial sequences together Kevin Corlette, Gauge theory and representations of Kahler with their groups groups and semigroup compositional properties. Generalized Bernoulli, Euler, and Stirling numbers associated Daniel R. Farkas, The Diophantine nature of some constructions with Bessel at infinity functions and the corresponding classes of polynomials are also studied. The book is intended Benjamin Fine and Gerhard Rosenberger, Complex for mathematicians and physicists at the research level in special representations and one-relator products of cyclics function theory. M. Gerstenhaber and S.D. Schack, Sometimes H1 is H2 and discrete groups deform William M. Goldman, Geometric structures on manifolds and varieties of representations William M. Goldman and Yoshinobu Kamishima, Topological rigidity of developing maps with applications to conforma/ly flat structures W. J. Harvey, Modular groups and representation spaces Alexander Lubotzky and Andy R. Magid, Local structures of representation varieties: examples John J. Millson, Deformations of representations of finitely generated groups Kent Morrison, Connected components of representation varieties Joyce O'Halloran, A characterization of orbit closure R. C. Penner, Calculus on moduli spaces Dennis M. Snow, Affine homogeneous spaces Christopher W. Stark, Deformations and discrete subgroups of loop groups

1980 Mathematics Subject Classifications: 20, 22, 30, 57 ISBN 0-8218-5082-2, LC 88-14490 BIOGRAPHY------­ ISSN 0271-4132 Frank M. Cholewinski is Professor of Mathematics at Clemson 328 pages (softcover), July 1988 University. He received a bachelor's degree in engineering physics Individual member $18, List price $30, and a master's degree in applied mathematics from Auburn University. Institutional member $24 He received his PhD in 1964 from Washington University under the To order, please specify CONM/74N supervision of I. I. Hirschman, Jr. Professor Cholewinski came to the University of North Carolina at Chapel Hill in 1964 before moving to Clemson in 1966.

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JULY/AUGUST 1988, VOLUME 35, NUMBER 6 905 New AMS Publications

Contents then move beyond the consideration of individual algebras to a study of locally finite varieties. A list of open problems The 11-umbral algebra closes the work. The 11-umbral field Contents The group of 11-delta functionals under composition Basic concepts and notation Generalized binomial p-'vnomial sequences Tight lattices The composition of polynomial sequences Tame quotients Compositions of Moebius delta functionals Abelian and solvable algebras Generalized shift invariant operators The structure of minimal algebras The generalized derivative of 11-shift invariant operators The types of tame quotients Generalized Sheffer polynomials Labeled congruence lattices Cross sets of polynomials Solvability and semi-distributivity A class of Laguerre type polynomials Congruence modular varieties The generalized heat polynomials Maleev classification and omitting types A primitive integral for the Euler operator Residually small varieties Bernoulli type polynomials and numbers Decidable varieties Generalized Euler polynomials and numbers Free spectra Generalized Stirling numbers and factor polynomials Tame algebras and E-minimal algebras 1980 Mathematics Subject Classifications: 33, 05, 12, 35, 39, 44 Simple algebras in varieties ISBN 0-8218-5083-0, LC 88-14597 ISSN 0271-4132 1980 Mathematics Subject Classification: 08 136 pages (softcover), July 1988 ISBN 0-8218-5073-3, LC 88-16712 Individual member $11, List price $19, ISSN 0271-4132 Institutional member $15 212 pages (softcover), August 1988 To order, please specify CONM/75N Individual member $16, Ust price $27, Institutional member $22 To order, please specify CONM/76N

THE STRUCTURE OF FINITE ALGEBRAS 5l1l~-ti~~~~~~t*~~i;i~lli!l~1l David Hobby and Ralph N. McKenzie NUMBER THEORY AND Mathematics, Volume 76) (Contemporary ITS APPLICATIONS IN CHINA Yuan, Yang Chung-chun, The utility of congruence lattices in revealing the structure of Wang general algebras has been recognized since Garrett Birkhoff's and Pan Cheng-biao, Editors pioneering work in the 1930s and 1940s. However, the results (Contemporary Mathematics, Volume 77) presented in this book are of very recent origin: most of them were developed in 1983. The main discovery presented here Of all modern mathematical forms, number theory is one of is that the lattice of congruences of a finite algebra is deeply the earliest to be explored in China and is the one to which connected to the structure of that algebra. The theory reveals the Chinese have made their greatest contributions. Yan ·a sharp division of locally finite varieties of algebras into six Wu-zhi first introduced number theory into China in the 1920s. interesting new families, each of which is characterized by the Particularly influential in the field was Hua Loo-keng, who behavior of congruences in the algebras. The authors use the studied with G. H. Hardy and made significant contributions theory to derive many new results that will be of interest not in the areas estimating complete exponential sums, Waring's only to universal algebraists, but to other algebraists as well. problems, Tarry's problems, and Vinogradov's method. Interest in number theory continued to flourish following and complete The authors begin with a straightforward the founding of the People's Republic of China. The most theory, a topic that development of basic tame congruence noted accomplishments by Chinese mathematicians were offers great promise for a wide variety of investigations. They

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906 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New AMS Publications

focused on the solution of Goldbach's Conjecture and on the the regular integral setting the category of Harish-Chandra sieve method. Although the Cultural Revolution interrupted modules and the category of highest weight representations. esearch in number theory for more than 10 years, the field One consequence of the main theorm is an algorithm to ~s now growing in China. A number of universities now have compute the ""lwasawa"" nilpotent cohomology groups of ~dvanced programs in the subject and a wide variety of the irreducible and induced representations in these two topics, including the applications of number theory. categories. Qualitatively, the author's results show that these cohomology groups are ""as small as possible,·· a notion made This volume contains nine survey articles and three precise through detailed spectral sequence analysis. These articles on current research. The collection emphasizes results are aimed toward an understanding of higher order the accomplishments of Chinese number theorists during extension groups. Although the final results concern rank one 1949-1979, a period when correspondence between China Lie groups, the ideas and approach are applicable to any and other countries was discouraged. The collection is semisimple matrix group. Directed at graduate students and intended not only to survey the significant contributions researchers in the representation theory of Lie groups, this of Chinese mathematicians, but also to reflect the latest book requires a basic first course in Lie group representations developments and current state of research in number theory and familiarity with the first paper in this series. in China. Contents Contents Jacquet modules and strategy Chen Jingrun and Pan Chengbiao, Analytic number theory in SL(2,R) China I Orthogonal cases Pan Chengdong, Pan Chengbiao, and Xie Shenggang, Unitary cases Analytic number theory In China II Exceptional case Wang Yuan, Number theoretic method in numerical analysis Symplectic cases Wang Yuan, Diophantine equations and Diophantine inequalities Figures in algebraic number fields An illustrative example: Sp(2, 1) Pei Dingyi and Feng Xuning, Some results of modular forms Classifications: 22; 20 Sun Qi, Some results in the application of the number theory to 1980 Mathematics Subject ISBN 0-8218-2450-3, LC 88-10415 digital signal processing and public-key systems ISSN 0065-9266 Sun Qi, Some results on Diophantine equations 108 pages (softcover), June 1988 Xu Guangshan, Diophantine approximation and transcentdental Individual member $8, List price $13, number theory Institutional member $10 Li Delang and Lu Hongwen, Quadratic forms and Hermitian To order, please specify MEM0/387N forms Llu Mlngchit and Tsang Kaiman, Small prime solutions of TOPOLOGICAL INVARIANTS OF linear equations and the exceptional set in Goldbach's problem Lal K. F., On the relative trace formula QUASI-ORDINARY SINGULARITIES Ye Yangbo, Kloosterman integrals and base change Joseph Lipman 1980 Mathematics Subject Classification: 11 ISBN 0-8218-5084-9 ISSN 0271-4132 184 pages (softcover), August 1988 EMBEDDED TOPOLOGICAL Individual member $12, List price $20, CLASSIFICATION OF QUASI-ORDINARY Institutional member $16 SINGULARITIES please specify CONM/77N To order, Yih-Nan Gau (Memoirs of the AMS, Number 388) ~--- '_ -~'f!'f'Wf~,~;m~tfi\WJ

REPRESENTATIONS OF RANK ONE This book contains two related papers of interest to LIE GROUPS II: n-COHOMOLOGY algebraic geometers and geometric topologists interested David H. Collingwood in singularities. The first paper establishes the existence (Memoirs of the AMS, Number 387) of a topological invariant of quasi-ordinary singularities on complex analytic hypersurfaces. This result is a main This paper is the second in a series aimed at a detailed ingredient in the second paper, which presents an embedded understanding of the representation theory associated with a topological classification of quasi-ordinary singularities simple Lie group of real rank one. The author investigates in using ""characteristic monomials:· The classification is a

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JULY/AUGUST 1988, VOLUME 35, NUMBER 6 907 New AMS Publications

higher-dimensional generalization of the classical topological topological triviality theorem provides sufficient conditions for treatment of plane curve singularities using characteristic a deformation to be topologically constant; and the versality pairs. Requiring a background in basic algebraic topology and theorem ensures that a family of deformations of a germ commutative algebra, this book will give readers an enhanced of a mapping contains, up to topological equivalence, all appreciation of the interaction between algebra a~d topology. possible perturbations of the germ. From these theorems Contents of Topological invariants of quasi-ordinary follow versions of the basic theorems of singularity theory but singularities for topological equivalence. Rational equivalence and local homology in codimension one Requiring a familiarity with the basic ideas of singularity Local fundamental class map theory, this book will provide readers with a number of results Codimension one cycles at quotient singularities which are useful in dealing with topological questions in Quasi-ordinary singularities virtually any area in which singularity theory can be applied. Presentation of the group Ad-1 ~ H2d-2 Contents The hypersurface case Stratified vector fields without Whitney conditions Characteristic monomials of quasi-ordinary parametriza­ An example illustrating the method tions Local integrability of stratified vector fields Topological invariance of the reduced branching sequence Sums of stratified vector fields Appendix: The singular locus Algebraic stratifications and systems of DA-algebras Contents of Embedded topological classification of Systems of DA-algebras quasi-ordinary singularities Operations and properties modelled on DA-algebras Statement of main results Algebraic stratifications and algebraically stratified vector fields Some plane sections of X and two key lemmas Several algebraic lemmas Topological invariants The main theorems Proofs of the main theorem A special class of geometric subgroups (and some examples} Topological triviality and versality theorems in the weighted Appendix by J. Lipman homogeneous case 1980 Mathematics Subject Classifications: 32, 14, 15 A reduction and the method for an example ISBN 0-8218-2451-1, LC 88-10559 Collections of vector fields ISSN 0065-9266 Filtered versions of the main theorems 136 pages (softcover), June 1988 Individual member $10, List price $16, 1980 Mathematics Subject Classifications: 58, 14 Institutional member $13 ISBN 0-8218-2452-X, LC 88-15554 To order, please specify MEM0/388N ISSN 0065-9266 106 pages (softcover), August 1988 Individual member $8, List price $13, t~~:~1lifJ,:,,;~:i~~:·:~z,,··).%1.:,,,;g;~,.J Institutional member $1 0 TOPOLOGICAL TRIVIALITY AND VERSALITY To order, please specify MEM0/389N FOR SUBGROUPS OF A AND K James N. Damon (Memoirs of the AMS, Number 389} IRREDUCIBLE SUBGROUPS OF EXCEPTIONAL ALGEBRAIC GROUPS Singularity theory plays an increasingly valuable role in Donna M. Testerman the analysis of problems which can be modeled using (Memoirs of the AMS, Number 390} nonlinear mappings. Much of the success of this approach results from the application of the infinitesimal methods This book is aimed at researchers in group theory and originally suggested by Thorn and developed by Mather. The particularly those interested in the structure and representation appearance of moduli in such problems requires the solution theory of algebraic groups. The author focuses on certain of corresponding problems for topological equivalence of subgroups of an exceptional algebraic group defined over mappings. an algebraically closed field of nonzero characteristic. The This book presents two theorems which permit infinitesimal main result classifies those semisimple, closed, connected methods to be applied to such topological problems. The subgroups which act irreducibly on some nontrivial rational

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908 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New AMS Publications

rnodule for the group. An extension of Dynkin's earlier work 1980 Mathematics Subject Classification: 4 7 with groups of characteristic zero, this result is combined with ISBN 0-8218-2454-6, LC 88-16693 the work of Seitz to provide a classification of the maximal, ISSN 0065-9266 closed, connected subgroups of the classical 137 pages (softcover), August 1988 algebraic groups Individual in nonzero characteristic. member $10, List price $16, In addition, the author develops Institutional member $13 techniques that are applicable to general questions in the To order, please specify MEM0/391 N study of embeddings of groups of Lie type. Contents Preliminary lemmas LECTURES ON INTEGRAL Parabolic embeddings TRANSFORMS N. I. Akhiezer y = F4 or G2 (Translations The one component theorem of Mathematical Monographs, Volume 70) Rank(A) ~ 3 Initial rank two results This book, which grew out of lectures given over the course A= 82 of several years at Kharkov University for students in the A=G2 Faculty of Mechanics and Mathematics, is devoted to classical Special cases integral transforms, principally the Fourier transform, and their applications. The author develops the general theory of the 1980 Mathematics Subject Classification: 20 ISBN 0-8218-2453-8, LC 88-15584 Fourier transform for the space L 1 (En) of integrable functions ISSN 0065-9266 of n variables. His proof of the inversion theorem is based 190 pages (softcover), August 1988 on the general Bochner theorem on integral transforms, a Individual member $11, List price $19, theorem having other applications within the Institutional subject area member $15 of the book. The author To order, please specify MEM0/390N also covers Fourier-Piancherel theory in L 2(En). In addition to the general theory of integral transforms, connections are established with other areas of mathematical analysis-such as the theory of harmonic and HILBERT'S PROJECTIVE METRIC analytic functions, the theory of orthogonal polynomials, and AND ITERATED NONLINEAR MAPS the moment problem-as well as to mathematical physics. Roger D. Nussbaum "This book is remarkable for its rigor, brevity, and systematic (Memoirs of the AMS, Number 391) expression which, together with the problems proposed in each chapter, make it extremely useful for students, This book is concerned with nonlinear maps which take the mathematicians, and physicists:· -Mathematical Reviews interior of a cone in a Banach space into itself. The kinds Contents of maps considered here are motivated by examples from Averaging operators and the Bochner theorem mathematical biology, the theory of means and their iterates, The Fourier transform in L 1 and the so-called D-A-D theorems. Using Hilbert's projective The inversion theorem in L 1 . The Poisson integral metric, the author proves a variety of theorems concerning Harmonic functions. The Dirichlet the asymptotic behavior of the problem for a ball and a iterates of these maps and half-space the structure of their sets of eigenvectors. The author also The Fourier transform in L 2 provides applications to means and their iterates. To make the Hermite functions paper as self-contained as possible, the author summarizes Spherical functions relevant facts and definitions about cones, cone mappings, Positive definite functions Hilbert's projective metric, and Thompson's variant on that metric. The Hankel transform Orthogonal polynomials and the moment problem Contents The class H2 . The Paley-Wiener theorem Basic properties of Hilbert's projective metric Boundary properties of functions analytic in the upper Uniqueness and global stability for eigenvectors half-plane and the Hilbert transform Iterates of unnormalized maps The Poisson summation formula and some of its applications Nonexpansive maps and Hilbert's projective metric Applications of the Laplace and Fourier transforms to the solution of boundary value problems in mathematical physics

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JULY/AUGUST 1988, VOLUME 35, NUMBER 6 909 New AMS Publications

Frederick W. Gehring Fourier transforms of increasing functions. The Wiener-Hopf Quasiconformal mappings technique Price $49 1980 Mathematics Subject Classifications: 44, 42; 33, 31 Code VIDGEHRING/N ISBN 0-8218-4524-1 ISSN 0065-9282 Mikhael Gromov 120 pages (hardcover), August 1988 Soft and hard symplectic geometry Individual member $ 27, List price $45 , Price $49 Institutional member $36 Code VIDGROMOV /N To order, please specify MMON0/70N Hendrik W. Lenstra Efficient algorithms in number theory Price $49 VIDEOTAPES OF THE INTERNATIONAL Code VIDLENSTRA/N CONGRESS OF MATHEMATICIANS 1986 Richard M. Schoen AUGUST 3-11, 1986 New developments in the theory of BERKELEY, CAUSA geometric partial differential equations Price $49 Now available on videotape are twelve of the Plenary Code VIDSCHOEN/N Addresses presented at the International Congress of Mathematicians, held in Berkeley, California in August 1986. Saharon Shelah Presented by world-renowned experts in a wide range of Classifying general classes Price $49 areas in the mathematical sciences, these addresses cover Code VIDSHELAH/N major concepts, problems, and trends in mathematics. Many of the addresses are broad surveys of interest to a wide Stephen Smale audience of nonspecialists. Also available on videotape are Complexity aspects of numerical analysis addresses by four eminent mathematicians who spoke on the Price $49 work of the recipients of the Fields Medals and the Nevanlinna Code VIDSMALE/N Prize, which are traditionally awarded during the Congress. (These four talks are on one tape.) These videotapes provide Elias M. Stein Problems in harmonic analysis related a valuable historical record of the Congress and of the current to oscillatory integrals and curvature research. state of mathematical Price $49 1980 Mathematics Subject Classification: 00 Code VIDSTEIN/N VHS format, approx. one hour each, 1988 David A. Vogan, Jr. Representations of reductive Lie groups ICM Plenary Addresses Price $49 Simon K. Donaldson Code VIDVOGAN/N Geometry of four-manifolds Edward Witten Price $49 String theory and geometry Code VIDDONALDSON/N Price $49 Louis de Branges Code VIDWITTEN/N Underlying concepts in the proof of the Bieberbach conjecture Price $49 ICM Addresses on the work of the 1986 Code VIDDEBRANGES/N Fields Medalists and Nevanlinna Prize Winner

Gerd Faltings Michael F. Atiyah, On the work of Simon K. Donaldson, Fields Recent progress in arithmetic algebraic geometry Medalist; Barry Mazur, On some of the mathematical contributions Price $49 of Gerd Faltings, Fields Medalist; John W. Milnor, On the Code VIDFALTINGS/N work of Michael Freedman, Fields Medalist; Volker Strassen, On the work of Leslie G. Valiant, Nevanlinna Prize Winner Price $49 Code VIDMEDAL/N

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910 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Reports and Communications

Recent Appointments (1990), Paul H. Rabinowitz (1989), in College Park, Maryland. Pres­ Thomas Crawford Spencer (1990), ident G. D. Mostow was in the and Robert B. Warfield, Jr. (1989). chair. Committee members' terms of Terms expire on February 28. The Council increased the size office on standing committees Roger E. Howe (AMS, 1992) of the Editorial Committee of expire on December 31 of the has been appointed to the AMS­ Mathematical Reviews from three year given in parentheses fol­ IMS-SIAM Committee on Joint to four. The additional position is lowing their names, unless oth­ Summer Research Conferences in to be filled in the next election. erwise specified. the Mathematical Sciences by Pres­ The Council approved the ident G. D. Mostow. Other mem­ change in name of the joint AMS­ bers of the committee are William MAA-AAAS Committee on Oppor­ V. I. Amol'd, Richard Askey, S. G. B. Arveson (AMS, 1989), Daniel tunities for Disadvantaged Groups Gindikin, N. K. Nikol'skii, and J. Kleitman (AMS, 1989), Mary to Committee on Opportunities in Allen Shields have been appointed Ellen Rudin (AMS, 1989), Les­ Mathematics for Underrepresented to a Subcommittee on Russian ley M. Sibner (AMS, 1990), and Minorities. Mathematical History by Presi­ Stephen G. Simpson (AMS, 1990). The Council approved a pro­ dent G. D. Mostow. Professor Terms expire on June 30. posal that the Society for Mathe­ Shields will serve as chairman. The name of the AMS.MAA­ matical Biology (SMB) join with Jane P. Gilman, Irwin Kra, AAAS Committee on Opportuni­ the Society and SIAM in the Com­ William P. Thurston, William A. ties in Mathematics for Disadvan­ mittee on Mathematics in the Life Veech, and James A. Voytuk have taged Groups has been changed Sciences. The SMB is the organiza­ been appointed by President G. D. to the AMS-MAA-AAAS Commit­ tion through which the AMS and Mostow to the Committee on Elec­ tee on Opportunities in Mathemat­ SIAM arrange symposia at meet­ tion Scheduling. Professor Thurs­ ics for Underrepresented Groups. ings of quantitative biologists. ton will serve as chairman. The Society for Mathemati­ The Council approved Society President G. D. Mostow has cal Biology has joined with the participation with the Mathemati­ appointed Tosio Kato ( 1992) to AMS and SIAM on the joint AMS­ cal Association of America in the the Committee to Select the Win­ SIAM-SMB Committee on Mathe­ possible launching of a newsletter ner of the Steele Prize. Contin­ matics in the Life Sciences. Mem­ on Collegiate Mathematics Educa­ uing members of the commit­ bers of the committee are Jack D. tion. tee are Frederick J. Almgren, Jr. Cowan (1991), Michael C. Mackey The Council recommended an {1989), Luis A. Caffarelli {1990), (1989), Hans G. Othmer (1988), amendment to the bylaws whereby William S. Massey ( 1989), Chair­ Richard E. Plant (1988), Chair­ institutional dues will be based on man, Frank A. Raymond (1989), man, and John M. Rinzel {1989). a broader definition of scholarly Neil J. A. Sloane (1990), Louis activity. See the program of the Solomon (1989), Richard P. S~an­ Centennial Meeting in August for ley ( 1989), and Michael E. Taylor Report of Past Meetings the text. ( 1990). Terms expire on June 30. The April Meeting The Council nominated some Haynes R. Miller ( 1991) and in College Park of the candidates for the election Raghavan · Narasimhan (1991) by the membership in the fall of have been appointed by President The Council 1988. They are as follows: G. D. Mostow to the Committee The Council met on 23 April Vice President on Summer Institutes and Special 1988 at 7:00 PM in the Prince (two for one position) Symposia. Other members of the George Room of the Quality Inn Sun-Yung Alice Chang committee are Steven L. Kleiman

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 911 AMS Reports and Communications

Treasurer Committee to Monitor the Executive Committee with the Franklin P. Peterson Problems in Communication approval of the Board of Trustees. (two positions) Associate Treasurer Everett Pitcher A~hur M. Jaffe Steve Armentrout Secretary Associate Secretary Bethlehem, Pennsylvania that the (two positions) The Council agreed the dues Joseph A. Cima "negative checkoff" on bill for the Centennial Fellowship 1988 Symposium on W. Wistar Comfort in the amount of $10.00 should Some Mathematical Questions Trustee continue. The terms in which the in Biology (two for one position) fellowship will be offered are to The Dynamics of Excitable Media Edwin E. Floyd be changed slightly by authoriz­ Paul J. Sally, Jr. ing the selection committee to The twenty-second annual Sym­ Member-at-large give preference to applicants who posium on Some Mathematical (five positions) have not had extensive postdoc­ Questions in Biology was held on Jonathan L. Alperin toral research support. The terms Wednesday, May 4, in Room L2 of Fan R. K. Chung and conditions of the fellowship the Las Vegas Convention Center, Lawrence J. Corwin are to be studied by an ad hoc during the annual meeting of the Hugo Rossi committee. Federation of American Societies George R. Sell The Council empowered the for Experimental Biology, May 1 - William Yslas Velez President to increase Society rep­ 6, 1988. The symposium is spon­ Robert J. Zimmer resentation in AAAS in such man­ sored by the American Mathemat­ ner as is deemed suitable. ical Society, the Society for In­ Editorial Committees The Council empowered the dustrial and Applied Mathemat­ ics, and the Society for Mathe­ American Journal President to appoint a commit­ matical Biology. The AMS-SIAM (two positions) tee to generate and review ideas Committee on Mathematics in the David Gieseker for cosponsoring sessions involv­ ing applications of mathematics Life Sciences served as the Or­ Bulletin with other organizations. ganizing Committee for the sym­ Murray H. Protter The Council agreed to change posium. The committee consisted Colloquium Publications the terms of the members of the of Gail A. Carpenter (Northeast­ Charles L. Fefferman Nominating Committee from two em University); Kenneth L. Lange to three years, with problems of (Massachusetts Institute of Tech­ Mathematical Reviews transition yet to be solved. It re­ nology); Hans G. Othmer (Uni­ (two positions) ferred these problems to an ad hoc versity of Utah); Alan S. Perel­ Leonard D. Berkovitz committee and further charged son (Los Alamos National Labora­ John L. Selfridge the committee to study a vari­ tory); Richard E. Plant, Chairman Mathematical Su"eys ety of problems about contested (University of California, Davis); David S. Kinderlehrer elections and lengths of terms for and John Rinzel (National Insti­ various offices. tutes of Health). Professor Othmer Mathematics of Computation was the organizer of the sympo­ Andrew M. Odlyzko The Council was made aware of a pending recommendation that sium. PrOceedings the level of professional income The theme of the symposium (six positions) separating the members paying was The Dynamics of Excitable William W. Adams lower dues from those paying Media. There were two half-day J. Marshall Ash higher dues be raised from $30,000 sessions, each including three one­ Maurice Auslander to $38,000. The Council had au­ hour lectures. Forty-two people Andreas R. Blass thorized the Executive Committee registered for the symposium. Sup­ Clifford J. Earle to act for it in this matter. The port was provided by a grant from Transactions & Memoirs level was subsequently raised by the National Science Foundation. Eugene F. Fabes

912 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Miscellaneous

Personal Items a Fellow of the Royal Society of age of 80. He was a member of London. the Society for 56 years. Berzsenyi, Professor of George Teo Sturm has been promoted Julia P. Kennedy, of Georgia at Lamar University, Mathematics to full Professor of Mathematics at State University, died on April 16, chairmanship of has accepted the the University of Natal, Durban, 1988, at the age of 49. She was Department at the Mathematics South Africa. a member of the Society for 14 of Tech­ Rose-Htilman Institute Michael J. Tierney, of the Vir­ years. in Terre Haute, Indiana, nology ginia Military Institute, has been Horst Leipholz, Professor 15, 1988. beginning August promoted to Professor of Mathe­ Emeritus of the University of Wa­ of the Uni­ Gustave Choquet, matics and Computer Science at terloo, died on March 22, 1988, at IV, was elected an versity of Paris that institution. the age of 68. He was a member Honorary Member of the London Deaths of the Society for 7 years. Society, in recogni­ Mathematical Sholom Arzt, of Cooper Union, Emmanuel 0. Okoronkwo, of to anal­ tion of his contributions New York, died on January 2, Loyola University, died on April potential theory, functional ysis, 1988, at the age of 58. He was 9, 1988, at the age of 41. He was a theory, and analysis, measure a member of the Society for 37 member of the Society for 3 years. convexity infinite-dimensional years. Owen G. Owens, Professor theory. Norman H. Blaufox, of Mon­ Emeritus of Wayne State Univer­ Fulton, of Hamp­ James Paul trose, New York, died on March sity, died on December 21, 1986, University, has received a ton 12, 1988, at the age of 62. He at the age of 71. He was a member Summer Fac­ 1988 NASA/ ASEF was a member of the Society for 7 of the Society for 43 years. Fellowship Award. ulty years. Myra Reed, of St. Bonaventure Green, Associate Profes­ Judy Richard C. Courter, of Con­ University, died on October 20, at Rutgers sor of Mathematics stantine, Michigan, died on June 1987, at the age of 51. He was been elected First University, has 15, 1987, at the age of 70. He was a member of the Society for 22 of the American Vice-President a member of the Society for 30 years. Profes­ Association of University years. Hugo B. Ribeiro, of Pennsyl­ sors for 1988-1990. Henry A. Dye, of the Univer­ vania State University, died on to Sung J. Lee has returned sity of California, Los Angeles, February 26, 1988, at the age 77. a the University of Florida after died on November 26, 1986, at He was a member of the Society at the University of one-year leave the age of 60. He was a member for 40 years. Delaware. of the Society for 37 years. Morris Schreiber, of Rocke­ to Paul Nevai has returned Ben Goldbeck, Professor Emer­ feller University, died on April from the Ohio State University itus of Texas Christian University, 30, 1988, at the age of 61. He was Carolina. University of South died on October 1, 1987, at the a member of the Society for 35 Patterson III, of Walter M. age of 71. He was a member of years. Lander College, South Carolina, the Society for 35 year~. Gordon L. Tindle, of London, at has been promoted to Professor George G. Harvey, Professor England, died on March 24, 1988, that institution, effective August Emeritus of the Massachusetts In­ at the age of 46. He was a member 1988. stitute of Technology, died on of the Society for 6 years. Professor of Philip Saffman, April 9, 1988, at the age of 80. William Wooton, of Lake San and Exec­ Applied Mathematics He was a member of the Society Marcos, California, died on Jan­ Applied Mathe­ utive Officer for for 53 years. uary 5, 1988, at the age of 68. He matics at the California Institute I. N. Kagno, of Bronx, New was a member of the Society for has been elected of Technology, York, died on April4, 1988, at the 29 years.

JULY /AUGUST 1988, VOLUME 35, NUMBER 6 913 Visiting Mathematicians

(Supplementary List)

This list of visiting mathematicians includes only foreign mathematicians visiting in the United States and Canada.

Visiting Foreign Mathematicians Name and Home Country Host Institution Field of SQecial Interest Period of Visit Al-Amrani, Abdellah Queen's University Algebra 3/89- 4/89 (Morocco) Brummer, G. L. C. (South University of Toledo Topology, Category Theory 9/88- 12/88 Africa) Byczkowski, Tom (Poland) Case Western Reserve University Probability 8/88- 12/88 Emery, Michel (France) University of British Columbia Stochastic Differential Equations 1/88- 6/89 Frigon, Marlene (Italy) University of British Columbia Differential Equations 9/88- 8/89 Giordano, Thierry Queen's University Operator Algebras 9/88- 3/89 (Switzerland) Gyorfi, Zoltan (Hungary) University of Toledo Information Theory 9/88- 9/89 Hermann, Manfred (West Queen's University Commutative Algebra 9/88- 10/88 Germany) Jin, Gyo T. (Korea) University of British Columbia Knot Theory 7/88- 12/88 Kashiwara, Masaki (Japan) Johns Hopkins University Algebraic Analysis 9/88- 12/88 Katsura, Toshiyuki (Japan) Queen's University Arithmetical Algebraic Geometry 8/88 Krajicek, Jan University of Illinois at Mathematical Logic 8/88- 5/89 (Czechoslovakia) Urbana-Champaign Kwapien, Stan (Poland) Case Western Reserve University Probability, Functional Analysis 6/88- 9/88 Lin, Shao-Shiung (Taiwan) University of Minnesota Combustion, Mathematical 6/88- 9/89 Physics, Differential Equations Matsuki, Toshihiko (Japan) Johns Hopkins University Algebraic Analysis 9/88- 5/89 Olagunju, David (Nigeria) Argonne National Laboratory Bifiguration Phenomena in 9/88- 8/89 Combustion Oshima, Toshio (Japan) Johns Hopkins University Algebraic Analysis 2/89- 4/89 Peletier, L.A. (Netherlands) Argonne National Laboratory Non-Linear Differential Equations 7/88- 8/88 Power, John (Australia) Case Western Reserve University Category Theory 8/88- 6/89 Rao, S. E. (India) University of Saskatchewan Lie Theory 9/88- 8/89 Sekiguchi, Jiro (Japan) Johns Hopkins University Algebraic Analysis 9/88- 5/89 Shimizu, Yuji (Japan) Johns Hopkins University Algebraic Analysis 9/88- 5/89 Sieverking, Malte (West University of British Columbia Differential Equations 9/88- 1/89 Germany) (J. W. G. University Hamburg) Suwa, Noriyuki (Japan) Queen's University Arithmetic Algebraic Geometry 7/88- 8/88 Szymanski, Jerzy (Poland) Case Western Reserve University Random Graphs 8/88- 12/88 Tutek, Z. (Yugoslavia) University of Saskatchewan Hyperbolic Systems 3/88- 8/89 Ueno, Kenji (Japan) Queen's University Algebraic Geometry, Superstring 8/88 Theory Womersley, Robert Argonne National Laboratory Numerical Optimization 9/88- ll/88 (Australia) 1/89- 2/89 Wu, Jianhong (People's Memphis State University Functional Differential Equations 8/88- 12/88 Republic of China)

914 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Visiting Mathematicians

Name and Home Country Host Institution Field of Special Interest Period of Visit Xiao, Changcheng (China) University of British Columbia Finite Group Theory 10/88- 9/89 Ycart, Bernard (France) Case Western Reserve University Probability 6/88- 8/88 Yokonuma, T. (Japan) University of Saskatchewan Lie Theory 9/88- 8/89

MATHEMATICAL QUANTUM FIELD THEORY AND RELATED TOPICS

Joel S. Feldman and Lon M. Rosen, Editors (Conference Proceedings, Canadian Mathematical Society, Volume 9)

Aimed at researchers and advanced graduate students in Recherches Mathematiques of the Universite de Montreal in mathematical physics, this book constitutes the proceedings September 1987. With articles by some of the top researchers of a conference on mathematical quantum field theory and in the field, this book will bring readers to the leading edge of related topics. The conference was held at the Centre de research in a number of areas of mathematical physics.

1980 Mathematics Subject Classifications: 81, 82 Shipping/Handling: 1st book $2, each ISBN 0-8218-6014-3, LC 88-1274 add'l $1, $25 max. By air, 1st book $5, ISSN 0731-1036 each add'l $3. $100 max. 276 pages (softcover), April 1988 Prepayment required. Order from AMS. Individual member $19, List price $32, P.O. Box 1571. Annex Station, Providence, Institutional member $26 Rl 02901-9930. or call 800-556-7774 to use To order, please specify CMSAMS/9 NA VISA or MasterCard.

FACTORIZATIONS OF b" ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12 UP TO HIGH POWERS, SECOND EDITION John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff, Jr. (Contemporary Mathematics, Volume 22, Second Edition) This book is a revised and updated edition of a work that occurred since 1925. An update to the introduction is included originally appeared in 1983. It gives a historical account of the in this edition and discusses the major advances that have various methods and machines that have been used to factor, been made in the five years since the first edition appeared. and prove prime, the numbers b" ± 1. It is a revised version The introduction also discusses the multiplicative structure of an extension of a rare .1925 worJ< by Cunningham and of b" ± 1 and explains the relation between the two kinds of brings together results going back to the seventeenth century. algebraic factorizations of these numbers. The factorizations and the very large primes of special form 1980 Mathematics Subject Classification: 11 ~\\EMA~ are useful in group theory, number theory, discrete Fourier ISBN 0-8218-5078-4, LC 83-12316 ~~TPHTOIMH !d<' [[]] transforms, random number generators, and cryptography. ISSN 0271-4132 qj 1 ~ ~ ~ The present edition contains more than 2000 large primes 320 pages (softcover), June 1988 ~\ ~ ~ II~ which have never been published before. Individual member $19, List price $31, "'~ Jl Institutional member $25 • ;('~ ~ To order, please specify CONM/22NA 0UNoED" The book contains complete factorizations of b" ± 1 for the given values of b and for all n :::; 100, and for many n > 100. Shipping/Handling: 1st book $2, each add'! $1, $25 max. Included is an extensive and valuable introduction which max. By air, 1st book $5, each add'! $3, $100 Prepayment required. Order from AMS, P.O. Box 1571, describes the developments in computing technology and Annex Station, Providence, RI 02901-9930, or call in methods of factoring and primality testing which have 800-556-7774 to use VISA or MasterCard.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 915 Reciprocity Agreements

Asia The American Mathematical Society has "reciprocity agree­ ments" with a number of mathematical organizations around ·Allahabad Mathematical Society* the world. A current list appears below. Apply to: P. Srivastava, Secretary, Allahabad Mathemat- provide for reduced dues for These Reciprocity Agreements . ical Society, 10, C.S.P. Singh Marg, Allahabad-211001, members of these organizations who choose to join the AMS and who reside outside of the U.S. and Canada. Reciprocally, India. members of the AMS who reside in the U.S. or Canada may join Dues: US $25 (annual), US $250 (life), US $125 (life for these organizations at a reduced rate. Summaries of the priv­ members of American Mathematical Society); payable ileges available to AMS members who join under the terms of to Allahabad Mathematical Society. Mem­ reciprocity agreements are given on the following pages. Privileges: The Journal of the Society is sent to members who join the AMS as reciprocity bers of these organizations regularly. members enjoy all the privileges available to ordinary mem­ bers of the Society. AMS dues for reciprocity members are $44 Officers: U. N. Singh (President), Vachaspati (Vice­ for 1988 and $44 for 1989. Each organization was asked to President), K. K. Azad (Treasurer), P. Srivastava review and update its listing in the Spring. An asterisk ( *) after (Secretary). the name of an organization indicates that no response to this request had been received when the August Notices went to press. A disc ( •) before the name of an organization indicates Calcutta Mathematical Society* that application forms for that organization may be obtained by writing the American Mathematical Society, P.O. Box 6248, Apply to: U. Basu, Secretary, Calcutta Mathematical Providence, Rhode Island 02940. Society, 92, Acharya Prafulla Chandra Road, Calcutta 700 009, India. Dues: $2; payable to U. Basu, Secretary. Africa Privileges: News Bulletin (free of cost). •Nigerian Mathematical Society* Officers: M. Dutta (President), P. C. Vaidya, B. R. Bhonsle, F. Harary, L. Debnath, S. P. Bandyopadhyay, Apply to: Christopher 0. Imoru (Secretary), Nigerian (Vice-Presidents), B. K. Datta, (Treasurer), U. Basu Mathematical Society, Department of Mathematics, (Secretary). University of Ife, lle-lfe, Nigeria. Dues: $1 0; payable to the Treasurer, Nigerian Mathe­ Indian Mathematical Society matical Society, Department of Mathematics, University of Benin, Benin City, Nigeria. Apply to: S. P. Arya, General Secretary, Indian Mathe­ Maitreyi Privileges: Journal of the Nigerian Mathematical Society matical Society, Department of Mathematics, New at the price normally charged to individual members. College, Bapu Dham Complex, Chanakyapuri, Delhi 110 021, India. Officers: A. Olubummo (President), J. 0. C. Ezeilo IMS, (Vice-President), C. 0. Nwachuku (Treasurer), C. 0. M. Dues: $20; payable to V. M. Shah, Hon. Treasurer, Baroda, Imoru (Secretary), S. A. llori (Assistant Secretary), H. 0. Department of Mathematics, M. S. University, Tejumola (Editor-in-Chief). India. Privileges: Journal of the Indian Mathematical Society or Mathematics Student.

916 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Reciprocity Agreements

Officers: M. K. Singal (President), V. Singh (Immediate Officers: Simon C. Hsieh (President), Jau-D. Chen past President), V. M. Shah (Treasurer), S. P. Arya (Treasurer), Liang-Chi Tsao (Secretary). (General Secretary), J. N. Kapur {Academic Secretary), 1. B. S. Passi (Editor of Journal of Indian Mathematical Punjab Mathematical Society Society), A.M. Vaidya (Editor of Mathematics Student). Apply to: M. Rafique, Secretary, Punjab Mathematical Korean Mathematical Society Society, Department of Mathematics, University of the Punjab, Quaid-1-Azam Campus, Lahore 54590, Pakistan. Apply to: Kun Soo Chang, 538 Dowha Dong, Mapo Ku, Dues: US $25 for life membership; payable to M. Sung Ji Building, Room 706, Seoul 121, Korea. Rafique, Secretary. Dues: $15; payable to the Korean Mathematical Society. Privileges: Society Newsletter, Proceedings of the Con­ Privileges: Free receipt of Bulletin (two issues per year) ferences, Symposia, and Seminars arranged by the and Journal of the Korean Mathematical Society (two Society. issues per year). Officers: F. D. Anjum Roomani {President), Ch. Abdul Officers: Jeong Dae Rim (President), Suk-Young Lee Hameed, S. M. Kalim (Vice-Presidents), Khalifa Rashid­ (Vice-President), Ha-Jine Kimn (Treasurer), Kun Soo ud-Din (Treasurer), M. Rafique (Secretary). Chang (Secretary). Southeast Asian Mathematical Society* .Malaysian Mathematical Society* Apply to: Boon-Yian Ng, Southeast Asian Mathematical Apply to: The Secretary, Malaysian Mathematical Society, Society, cfo Department of Mathematics, University of cfo Department of Mathematics, University of Malaya, Malaya, Kuala Lumpur, Malaysia. Kuala Lumpur, Malaysia. Dues: US $5; payable to Boon-Yiang Ng, Southeast Dues: $5; payable to Malaysian Mathematical Society. Asian Mathematical Society. Privileges: MMS Newsletter, Bulletin of the Malaysian Privileges: SEAMS Newsletter, Southeast Asian Bulletin Mathematical Society (two issues per year), reduced rate of Mathematics. for Menemui Matematik (three issues per year). Officers: Lim Chong Kong (President), Chong Chi Officers: Sin-Leng Tan (President), Abdul Razak Salleh, Tat, Mari-Jo Ruiz (Vice-Presidents), Gek-Ling Chia Gek-Ling Chia (Vice-Presidents), Nik Ahmad Kamal {Treasurer), Boon-Yian Ng (Secretary). {Treasurer), Boon-Yian Ng (Secretary). · • Vijnana Parishad of India* •Mathematical Society of Japan Apply to: H. M. Srivastava, Foreign Secretary, VPI, Apply to: Setsuko Izawa, Secretary, Mathematical Society Department of Mathematics, University of Victoria, of Japan, 25-9-203, Hongo 4-chome, Bunkyo-ku, Tokyo Victoria, British Columbia, Canada, V8W 2Y2 or 113,Japan. · R. C. Singh Chandel, Secretary, VPI, Department of Dues: US $40; payable to Mathematical Society of Japan. Mathematics, D. V. Postgraduate College, Orai-285001, U. P., India. Privileges: Journal of the Mathematical Society of Japan; Sugaku (in Japanese) for US $8 additional dues. Dues: US $7.50 (annual), US $75 (life); payable to Vijnana Parishad, o Department of Mathematics, Officers: Seizo Ito (President), Takuo Isa (Treasurer), cj Setsuko Izawa (Secretary). · · D. V. Postgraduate College, Orai-285001, U. P., India. Privileges: Jftanabha (an interdisciplinary mathematical journal currently published once a year); back volumes Mathematical Society available at 25% discount. of the Republic of China* Officers: J. N. Kapur (President), R. P. Agrawal, M. Apply to: Mathematical Society of the Republic of China, K.. Singal, U. N. Shukla (Vice-Presidents), R. C. Singh P.O. Box 23-3, Taipei, Taiwan, Republic of China. Chandel (Secretary-Treasurer), H. M. Srivastava (Foreign Secretary). Dues: N.T. $200 (US $5.00); payable to Mathematical Society of the Republic of China. Privileges: Chinese Journal of Mathematics (two to four issues per year).

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 917 Reciprocity Agreements

Europe ·Deutsche Mathematiker-Vereinigung e. V. Asociacion Matematica Espanola* Apply to: Deutsche Mathematiker-Vereinigung e.V., Al­ bertstraBe 24, 7800 Freiburg, Federal Republic of Apply to: Miguel de Guzman, President, Asociaci6n Germany. Matematica Espanola, Facultad de Matematicas, Uni­ Dues: DM 30.- (for reciprocity members); payable versidad Complutense, Madrid 3, Spain. to Kreissparkasse Tubingen 16269 (BLZ 641 500 Dues: US $15 for members of the Am-erican Math­ 20), Federal Republic of Germany or Postscheckamt ematical Society; payable to Asociaci6n Matematica Stuttgart 18517-706 (BLZ 600 100 70), Federal Republic Espanola. of Germany. Privileges: Boletin de Ia Asociaci6n M atemdtica Espanola; Privileges: Mitteilungen der Deutschen Mathematiker­ Publicaciones de Ia Asociaci6n Matemdtica Espanola (at Vereinigung (four issues a year), Jahresbericht der reduced prices). Deutschen Mathematiker-Vereinigung (four issues a Officers: Miguel de Guzman, (President), Ireneo Peral year). (Treasurer), Maria T. Carrillo (Secretary). Officers: W. Tornig (President), K. P. Grotemeyer (Treasurer), J. Plum (Secretary) . • Berliner Mathematische Gesellschaft e.V. Apply to: R. D. Grigorietf, Fachbereich Mathematik, .Edinburgh Mathematical Society TU Berlin, StraBe des 1 7.Juni 136, D-1 000 Berlin 12, Apply to: The Honorary Secretary, Edinburgh Mathe­ Federal Republic of Germany. matical Society, James Clerk Maxwell Building, King's Dues: DM 12; payable to G. Preuss, lnstitut fur Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scot­ Mathematik I, FU Berlin, Arnimallee 3, D-1 000 Berlin land. 33, Federal Republic of Germany. Dues: $6 (preferably £3 sterling); payable to the Honorary Privileges: One free copy of"Sitzungsberichte der BMG". Treasurer. Officers: E.-J. Thiele (President), H.G.W. Begehr (Vice­ Privileges: Proceedings at reduced rate of $12 (preferably President), G. Preuss (Treasurer), R. D. Grigorietf £6 sterling) per annum. (Secretary). Officers: R.M.F.Moss (President), B. D. Sleeman (Vice­ President), A. C. McBride (Treasurer), J. Martin, C. J. • Dansk Matematisk Forening Shaddock (Secretaries) . Apply to: Mogens Esrom Larsen, Secretary, Dansk Matematisk Forening, Universitetsparken 5, 2100 K..aben­ eGesellschaft fur Angewandte Mathematik havn 0, Denmark. und Mechanik (GAMM) Dues: D.kr. 38; payable to Erik Christensen, Treasurer. Address for mail: Reinhard Mennicken, University of Privileges: Mathematica Scandinavica (D.kr. 183 per Regensburg, D-8400 Regensburg, Federal Republic of volume), Nord. Mat. Tidss. (Normal) (N.kr.120 per Germany. volume). (Members of the American Mathematical Apply to: R. Ansorge, Institut fur Angewandte Math­ Society do not have to join Dansk Matematisk Forening ematik, Universitat, Hamburg, Bundesstr. 55, D-200 to obtain the journals. Subscription orders should Hamburg 13, Federal Republic of Germany. Univer­ be sent directly to the journals: Normal, Dues: 25.-DM; payable to J. Siekmann, FB 12 - Maschi­ Postbox 2959 Teyen, sitetsforlaget, Avd. for tidsskrifter, nentechnik, Universitat- GHS Essen, Schutzenbahn 70, Scandinavica, Matematisk Oslo 6, Norway; Mathematica D-4300 Essen, Federal Republic of Germany. Institut, Aarhus Universitet, 8000 Aarhus C, Denmark.) Privileges: Regular publications of GAMM and partici­ (President), Mogens Esrom Officers: Lars-Erik Lundberg pation in scientific meetings at a reduced rate. Larsen (Vice-President), Erik Christensen (Treasurer), Zierep (Vice­ Mogens Esrom Larsen (Secretary), Ebbe Thue Poulsen, Officers: W. Walter (President), J. Sten Markvorsen. President), J. Siekmann (Treasurer), R. Mennicken (Secretary).

918 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Reciprocity Agreements

Glasgow Mathematical Association Officers: E. C. Zeeman {President), J. H. Coates, K. W. Gruenberg (Vice-Presidents), J.D.M. Wright (Treasurer), ,Apply to: R. J. Steiner, Glasgow Mathematical As­ C. J. Mulvey, A. R. Pears (Secretaries), D. A. Brannan sociation, Department of Mathematics, University of (Publications Secretary). Glasgow, University Gardens, Glasgow G12 8QW, Scotland. Forening Dues: £22; payable to Glasgow Mathematical Association. Norsk Matematisk Privileges: Glasgow Mathematical Journal. Apply to: Gerd Salter, Norsk Matematisk Forening, Officers: K. A. Lindsay (President), R. J. Cole, R. Bailey Matematisk Institutt, Postboks 1053 Blindern, N-0316 (Vice-Presidents), R. J. Steiner (Treasurer), P. A. Boyle Oslo 3, Norway. (Secretary). Dues: N.kr. 20 or N.kr. 200 for permanent membership; payable to Gerd Salter, Norsk Matematisk Forening. .Irish Mathematical Society Privileges: Reduced subscription rate on Mathematica Scandinavica and NORMAT (Nordisk Matematisk Tid­ Address for mail: A. G. O'Farrell, Secretary, Maynooth skrift), free monthly information bulletin "infomat" College, County Kildare, Ireland. about the activites of the Society. Apply to: G. M. Enright, IMS Treasurer, Department Officers: Bernt 0ksendal (President), Ragni Piene of Mathematics, Mary Immaculate College, Limerick, (Vice-President), Geir Ellingsrud (Treasurer), Jan Tore Ireland. Lenning (Secretary). Dues: US $4 (1989), $6 (1990); payable to G. M. Enright. .osterreichische Mathematische Privileges: Right to receive the Bulletin ofthe Irish Math­ Gesellschaft ematical Society; right to subscribe to the Proceedings of the Royal Irish Academy (Section A) at a reduced rate. Apply to: Werner Kuich, President, Osterreichische Math­ Officers: S. Dineen (President), F. Gaines (Vice­ ematische Gesellschaft, Technische Universitat Wien, President), G. M. Enright (Treasurer), A. G. O'Farrell Wiedner HauptstraBe 8-10, A-1 040 Wien, Austria. (Secretary). Dues: oS 150; payable to lnge Troch, Treasurer. Privileges: Internationale Mathematische Nachrichten islenzka Strerdfrredafelagid * (IMN). Officers: Werner Kuich (President), Ludwig Reich (Vice­ Apply to: President, Raunvisindastofnun Haskolans, President), lnge Troch (Treasurer), Hans Reichel (Secre­ Dunhaga 3, 107 Reykjavik, Iceland. tary). Dues: $1 0; payable to islenzka Strerdfnedafelagid. Privileges: Newsletter (yearly). Polskie Towarzystwo Matematyczne* Officers: Halldor I. Eliasson (President), Kristian Jonasson (Treasurer), Ragnar Sigurds.son (Secretary). Apply to: Polskie Towarzystwo Matematyczne, Sniadec­ kich 8, 00-950 Warszawa, Poland. London Mathematical Society Dues: $8; payable to Po1skie Towarzystwo Matematyczne. Privileges: Participation in scientific conferences orga­ Apply to: The Administrator, London Mathematical nized by the Polish Mathematical Society and in its sci­ Society, Burlington House, Piccadilly, London W1V entific sessions; in addition, members receive one of the ONL, United Kingdom. · · following five series of the publication Annales Societatis Dues: £5; payable to London Mathematical Society. Mathematicae Po/onae: Commentationes Mathemati­ (New members should not send payment until elected.) cae in congress languages, Wiadomosci Matematyczne Privileges: LMS Newsletter. Reduced rates for the Bul­ (Mathematical News) in Polish, Matematyka Stosowana letin, Journal, and Proceedings ofthe LMS; Nonlinearity; (Applied Mathematics) in Polish, Fundamenta Infor­ Journal ofApplied Probability; Mathematika; Mathemat­ maticae in congress languages, Dydaktyka Matematyki ical Proceedings of the Cambridge Philosophical Society; (Didactics of Mathematics) in Polish. Quarterly Journal of Mathematics; LMS Lecture Notes; Officers: Wlesllaw Zelazko (President), Jozef Siciak, LMS Student Texts; LMS Monographs. {Please write to Marian Kwapisz (Vice-Presidents), Andrzej Hulanicki the LMS for complete details.) (Secretary), Maciej Brynski (Vice Secretary), Maciej MQ.czynski (Treasurer).

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 919 Reciprocity Agreements

Real Sociedad Matematica Espanola Societe Mathematique de France Apply to: J. Llovet, Secretario General de Ia Real Address for mail: Societe Mathematique de France, B.P. Sociedad Matematica Espanola, Serrano 123, Madrid 126-05, F 75226 Paris, Cedex 05, France. 28006, Spain. Apply to: Madame Janine Le Peintre, S.M.F., B.P. Dues: $30; payable to Secretario, R. S.M. E. 126-05, F 75226 Paris, Cedex 05, France. Privileges: Boletin Real Sociedad Matemdtica Espaftola Dues: $40 or $58; payable to American Mathematical and Revista Matemdtica Ibero Americana. Society or S.M.F. Officers: P. L. Garcia Perez (President), Juan Llovet Privileges: Individuals who pay dues of $40 are entitled Verdugo (Secretary). to receive Officiel and Gazette. Individuals who pay dues of $58 are entitled to Officiel, Gazette, and Bulletin. de Matematica Asterisque may be purchased at a discount price. •Sociedade Portuguesa (Members in the U.S., Canada, or Mexico should order Apply to: J. E. Valenca, Sociedade Portuguesa de their copies from the AMS. See the AMS Catalogue of Matematica, Av. da Republica 37 4°, 1000 Lisboa, Publications.) Portugal. Officers: M. Demazure (President), J. P. Bourguignon, Dues: 600 Portuguesa Escudos; payable to Sociedade J. M. Lemaire, J. M. Deshouillers (Vice-Presidents), Portuguesa de Matematica. C. Gourieroux (Treasurer), M. Chaleyat-Maurel, B. Helffer (Secretaries). Privileges: Boletim da Sociedade Portuguesa de Matemdti­ ca, free; discount of 70% in the subscription fees for Portuga/iae Matemdtica. .societe Mathematique Suisse Officers: A. Ribeiro Gomes (President), J. E. Valenca, J. Apply to: H. Holmann, Secretary SMS, Institut de C. Silva, J. F. Queir6 (Vice-Presidents), A. M. Cadete Universite de Fribourg, CH-1700 Fri­ (Treasurer). Mathematiques, bourg, Perolles, Switzerland. Dues: SFr. 15.- for members of the AMS residing outside Societat Catalana de Matematiques Switzerland; payable to H. Holmann. Apply to: Secretari de Ia Societat Catalana de Matema­ Privileges: Commentarii Mathematici Helvetici (reduced tiques, Carrer del Carme 4 7, 08001 Barcelona, Spain. price); information concerning activities of SMS. Dues: 1000 pessetes for members of the AMS, payable Officers: N. A'Campo (President); U. Stammbach (Vice­ to the Societat Catalana de Matematiques. President); H. Holmann (Treasurer-Secretary). Privileges: Butletti de Ia Societat Catalana de Matema­ tiques, (2 numbers a year). Societe de Mathematiques Appliquees Officers: J. Girbau (President), Carles Perell6 (Secretary), et Industrielles Rubi Corber6 (Associated Secretary). Apply to: Societe de Mathematiques Appliquees et Industrielles, Centre de Mathematiques Appliquees, Societe Mathematique de Belgique* Ecole Polytechnique, 91128 Palaiseau, France. Apply to: Guy Hirsch, Secretary, Societe Mathematique Dues: US $25; Societe de Mathematiques Appliquees et de Belgique, 317, Avenue Charles Woeste, 1090 Brussels, Industrielles. Belgium. Privileges: Free subscription to the News Bulletin, Dues: $16; payable to Societe Mathematique de Bel­ reduced rates at conferences organized by SMAI. gique, preferably by International Money Order, VISA, Officers: J. C. Nedelec (President), P. Lascaux, J. Periaux MasterCard, or American Express. (Vice-Presidents), M. Lenoir (Treasurer), G. Meurant Privileges: Bulletin de Ia Societe Mathematique de (Secretary). Belgique, Series A (two numbers per year) and Series B (two numbers per year), about 450 pages a year. .suomen Matemaattinen Yhdistys Officers: P. Van Praag {President), F. Van Oystreyen (Vice-President), G. Hirsch (Secretary-Treasurer). Apply to: Ilkka Holopainen, Secretary, Department of Mathematics, University of Helsinki, Hallituskatu 15, SF-00 100 Helsinki, Finland.

920 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Reciprocity Agreements

Dues: 60 FIM; payable to Aatos Lahtinen, Treasurer, edelingen (nine issues a year containing announcements Department of Mathematics, University of Helsinki, and book reviews), Proceedings of the Royal Academy of Hallituskatu 15, SF-00100 Helsinki, Finland. Sciences- "Indagationes Mathematicae" (can be obtained Privileges: Arkhimedes. at a reduced subscription rate of Hfl 120.-). Officers: Seppo Rickman (President), Olli Martio (Vice­ Officers: G. Y. Nieuwland (President), J. P. Murre President), Aatos Lahtinen (Treasurer), Ilkka Holopainen (Vice-President), A. van Harten (Treasurer), R. W. (Secretary). Goldbach (Secretary).

Svenska Matematikersamfundet Latin America Apply to: Svenska Matematikersamfundet, Matematiska .sociedad Colombiana de Matematicas Institutionen, Umea Universitet, S-901 87 Umea, Swe­ Apply to: Sociedad Colombiana de Matematicas, den. Apartado Aereo 2521, Bogota, D.E., Colombia. Dues: 75 Skr. or 100 Skr. for permanent membership; Dues: $16; payable to Sociedad Colombiana de payable to Svenska Matematikersamfundet, Ingegerd Matematicas. Palmer, Saningsvagen 54, S-175 45 Jarfalla, Sweden. Privileges: Either Revista Colombiana de Matemdticas Privileges: Mathematica Scandinavica and Nordisk (four issues a year) or Lecturas Matemdticas (three Matematisk Tidskrift at reduced rate. Information issues a year). about the meetings of the Society. Officers: Myriam Munoz de Ozak (President), Jairo Officers: Urban Cegrell (President), Gert Almkvist (Vice­ Charris (Vice-President), Luis Giraldo (Treasurer), President), Ingegerd Palmer (Treasurer), Tord Sjodin Myriam Leonor Campos (Secretary). (Secretary). Sociedad de Matematica de Chile •Unione Matematica Italiana Apply to: Secretario, Sociedad de Matematica de Chile, Apply to: Segreteria della Unione Matematica ltaliana, Casilla 110-V, Valparaiso, Chile. Dipartimento di Matematica, Piazza Porta S. Donato, 5, Dues: US $1 0; payable to Sociedad de Matematica de 40127 Bologna, Italy. Chile. Dues: 40,000 lire; payable to · Unione Matematica Privileges: Receive: Gaceta de Ia Sociedad (three issues Italiana. per year), Circular de Informaciones (ten issues per year); Privileges: Free Notiziario del/'UMI (monthly), Bollettino Notas de Ia Sociedad de Matemdtica de Chile. dell'UMI, ser. A (3 issues a year), and membership list. Officers: Rubi Rodriguez (President), Rolando Rebolledo Reduced fees for subscriptions to Bollettino del/'UMI, (Vice-President), Jaime Figueroa (Treasurer), Victor Ser. B. and discounts for other UMI publications. Gonzalez (Secretary). Officers: Alessandro Figa-Talamanca (President), Benedetto Scimemi (Vice-President), Enrico Obrecht (Treasurer), Giuseppe Anichini (Secretary). Sociedad Matematica de Ia Republica Dominicana Wiskundig Genootschap. Apply to: Eduardo Luna, Sociedad Matematica de Ia Re­ publica Dominicana, Apartado 797-2, Santo Domingo, Address for mail: Wiskundig Genootschap, Delft Uni­ Dominican Republic. versity of Technology, Department of Mathematics Dues: US $6; payable to Isidro Rodriguez, Sociedad and Informatics, P.O. Box 356, 2600 AJ Delft, The Netherlands. Matematica de la Republica Dominicana. Privileges: Right Apply to: Membership Department, Wiskundig Genoot­ to receive Notimat (bimonthly newslet­ ter) and Revista schap, University of Utrecht, Postbus 80010, 3508 TA Matemdtica Dominicana (twice a Utrecht, The Netherlands. year). Dues: Hfl 40.-; payable to Amro Bank, Utrecht, Officers: Eduardo Luna (President), Pedro Suarez (Vice­ The Netherlands, Account 45.65.88.167, Penningmeester President), Isidro Rodriguez (Treasurer), David Castillo Wiskundig Genootschap. (Secretary). Privileges: Nieuw Archie/ Voor Wiskunde (three issues a year containing articles and a problem section), Med-

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 921 Reciprocity Agreements

•Sociedad Matematica Mexicana* Officers: C. Pereira da Silva (President), R. Petronzelli (Vice-President), E. Andretta (Treasurer), M. G. Borges Apply to: Sra. Hilda C. de Villa, Apartado Postal 70-450, (Secretary). Mexico 20, D.P. Mexico 04510. Dues: US $1 0; payable to Sociedad Matematica Mexi­ Union Matematica Argentina cana. Privileges: To be a regular member paying half of the Apply to: Secretary of the Union Matematica Argentina, regular fee for persons living outside of Mexico. Casilla de Correo 3588, 1000-Correo Central, Buenos Officers: Alejandro Lopez-Yanez (President), Diego B. Aires, Argentina. Hernandez (Vice-President), Juan Morales {Treasurer), Dues: US $6; payable to Union Matematica Argentina. Luis Rivera-Gutierrez (Secretary). Privileges: Revista de Ia Union Matemdtica Argentina (two issues per year). •Sociedade Brasileira de Matematica Officers: Roberto L. Cignoli (President), Juan A. Tirao, Roberto A. Macias (Vice-Presidents), Graciela Ana Apply to: Cesar Camacho, Estrada Dona Castorina, Canziani (Treasurer), Susana Elena Trione (Secretary). 110-Jardim Botanico, Rio de Janeiro, RJ, Brazil, 22460. Dues: US $6; payable to Sociedade Brasileira de Mate­ Middle East matica. Privileges: 50% reduced membership rate; Boletim da •Iranian Mathematical Society discount of Society publications. SBM; and up to a 25% Apply to: Secretary, Iranian Mathematical Society, P. 0. Officers: Cesar Camacho (President), Paulo Roberto G. Box 13145-418, Tehran, Iran. Carlos F. B. Palmeira {Treasurer), Sad (Vice President), Dues: $10; payable to Iranian Mathematical Society, M. Jorge D. Carneiro (Secretary). Mario Radjabalipour. Privileges: Bulletin of the Iranian Mathematical Society Sociedade Brasileira de Matematica and reduced rate for participation in the annual Iranian Aplicada e Computacional Mathematics conferences. Officers: M. Toomanian (President-Secretary), M. Rad­ Apply to: Comissao de Admissao da SBMAC, Rua Lauro jabalipour (Treasurer). Muller, 455, 22.290, Botafogo, Rio de Janeiro, RJ, Brasil. Dues: $16; payable to Sociedade Brasileira de Matematica Israel Mathematical Union Aplicada e Computacional. Apply to: Israel Mathematical Union, c/o Abra­ Privileges: Vote in election for officers of the Society, ham Berman, Secretary, Department of Mathematics, the present papers at meetings of the SBMAC, receive Technion-Israel Institute of Technology, Haifa 32000, and Boletim and Matemdtica Ap/icada e Computacional, Israel. reduced registration fees at SBMAC Congress. Dues: $10; payable to Israel Mathematical Union. Officers: Carlos A. de Moura (President), Leon R. Sinay Privileges: Newsletter; may attend and present papers at (Vice-President), Joaquim Pereira Neto (Treasurer), Hilton V. Machado (Secretary). meetings. Officers: Zvi Ziegler (President), Moshe Goldberg (Trea­ surer), Abraham Berman (Secretary) . • sociedade Paranaense de Matematica Apply to: C. Pereira da Silva, Sociedade Paranaense de South Pacific Matematica, Caixa Postal 1261, 80001, Curitiba-PR, Mathematical Society Brasil. Australian Dues: US $12; payable to Sociedade Paranaense de Address for mail: W. R. Bloom, Murdoch University, Matematica. Murdoch, WA 6150 Australia. Privileges: Boletim da Sociedade Paranaense de Mate­ Apply to: B. D. Jones, Department of Mathematics, mdtica (two issues per year), Monografias da Sociedade University of Queensland, St. Lucia, Queensland 4067, Paranaense de Matemdtica (one issue per year). Australia. Dues: $A25; payable to B. D. Jones.

922 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Reciprocity Agreements

Privileges: Free copies of The Gazette and copies of •New Zealand Mathematical Society Journal Series A and B, The Bulletin and the Lecture Series at members' rates. Address for mail: Department of Mathematics, University Officers: N. S. Trudinger (President), R. S. Anderssen, of Canterbury, Christchurch, New Zealand. G. I. Gaudry, J. R. Blake (Vice-Presidents), B. D. Jones Apply to: J. A. Shanks, Treasurer, Department of (Treasurer), W. R. Bloom (Secretary). Mathematics, University of Otago, Dunedin, New Zealand. Mathematical Society of the Philippines* Dues: $NZ 13.50; payable to J. A. Shanks, Treasurer, Department of Mathematics, University of Otago, Apply to: Membership Committee, Mathematical Society Dunedin, New Zealand. of the Philippines, Department of Mathematics, Ateneo Privileges: Newsletter of the NZMS (3 per year); de Manila University, P.O. Box 154, Masophman, subscription to Mathematical Chronicle at reduced rate. Philippines. Officers: B. A. Woods (President), I. L. Reilly (Vice­ Dues: $5; payable to Mathematical Society of the President), J. A. Shanks (Treasurer), D. R. Breach Philippines. (Secretary). Privileges: Publications and newsletter of the Mathemat­ ical Society of the Philippines. Officers: Jose Marasigan (President), Rene P. Felix (Vice-President), Leticia Castillo (Treasurer), Norman Quimpo (Secretary).

THEORY AND APPLICATIONS OF 1980 Mathematics Subject Classifications: DIFFERENTIABLE FUNCTIONS OF SEVERAL 26, 30,33,35, 39, 41, 42, 46, 47, 15 and others VARIABLES. XI ISBN 0-8218-3117-8 S. M. Nikol'skii, Editor ISSN 0081-5438 (Proceedings of the Steklov Institute, Volume 173) 290 pages (softcover), March 1988 Individual member $70, List price $116, Institutional member $93 This collection focuses on various problems in the theory To order, please specify STEKL0/173NA of differentiable functions and its applications to partial differential equations. Among the topics covered are: ~~~!d t;;~~~ • imbedding, extension, and approximation theorems; (~\~~;j~~ • traces for spaces of smooth functions of several variables • ;('~ ,...__ __./ ;/a!b defined on subsets of Euclidean space; ou"NDf.o\'D" • new inequalities for differentiable functions of several Shipping/Handling: 1st book $2. each variables; add'l $1. $25 max. By air. 1st book $5. • new sufficient conditions for multipliers of Fourier integrals; each add'l $3. $100 max. Prepayment required. Order from AMS. P.O. Box • fractional difference operators; 1571. Annex Station. Providence. Rl 02901-9930. or call 800-556-7774 to use • coercive properties of degenerate elliptic equations and VISA or MasterCard. difference equations.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 923 New Members of the AMS

ORDINARY MEMBERS George R Barnes Neculai Curteanu University of Louisville Universitatea 'AI. I. Cuza' Bassam Youssef Al-Nashef Louisville, Kentucky Iasi, Romania Yarmouk University Irbid, Jordan Cornelia-Livia Bejan Ronald J Czochor Universitatea 'AI. I. Cuza' Glassboro State College Porald C Alexander lasi, Romania Glassboro, New Jersey .· Norwood, Massachusetts Lawrence S Braden Dennis E Davenport Sara Margaret Allan Honolulu, Hawaii Oxford, Ohio Sarnia, Ontario Canada Joanne B Brooks Z I Dimitrov Jack MAnderson Arkansas College Plovdiv, Bulgaria University of Guam Batesville, Arkansas Mangilao, Guam Anabeth Dollins Susan Ann Brown Pittsburgh, Pennsylvania Paul Andrew Savannah, Georgia Goodwood, Australia Gregory Thomas Dougherty Thomas Eric Brunner Los Angeles, California Sigurd B Angenent Mountain View, California University of Wisconsin Renaud Dreyer Madison, Wisconsin Peggy Capell Ottawa, Ontario Canada Forest Park, Georgia Betty J Arnold James W Dunion Big Rapids, Michigan Grace C Cascio Taftville, Connecticut Northeast Louisiana University Gene W Arnold Martin Ehde Jr Monroe, Louisiana Big Rapids, Michigan William Carey College Donald R Chalice Hattiesburg, Mississippi Milos Arsenovic Western Washington University Berkeley, California George R Exner Bellingham, Washington Wooster, Ohio Florence H Ashby Dianna Chen Rockville, Maryland Vasiliki Achileas Farmaki Queens College Athens, Greece ldris Assani Flushing, New York University of Toronto Joan Feigenbaum Gui-Qiang Chen Toronto, Ontario Canada A T & T Bell Labs New York University, Courant Murray Hill, New Jersey Eric Bach Institute of Mathematical Sciences University of Wisconsin New York, New York Ivan Filippenko Madison, Wisconsin Redondo Beach, California Donald Cook Jacqueline Lee Bacon Darton College Marcia K Fitzmaurice Moraga, California Albany, Georgia University of Tennessee at Chattanooga Keith Ball Linda K Cooke Chattanooga, Tennessee Texas A & M University Spartanburg, South Carolina College Station, Texas Daniel Joseph Flynn Jr Donna Lynne Cooper Phillipsburg, New Jersey Rob S Ballantyne Wright-Patterson AFB, Ohio Delta, British Columbia Canada Athanassios S Fokas Thomas E Covington Clarkson University Baby Varkey Balliappadath Natchitoches, Louisiana Potsdam, New York International Centre for Theoretical B Eugene Crumrine Jr Physics Frederick C Fowler Media, Pennsylvania Trieste, Italy Atlanta, Georgia

924 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New Members of the AMS

Barbara Gail Frazier Neil Immerman Jack J LeTourneau Kennesaw College Yale University Oakland, California Marieta, Georgia New Haven, Connecticut Wen-Chien Lee Jane M Fugate Abdulcadir S lssa West Lafayette, Indiana Maryland Lincoln Memorial University Takoma Park, Kathryn E Lenz Harrogate, Tennessee Jack Jan Minneapolis, Minnesota West Florida John Burdette Gage University of Xiao Xin Liao Berkeley, California Pensacola, Florida Hiazhong Normal University Jean-Marc Gambaudo Brian Raymond Jefferies Wuhan Hubei, People's Republic of University of Nice University of Wollongong China Australia Nice, France Wollongong, Xinzhi Lili Josefa I Garcia Clark Jeffries University of Texas at Arlington Baton Rouge, Louisiana Clemson University Arlington, Texas Clemson, South Carolina Octavio C Garcia Gordon E MacKay Knox College Xinhua Ji Scottsdale, Arizona Galesburg, Illinois Academia Sinica Jana D Madjarova Beijing, People's Republic of China Bert Dee Garrett Institute of Mathematics Tennessee State University Peter D Johnson Jr Sofia, Bulgaria Nashville, Tennessee Auburn University Zoltan Magyar Auburn University, Alabama Seth I Goldberg Iowa City, Iowa Julius Junevicus AFP Corp Gerald Andrew J Majda Manhasset, New York Eckerd College Princeton University Saint Petersburg; Florida Janusz S Golec Princeton, New Jersey A Jur University of Texas at Arlington Barbara Abram A Malitsky at Arlington, Texas University of Tennessee San Francisco, California Chattanooga Jose M Gonzalez-Fernandez Chattanooga, Tennessee Charles J Martin National Institute of Health Cullowhee, North Carolina Bethesda, Maryland Louis J Jurca Jr Texas City, Texas Geoffrey K Martin William J Haboush Davis, California University of Illinois Haniya Kakakhail Urbana, Illinois Fort Collins, Colorado Honore Panzu Mavinga University of the South OmarMHamed Efthimios Kappas Sewanee, Tennessee King Saud University Brighton, Massachusetts Riyadh, Saudi Arabia Terral L McKellips Shannon J Kast Cameron University State University of Therese A Hart Northwestern Lawton, Oklahoma National Academy of Science Louisiana Washington, District of Columbia Natchitoches, Louisiana Robert D McMillan Edmond, Oklahoma GholamAli Hedayat Michael J Keller Oakland, California University of North Florida Gilcin F Meadors Jacksonville, Florida Flinstone, Maryland Dietrich Helmer University of Konstanz Sharad V Keny Michael Mesterton-Gibbons Konstanz, Federal Republic of Whittier College Florida State University Germany Whittier, California Tallahassee, Florida Hugh Robert Hind Ahmed Ali Khammash Abolghassem Miamee Sutton, England Makkah, Saudi Arabia University of Hampton Hampton, Virginia John G Hollingsworth Mefbaret N Kocatepe University of Georgia Bilkent University EdgarHMims Athens, Georgia Ankara, Turkey New Orleans, Louisiana E John Hornsby Jr Patricia C Ladd Aliakbar Montazer-Haghighi Mandeville, Louisiana Jacksonville, Florida Irmo, South Carolina Joyce Stevens Huesemann MarkDLauer George Stephen Morse Houston, Texas River Vale, New Jersey Milo, Maine

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 925 New Members of the AMS

William A Smith William R Moser Michael John Romanowski Tucker, Georgia Norwood, Massachusetts Capistrano Beach, California Soedirman Marian Muresan Robert L Roth Jr Massy France Institutul Politehnic Cluj-Napoca · Emory University Paris, Cluj-Napoca, Romania Atlanta, Georgia Ton Gei Sol Academia Sinica Elizabeth A C Newton Andrzej Rucinski Taiwan Massachusetts Institute of Technology Adam Mickiewicz University Taipei, Cambridge, Massachusetts Poznan, Poland Carol Ann Spiegel Dubuque, Iowa Lyn A Noble Leila Y Ruyan Jacksonville, Florida Pensacola, Florida John Greenwood Stell University of Manchester Benjamin H Noh Jurgen W Sander Manchester, England Santa Barbara, California University of Hannover Hannover, Federal Republic of Francis J O'Brien Jr I Ya Subbotin Germany Polytechnical Institute 1014 NUSC USSR Newport, Rhode Island Klaus Schmidt Kiev, Institute for Advanced Study Edward D Onstott Antoni Sym Princeton, New Jersey University Hawthorne, California Warsaw Robert Henry Schmidt Warsaw, Poland Jim R Otto Cumberland, Maryland Sztajnic Palo Alto, California Jerzy Greggory Michael Scible Lodz University George R Parks Pensacola, Florida Lodz, Poland University of Southern Maine Portland, Maine James A Sethian G Yu Tamanyan University of California, Berkeley Erevan State University Clifford R Paulson Berkeley, California Erevan, U S S R Silver Spring, Maryland Lere Shakunle Lorraine Mignon Tawfik Ronald I Perla Matran Software International Islip Terrace, New York Canmore, Alberta Canada Gottingen, Federal Republic of Liz S Tiu Peto P Petkov Germany Richardson, Texas Sector of Logic Boris L Shekhter Sofia, Bulgaria Douglas A Trojanowski IN Vekua lnst of Appl Mathematics Northville, Michigan Robert H Plummer Tbilisi, U S S R Cincinnati, Ohio Dvora Tzvieli Ching-Kuang Shene Louisiana State University Peter R Popivanov Johns Hopkins University Baton Rouge, Louisiana Bulgarian Academy of Science Baltimore, Maryland Sofia, Bulgaria Harimohan Ulaganathan Ding Hua Shi Scarborough, Ontario Canada Amy M Potter Shanghai Railway Institute Macon, Georgia Shanghai, People's Republic of China Pawel Urbanski University of Warsaw Nikolaos S Pouliezos Abdullah Shidfar Warsaw, Poland University of Manchester Institute of Iran University of Science and Science and Technology Technology Anthony N Valle Manchester, England Tehran, Iran LaGrange, Georgia Wallace C Pye Angela B Shiflet Robert M Vancko Hattiesburg, Mississippi Wofford College Ohio University Spartanburg, South Carolina Athens, Ohio Jeffrey M Rabin University of California San Diego Andrew J Simoson John E Veal II La Jolla, California King College Brookline, Massachusetts Bristol, Tennessee Miodrag D Raskovic Erik I Verriest Kragujevac, Yugoslavia R Sivaramakrishnan Georgia Institute of Technology University of Calicut Atlanta, Georgia Patrick Reardon Calicut, India Auburn, Alabama Al R Vilcius Penny D Smith Toronto, Ontario Canada Zoltan Reti Lehigh University Carolina Sarah H Voss University of South Bethlehem, Pennsylvania Columbia, South Carolina College of St Mary Omaha, Nebraska

926 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New Members of the AMS

Roy A Wagner Jr Israel Mathematical Union California State University, Hayward Embry-Riddle Aeronautical David Gilat Paula B Albert University Nancy K Grant Prescott, Arizona Korean Mathematical Society Ihn Sue Kim Myango Witanene Kapuku Jacqueline G Wells Charlene Ann Wieser McKeesport, Pennsylvania London Mathematical Society Geoffrey Raymond Robinson Catholic University ofAmerica Stewart Chalmers Welsh Margaret J Benetiel Texas A & M University Mathematical Society ofJapan Haile B Mariam College Station, Texas Mitsuhiro Itoh Yoshihiro Kubokawa Clark University Alice T Williams Tetsuro Miyakawa Susan M Lariviere Lexington, Virginia Hisao Sekigawa Cornell University Christopher J Williams Yuzo Utumi Walter Bergweiler University of Georgia Norsk Matematisk Foreningros Goucher College Athens, Georgia Tore M Jonassen Lisa M Merritt David Barrett Wilson Societe Mathematique de France Gustavus Adolphus College Boulder, Colorado Aline G Bonami Lance Nielsen Nathaniel R Withers Lawrence S Breen Xue-Zao Zhang Eric Louis Detrez University of Richmond Harvard University Richmond, Virginia Philippe Flajolet Bernard Lapeyre Stefano Trapani Elizabeth G Yanik Jean-Jacques Sansuc City University ofNew York, Hunter Virginia Commonwealth University Joseph A Tapia College Richmond, Virginia Michel A Thera Chao Li Qi-Xiao Ye Sociedad de Matemdtica de Chile Indiana State University Beijing Institute of Technology Hector J Roho Howard V Francis Beijing, People's Republic of China Sociedade Portuguesa de Matemdtica Donald L Ransford Thomas M Zachariah Fernando Pestana Da Costa Indiana University, Southeast North Haven, Connecticut Societal Catalana de Ciencies Fisiques Michael D May Lan Zhao Quimiques i Matematiques Richard L Ross Brown University Daniel Pascuas Indiana-Purdue University, Indianapolis Providence, Rhode Island Suomen Matemaattinen Yhdistys Vicumpriva S Perera John Alan Ziegler Sirkka-Liisa A Eriksson-Bique Jian Wang Marietta, Georgia Unione Matematica Italiana Indiana-Purdue University, Fort Wayne Ben F Zirkle Andrea Bacciotti Joseph D Lawrence Virginia Western Community College Luisa Carini Rebecca A Patterson Roanoke, Virginia Diego M Pallara Laurie E Williams Henry A Zwick Giancarlo Troi Marquette University Ferris State University Wiskundig Genootschap Jean E Pelkey Big Rapids, Michigan Herman Bavinck McMaster University John R Harding RECIPROCITY MEMBERS NOMINEE MEMBERS Northwest Missouri State University Martin J Nish Deutsche Mathematiker-Vereinigung e.· V. Arizona State University Dorena L Vivian Rudiger Winfried Braun Youngik Kim Theo Grundhofer Occidental College Arkansas State University Jens Carsten Jantzen Katherine A Clark Carrie R Clifton Claudia Khippelberg Renee Lynn Cook Lane Lea Harvill Daniel E Cronk Gesel/schaft jar Angewandte Mathematics Eldon R Kerr und Mechanik W Andy McCracken Ohio State University, Columbus Georg Peter Ostermeyer Yining Xia Auburn University Iranian Mathematical Society Rebecca S Calahan Portland State University Behzad Djafari-Rouhani Robert B Lund John Joseph Peterkort Timothy Irish Mathematical Society E Olson Roc/iford College John Brian Twomey Brandeis University Rena Lynn Ashton Songqing Ding Andreas Lauer Lori B Nelson

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 927 New Members of the AMS

Saint Olaf College University of Illinois at Urbana- University of New Brunswick, Fredericton Jon D Westbrock Champaign Rogerio Enriquez Charles J Nainan Perry J Meade State University of New York at Anne Marie Warner Binghamton University of Iowa Matthew P Fisher Francisco Alarcon University of New Orleans David J Rosher Qing Lin Gao Temin Chao Edwin E Hardee New York at Buffalo University of Texas at El Paso State University of Knopp Kent Cheng Cheng Xiaoluo Zhou M Koopman Anne Muralidharan Subramanian Texas A & M University Clement Sau Wai Lam Michael H Crawford Yixun Shi University of Washington Vincent E Dimiceli Virginia Vera de Serio Santiago Del Castillo Texas Tech University University of Lowell University of Waterloo Guanghwa Chang Keith Peters Christopher Kumar Anand Ruey-Jen S Jang University of Massachusetts, Boston Charles Dennis Sissel Mohammed Q Ashab INSTITUTIONAL ASSOCIATE University of Guelph Dayanand Birju Beatrice C Aebersold Geoffrey M Dixon Institute for Defense Analyses Susan L Gray Guozhen Duan Supercomputing Research Center, T Gordon Prior Bruce R Larson Lanham Maryland Anthony J Zito University of Hawaii Megumi Shimizu University of Minnesota-Duluth Patricia J Yeager University ofIllinois at Chicago Isabel M Ferreirim University ofMississippi Elizabeth L Kochneff Chris Bradford Abdes Samad Mortabit

FREE GROUP RINGS Narain Gupta (Contemporary Mathematics, Volume 66)

This book deals with some aspects of linear Contents techniques in combinatorial group theory having Magnus embeddings and free differential calculus their origin in the work of Wilhelm Magnus in the 1930s. The central theme is the identification Applications of Magnus embedding of free groups and properties of those subgroups Fox subgroups of free groups which are induced by certain ideals of the integral group rings of free groups. This subject has Dimension subgroups been developed extensively, and the author seeks Generalized Magnus embeddings to present, in contemporary style, a systematic and comprehensive account of some of its 1980 Mathematics Subject Classifications: 20C05. 20C07. 20F05. 20F26. 20H25. 20F99. developments. Included in the book are a solution 20F14. 16A26. 16A27. 20015. 20F10. 20F12 of the Fox subgroup problem and an up-to-date ISBN 0-8218-5072-5. LC 87-12427 development of the dimension subgroup problem. ISSN 0271-4132 and researchers in 144 pages ( softcover). June 1987 Aimed at graduate students Individual member 113. List price $22. combinatorial group theory, the book requires a Institutional member $18 familiarity with the general terminology of free To order. please specify CONM/66 NA groups and group rings.

Shipping/Handling: 1st book $2, each additional $1, maximum $25; by air, 1st book $5, each additional $3, maximum $100 Prepayment required. Order from American Mathematical Society, P.O. Box 1571, Annex Station Providence, Rl 02901-9930, or call toll free 800-556-7774 to charge with VIsa or MasterCard

928 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY INSTITUTIONS NOT SUBJECT TO U.S. LAWS may be required to execute and submit a copy of this form with the text of recruitment advertisements for the Notices (see Policy State­ ment below). Publication of recruitment advertisements not accompanied by the completed form may be delayed or denied.

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Please return this form to: Advertising Department Notices of the American Mathematical Society P.O. Box 6248 Providence, RI 02940, U.S.A.

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No listing for a position will be printed if it expresses or implies any preference, limitation, specification, or discrimination based on age, color, race, religion, sex, or national origin in contravention of any federal or state statute prohibiting discrimination in employment on the basis of these criteria. The publisher reserves the right to edit any listing prior to publication which does not conform with federal or state law on discrimination in employment. All employers in the United States are required to abide by the requirements of Title VII of the Civil Rights Act of 1964, announcing a national policy of equal employment opportunity in private employment, without discrimination because of color, race, religion, sex, or national origin. All U.S. listings are accepted with the understanding that the employer complies with federal requirements. Advertisers not subject to U.S. laws may be required to sign a statement that they do not discriminate in employment on grounds of age, color, race, religion, sex, or national origin. Applicants should be aware that institutions outside the U.S. may not be legally bound to conform to these or similar requirements and the AMS cannot assume responsibility for enforcing compliance. Applicants are advised to inform themselves of conditions that may exist at the institutions to which they apply. In particular, readers should note that the Equal Employment Opportunity Act (42 U.S.C., §§2000e et seq.), which prohibits discrimination in employment on the basis of race, religion, sex, or national origin, contains (in §2000e-l) an exception from the provisions of the Act for any religious corporation, association, educational institution, or society with respect to employment of individuals of a particular religion to perform work connected with the carrying on by such corporation, association, educational institution, or society of its activities. The Age Discrimination Act of 1967 (29 U.S.C., §§621 et seq., as amended), makes it unlawful for an employer to discriminate against any individual between the ages of 40 and 70 because of age. Thus it is legal to seek as an employee someone who is "over 30," but not one "ove1 50"; neither is it legal to express a preference for someone who is "young," or is a "recent graduate," since the latter tend (on statistical grounds) to be young.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 929 Classified Advertisements

SUGGESTED USES for classified advertising are books or lecture notes for sale, books being University of Toronto sought, positions available, exchange or rental of houses, and typing services. Department of Mathematics THE 1988 RATE IS $.85 per word with a minimum of $42.50. The same ad in 6 consecutive Issues is $4.00 per· word. The same ad In 10 consecutive issues is $7.00 per word. Type The Department of Mathematics, Uni­ will be set solid unless centering and spacing are requested. A centered line of any length versity of Toronto is looking for strong or the equivalent in white space is $10.00 (one-time charge). A word is defined as a group applicants in pure or applied mathe­ of individuals but not of of characters with space at each end. Prepayment is required matics to nominate as candidates for Institutions. For an additional $10.00 charge, announcements can be placed anonymously. NSERC Research Fellowships for 1989- Correspondence will be forwarded. DEADLINES are listed on the inside front cover. 90. These are five year research posi­ U. S. LAWS PROHIBIT discrimination in employment on the basis of color, age, sex, race, tions (subject to a review in the third religion or national origin. "Positions Available" advertisements from institutions outside the year) with a teaching load of at most one U. S. cannot be published unless they are accompanied by a statement that the Institution course per year. One of the five years does not discriminate on these grounds whether or not It Is subject to U. S. laws. Details and may usually be taken as a sabbatical. specific wording may be found following the Classified Advertisements in the January and Successful candidates may, in special July/August issues of the Notices. circumstances, be considered directly SITUATIONS WANTED ADVERTISEMENTS from involuntarily unemployed mathematicians for a tenure-stream position. 800-556-7774 and are accepted under certain conditions for free publication. Call toll-free Applicants should have a relatively speak to Paula Montella for further information. recent doctorate and have demonstrated SEND AD AND CHECK TO: Advertising Department, AMS, P.O. Box 6248, Providence, Rhode Island 02940. Individuals are requested to pay in advance, institutions are not required their ability with some substantive post­ to do so. thesis research. They must be Canadian citizens or landed immigrants by Novem­ ber 1, 1988. The University of Toronto women to POSITIONS AVAILABLE encourages both men and University of Victoria apply. Applicants should send an up-to-date Department of Mathematics The Department of Mathematics at the of University of Alberta University of Victoria is seeking strong curriculum vitae, a short description and arrange to candidates for the 1988-89 NSERC Uni­ their research program, Applications are invited for tenure-track sent di­ versity Research Fellowship competition. have three letters of reference positions, subject to budgetary approval, Associate These research fellowships have an ini­ rectly to Professor D. Masson, in Approximation Theory (File AP-1), Mathematics, tial term of up to five years and are Chairman, Department of Numerical Optimization or Partial Differ­ Ontario, awarded on a competitive basis by the University of Toronto, Toronto, ential Equations (File NP-1), in Number Natural Sciences and Engineering Re­ Canada MSS 1 A1. This material should Theory (File NT-1), or closely related ar­ The De­ search Council of Canada. The duties of arrive before October 1, 1988. eas and Algebraic or Differential Topol­ will be a University Research Fellow will include partment's choice of candidates ogy (File AT-1) at the Assistant Pro­ teaching one course per term in Mathe­ made in October and the final decision fessor level, beginning July 1, 1989. Re­ in the matics and participation in the graduate will be announced by NSERC quirements are a Ph.D. and proven ability program. Areas of research strength in spring. or demonstrated potential for research the Department include discrete mathe­ and teaching. Current salary range is matics, non-linear POE's in mathematical from $33,144 (Canadian) per annum de­ physics, and operator theory. pending upon qualifications. Send vitae VIRGINIA POLYTECHNIC INSTITUTE AND Applicants should submit, not later and arrange for three letters of refer­ STATE UNIVERSITY-DEPARTMENT OF than August 30, 1988, a curriculum vitae ence to be sent to: Professor L. H. MATHEMATICS. We anticipate making and the name of at least 3 referees to: Erbe, Chairman, Department of Mathe­ one or more tenure-track appointments Dr. C. R. Miers, Chairman matics, University of Alberta, Edmonton, at the assistant professor level or above Department of Mathematics Canada, T6G 2G1. In accordance with beginning in the fall of 1989. A Ph.D. is University of Victoria Canadian Immigration requirements, pri­ required. Very strong research potential Victoria, B.C. ority will be given to Canadian citizens required for junior-level appointments V8W2Y2 and permanent residents of Canada. and demonstrated outstanding record The University of Victoria offers equal Closing date for applications is Octo­ for senior-level appointments. Applica­ employment opportunities to qualified ber 31, 1988. Please quote file numbers tions will be accepted until March 15, male and female applicants. NSERC reg­ when responding to this advertisement. 1989, or until the positions are filled. ulations require that University Research The University of Alberta is committed to Applicants should send vita and three Fellowship nominees be Canadian citi­ the principal of equity in employment. letters of reference to: Chairman, Search at the time of zens or landed immigrants Committee, Department of Mathematics, nomination. Virginia Tech, Blacksburg, VA 24061- 0123. Virginia Tech is an Equal Opportu­ nity/Affirmative Action Employer.

930 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Classified Advertisements

POSITIONS AVAILABLE VIRGINIA POLYTECHNIC INSTITUTE AND The Department of Mathematics is ac­ STATE UNIVERSITY-DEPARTMENT OF tively seeking applications in the area The Department of Mathematics is ac­ MATHEMATICS. One position for a se­ of computational mathematics and nu­ tively seeking applications in the area nior visitor (Associate or Full Professor) merical analysis. We anticipate making of discrete mathematics and combina­ in Analysis is anticipated for the aca­ several tenure-track appointments at the torics. We anticipate making several demic year 1989-1990. The position can assistant professor level or above be­ tenure-track appointments at the assis­ be split into two one-semester appoint­ ginning in the fall of 1989. A Ph.D. is tant professor level or above beginning ments. Areas of expertise of potential required. Applications will be reviewed in the fall of 1989. A Ph.D. is required. interest to current faculty members are as they are received and will be accepted Applications will be accepted until March preferred. Duties are expected to include until the positions are filled. A formal let­ 15, 1989 or until a successful candi­ teaching of a topics course and inter­ ter of application expressing interest, a date is found. A formal letter of ap­ action with graduate students. Salary resume, and the names, addresses, and plication expressing interest, a resume, is negotiable. Applications received by telephone numbers of three references and names, addresses, and telephone September 1, 1988 will be consid­ should be sent to Chairman, Numerical numbers of three references should ered. To apply, send curriculum vitae Analysis Search Committee, Department be sent to Chairman, Discrete Math­ to Joseph A. Ball, Chairman, Analysis of Mathematics, Virginia Tech, Blacks­ ematics Search Committee, Department Visitor Search Committee, Department burg, VA 24061-0123. Virginia Tech is of Mathematics, Virginia Tech, Blacks­ of Mathematics, Virginia Tech, Blacks­ an Equal Opportunity1 Affirmative Action burg, VA 24061-0123. Virginia Tech is burg, VA 24061. Virginia Tech is an Equal Employer. an Equal OpportunityI Affirmative Action Opportunity/Affirmative Action employer. Employer. McGill University Department of Mathematics and Statistics The Department of Mathematics and Statistics at McGill University is seeking to fill a tenure-track Assistant or As­ UNIVERSITY OF COLORADO AT COL­ sociate Professorship in Mathematical ORADO SPRINGS, DEPARTMENT OF Statistics in September 1989. MATHEMATICS, COLORADO SPRINGS, The applications should be sent to: The University of Michigan, Ann Ar­ co 80933-7150 Professor K. P. Russell, Chairman bor, seeks a Director for its Residen­ Applications are invited for several Department of Mathematics and tial College, an innovative unit of its tenure-track positions starting Fall 1989. Statistics College of Literature, Science and the Preferred areas: Algebra, Differential McGill University Arts distinguished by its concern for Equations, Harmonic Analysis, Proba­ 805 Sherbrooke Street West undergraduate teaching in the arts, hu­ bility and Statistics. However, candi­ Montreal, Quebec, Canada nanities, social and natural sciences. dates with research interests outside H3A2K6 The Director is expected to have a of these areas are also encouraged Please include a statement of research genuine interest in the opportunities to to apply. Applicants should have sig­ accomplishments and plans along with develop imaginative programs in a res­ nificant research accomplishments or your letter of application, and arrange idential college environment, as well as exceptional research promise and evi­ for 3 letters of recommendation to be a distinguished record of research. Ap-. dence of good teaching ability. Ph.D. is forwarded. McGill University is an equal pointment as Director is for ·five years., required. Salary and rank are negotiable. opportunity employer, but in accordance effective July 1, 1989. Appointment car~ The average weekly teaching load is 7 with the Canadian Immigration require­ ries tenure, and will be made jointly with · 1/2 hours. There is generous support ments priority will be given to Canadian the university department appropriate to for faculty development like conference citizens and permanent residents. The the Director's research specialization. travel, teaching off-loads and summer deadline for the receipt of completed Please send letter of application or nom­ research. Please arrange to send a re­ applications is October 28, 1988. ination by September 30 to Associate sume and 3 letters of reference to Dr. Dean, Jack W. Melland, 2522 LSA Bldg., K. M. Rangaswarny, Chairman. Initial University of Michigan, Ann Arbor, Ml screening will begin on December 1, 48109. The University of Michigan is 1988 and continue through March 1989 an Equal Opportunity1 Affirmative Action or until all positions are filled. UCCS is Employer. an Equal Opportunity 1Affirmative Action Employer.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 931 Classified Advertisements

POSITIONS AVAILABLE Massachusetts Avenue, NW, Washing­ Head, School of Civil Engineering ton DC 20016. Purdue University UNIVERSITY OF CAPE TOWN The University's stand against apart­ POSTS IN THE DEPARTMENT OF The Schools of Engineering at Purdue heid and all racially discriminatory legis­ APPLIED MATHEMATICS University invite nominations and appli­ lation is on record. Information on this as cations for the position of Head, School following well as on the University's policy not to Applications are invited for the of Civil Engineering. The successful can­ discriminate in the appointment of staff posts (starting dates negotiable): didate will possess outstanding leader­ or the selection of students on grounds PROFESSOR ship qualities and administrative abilities. of sex, race or religion is obtainable on Priority will be given to applications The candidate shall be eligible for ap­ interests request. from scholars whose major pointment as full professor with imme­ (solid and activities are in mechanics diate tenure based on a distinguished and/or fluid), but consideration will also record of scholarly activity to include in be given to applicants of standing teaching, research and service in the mathematical modelling of complex sys­ SYDNEY/ AUSTRALIA engineering profession. tems, such as biological, technological or THE UNIVERSITY OF Purdue is a land grant institution. math­ socio-economic systems. Applied NEW SOUTH WALES The Schools of Engineering constitute are ematicians whose major interests SCHOOL OF MATHEMATICS one of the largest and highest quality analy­ in areas (for example, numerical LECTURER IN PURE MATHEMATICS engineering instructional and research which are sis or differential equations) (Tenured or tenurable appointment) organizations in the United States. The and math­ complementary to mechanics (REF 1039) School of Civil Engineering currently has invited to ematical modelling, are also 58 faculty members and over 600 gradu­ apply. Applicants should have a PhD or equiv­ ate and undergraduate students exclud­ ASSOCIATE PROFESSOR, SENIOR alent qualification, and active research interests in an area of pure mathematics. ing freshmen. Research activities cover LECTURER AND LECTURER a broad range of topics and account Applicants should be well-qualified, Preference will be given to candidates for an annual expenditure of approxi­ interests in whose research reinforces and extends with teaching and research mately $3 million. The curriculum spans in­ existing strengths in the School. The · any branch of Applied Mathematics, a wide spectrum of Civil Engineering general successful applicant will be expected cluding mathematical modelling, disciplines, with emerging technologies relativity and cosmology, and mechanics. to participate in the general teaching being continuously integrated into both The salary packages Include attractive activities of the School as well as in instruction and research. The candidate staff benefits such as medical aid, a the teaching of pure mathematics at all levels and in the supervision of grad­ selected shall be an innovative individual housing subsidy in certain circumstances with a firm grasp and understanding of Further uate students. The School is currently and a good pension scheme. the current and future needs of the Civil details concerning remuneration will be committed to developing its strengths in Engineering profession. on request. mathematical computer science. supplied The position will be available as early The ranks of the successful appli­ The position will be available from as 1 July 1989. Applications will be cants will be decided upon the basis February 1989, and appointment will be considered until the position is filled. of qualifications and experience; candi­ either with tenure, or on the basis of a Screening of applications will begin 1 dates should indicate in their letters of contract with provision for conversion to August 1988. Nominations and applica­ application for which postfs they wish to tenure. Further information from Professor 1 tions should be sent to: be considered. Dr. Henry T. Yang submit a full curriculum vitae H Sloan, Head of School [(02) 697 2957]. Please Dean, Schools of Engineering names and addresses of three Applications close on 16 September and the Purdue University referees (who should be supplied with a 1988. West Layfayette, IN 47907 of the curriculum vitae and list of Salary: $A28,694 range $A37,435 copy Purdue is an Equal Opportunity1 Affirmative publications) to the Registrar (Attention: Commencing salary according to qual­ Action employer. '' Appointments Office, Ref: DP), Univer­ ification and experience. sity of Cape Town, Rondebosch 7700, Applicants should forward two copies Republic of South Africa (telex: 5-22208 of their application, including curriculum UCTADSA; telefax: 27-21-6502138). The vitae, telephone number during business closing date for applications is 15 Au­ hours, transcripts of academic record gust 1988, but late applications will be and the names and addresses of two considered. referees to the Academic Staff Office Further information may be obtained PO Box 1, Kensington, NSW, 2033: from the above, or from the Coun­ Australia. sellor, Science and Technology, South Equality of employment opportunity is African Embassy, Suite 350, 4801 University policy.

932 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Classified Advertisements

POSITIONS AVAILABLE McGill University NEW SENIOR FACULTY Department of Mathematics and in NEW FACULTY Statistics COGNITIVE AND NEURAL SYSTEMS in at COGNITIVE AND NEURAL SYSTEMS The Department of Mathematics and BOSTON UNIVERSITY at Statistics at McGill University wishes to BOSTON UNIVERSITY sponsor a strong candidate for the Nat­ Boston University seeks a full professor ural Sciences and Engineering Research or associate professor starting in Fall, Boston University seeks a tenure track Council of Canada (NSERC) 1988-89 1989 to act as Co-Director for its M.A. assistant professor starting in Fall, 1989 University Research Fellowship Compe­ and Ph.D. Program in Cognitive and Neu­ for its M.A. and Ph.D. Program in Cogni­ tition. These Fellowships are five year ral Systems. The Co-Director will play tive and Neural Systems. This program research positions (with a review in a major role in curriculum development, offers an intergrated curriculum offering the third year), in the nature of Re­ administration, and training of mathe­ the full range of psychological, neuro­ search Assistant Professorships, and matically advanced graduate students. biological, and computational concepts, carry a teaching load of at most one The program curriculum offers the full models, and methods il") the broad field course throughout the academic year. range of psychological, neurobiological, variously called neural networks, con­ Applicants should have shown some and computational concepts, models, nectionism, parallel distributed process­ substantial research ability beyond their . and methods in the broad field variously ing, and biological information process­ doctoral thesis. They should be Cana­ called neural networks, connectionism, ing, in which Boston University is a dian Citizens or landed immigrants by parallel distributed processing, and bio­ leader. Each faculty member will have a November 1, 1988. logical information processing, in which joint appointment in the Ph.D. program Interested candidates should send Boston University is a leader. The faculty and in one or more of the departments of their curricula vitae to:- member will have a joint appointment in mathematics, biology, computer science, Professor K. P. Russell, Chairman the Ph.D. program and in one or more and psychology. Candidates should have Department of Mathematics and of the departments of mathematics, bi­ extensive analytic or computational re­ Statistics ology, computer science, and psychol­ search experience modelling a broad McGill University ogy. Candidates should have extensive range of real-time nonlinear neural net­ 805 Sherbrooke Street West analytic and computational research ex­ works, especially in one or more of Montreal, Quebec, Canada perience modelling a broad range of the areas: adaptive pattern recogni­ H3A2K6 real-time nonlinear neural networks, en­ tion, speech and language, cognitive in­ They should arrange for at least two abling them to teach graduate courses formation processing, self-organization, letters of reference from competent ref- including the areas of adaptive pat­ and conditioning and attention. Send . erees to be sent directly to the same tern recognition and self-organization. a complete curriculum vitae and three address. All documentation should reach Send a complete curriculum vitae and at letters of recommendation to Profes­ the department by September 9, 1988. least three letters of recommendation to sor Stephen Grossberg, Chairman, CNS The department will make its recom­ Professor Stephen Grossberg, Director, Program, Center for Adaptive Systems, mendations to NSERC early in October CNS Program, Center for Adaptive Sys­ 111 Cummington Street, Boston Uni­ 1988. NSERC will announce its decision tems, 111 Cummington Street, Boston versity, Boston, MA 02215 by January in March 1989. University, Boston, MA 02215 by January 1, 1989. Boston University is an Equal 1, 1989. Boston University is an Equal OpportunityI Affirmative Action employer. OpportunityI Affirmative Action employer.

JULY/AUGUST 1988, VOLUME 35, NUMBER 6 933 Classified Advertisements

POSITIONS AVAILABLE PUBLICATIONS ANNOUNCEMENTS

FOR SALE MATH SCI PRESS, 53 Jordan Rd., CALL FOR PAPERS Brookline, MA 02146, (617)738-0307. PRIZE ($1 ,000) for PHILOSOPHIA .PASCAL-SC: Compilers for personal SPECIAL SALE: INTERDISCIPLINARY MATHEMATICA (circulating in some 50 computers, documentation, software. MATHEMATICS, Vols. 1-9, 11-16, 18, nations since 1964) on such topics as: FBSoftware, 5101 Odana Road, Madi­ 19. LIE GROUPS: 1-3, 6-8, 10, 11. SYS­ 1. Ethics and Mathematics (in particu­ son, Wisconsin 53711. (608) 273-3702. TEMS AND CONTROL: 1, 2. $10 plus lar, Morals of Knowledge in general); 2. postage for each volume. Sociology or Social Theory of Mathemat­ Journals from 1961, American Mathe­ ics; 3. The Topological vs the Algebraic; matical Monthly, Bulletin AMS. 4. Others, related to the study on the For detailed information write Nature of Mathematics (definitely not Professor Bert Ross Just appeared: via Logic alone). For details, write to: Mathematics Dept. [Prof.] J. Fang, ODU (Phil), Norfolk, VA (ed): General Topology and its University of New Haven Z. Frolik 23529-0083. Relations ..., Proc. of the Sixth Prague West Haven, Conn. 06516 Topol. Symp. 1986, 736 pp., OM 108.­ or $66.- K. G. Murty: Linear complementarity, POSITIONS WANTED linear and nonlinear programming, 680 pp., OM 148.- or $89.- Position in mathematics required for fall 0rder directly from: Heldermann Ver­ 1988. UK citizen. lan D. Macdonald. lag, Nassauische Str. 26, D-1000 Berlin­ Please call 011-44-259-61601. West 31.

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934 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY U.S. Department of Commerce BUREAU OF THE CENSUS The Senior Mathematical Statistician

The Bureau of the Census, the primary collector of economic and demographic data in the United States, has an important vacancy as the Senior Mathematical Statistician. We seek an outstanding individual with an extensive record of accomplishment in any area of statistics or any interface area between statistics and other disciplines.

A distinguished scientific record including published original work and application of advanced methodology to real world problems and excellent communication skills are essential. Applicant must demonstrate potential to formulate technical policy and direct teams on problems of national statistical importance. The individual will serve as a principal consultant to Census Bureau executives and serve as a consultant to other government agencies and to governments of other nations.

Salary ($64,397 - $72,500) commensurate with qualifications.

Candidates should send a curriculum vitae, several recent unpublished papers, and three reference letters to: Dr. Kirk M. Wolter, Bureau of the Census, Washington, D.C. 20233. An Equal Opportunity Employer, U.S. citizenship required.

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3701 W. Alabama, Suite 450-130 Houston, TX 77027 (U.S.A.) -- -.-r I on Conference Themes Domain Decomposition and Mixed SIAM Conference • Mathematical Foundations of Finite Element Methods for Para­ Domain Decomposition bolic and Hyperbolic Problems Methods Mary F. Wheeler, University of Domain Houston • Coupling Domain Decomposi­ On the Mathematical Foundations tion Methods of Multigrid or of Domain Decomposition Methods Decomposition Finite Element Methods Pierre-Louis Lions, Universite de • Numerical Analysis of Domain Paris-Dauphine, France Intercontinental Hotel, Decomposition Methods Multigrid and Domain Methods on Parallel Decomposition Methods Houston, Texas • Implementation Yuri Kuznetsov, U.S.S.R. Academy of Computers Sciences, Moscow March 20-22, Cosponsored by Societe de • Hyperbolic Problems and Conjugate Gradient and Lanczos Mathematiques Appliquees Computational Fluid Dynamics Strategies for Domain • Finite Element Methods Decompositions: Comparisons 1989 et Industrielles Thomas J. R. Hughes, Stanford • Spectral Methods University The conference will present state of the art numerical • Engineering Applications On Zonal Methods in equations using Computational Fluid Dynamics solution methods for partial differential Joseph Steger, NASA-Ames Research domain decomposition. By breaking the original problem Invited Presentations Center (Tentative) into smaller subdomains of the original problem, if cor­ Domain Decomposition and require fewer Software for Parallel Computers Spectral Methods rectly implemented, such an approach will Kenneth W. Kennedy, Rice University to better utilization. Alfio Ouarteroni, CNR, Pavia, Italy computer resources and will lead Domain Decomposition Strategies systems of partial differential for Hyperbolic Problems: Parallel Who Will Attend Organizing Committee The conference will bring to­ equations. Implementations William F. Symes, Rice University gether specialists in areas such Roland Glowinski, Chairman For additional information, con­ of Houston as partial differential equations, On the Coupling of Viscous and University tact SIAM Conference Coordina­ Inviscid Models Via Domain analysis, computer Garret J. Etgen numerical tor: 117 South 17th Street, 14th Decomposition geophysics, and en­ University of Houston science, Floor, Philadelphia, PA 19103-5052. Jacques Periaux, Avions Marcel oil, Dassault/Breguet Aviation, Kenneth W. Kennedy gineers from the aerospace, Telephone (215) 564-2929. E-Mail chemical industry, and more St. Cloud, France Rice University and to [email protected] anyone who is involved Preconditioners Mary Wheeler generally, FAX Number: 215-564-4174 Multilevel in the numerical solution of large James H. Bramble, Cornell University University of Houston

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Honorary Editor-in-Chief: Paul Erd6s, Budapest Some of the papers to appear iD Editor-in-Chief: Uszl6 Lovasz, Budapest/Princeton COMBINATORICA, Volule 8 (Jllllll) Executive Editor: Uszl6 Babai, Budapest, Chicago A. AaanraJ, A. Alldenoa: A random NC algorithm for depth lint search R. AMroai, P. Enliis, N. Lillllll: Optima Combinatorica is an international quarterly of the J. Bolyai of dual integer linear prognms 1. Beck: On a lattice problem of Mathematical Society published jointly by Akademiai Kiad6, L. Moser, I B. Bollobols: The chromatic number of Budapest, and Springer-Verlag.* random gnphs 1. BaUd: Dual vectors and lower It covers a variety of areas in combinatorics and the theory of bouods for the nearest lattice point computing, with particular emphasis on general techniques and problem B. LilldslroJa: A generalization of the unifying principles. 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Thnrilliger: The classification of by North-Holland publishers distance-regolar graphs of type JIB Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HeidelbergerPiatz3, D-HXXJ Berlin 33 · 175 Fifth Ave., New York, NY10010, USA· 28, Lurke Street, Bedford MK403HU, England· 26, rue des Carmes, F-75005 Paris· 37-3, Hongo 3-cl!ome, Bunkyo-ku, Thkyo 113, Japan· Room 1603, Citicorp Centre, 18Whitfield Rnad, Causeway Bay, Hong Kong A new journal in the area that is producing some of the moat innovative work in modern applied - mathematics . • . Mathematics IICSS Vd="'i= ol c.lnl, Siplls ... Syslals Managing Editors: ...... Princeton; Journals E.l. Salll, Rutgers lltSS emphasizes high-quality, mathematically rigor­ ous contributions in this broad field. Researchers interested in this vital area of applications of mathe­ from matics to problems in engineering with lind lltSS an inwluable resource. 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Volume 1 · 1989 Forum Mathematicum An international journal devoted to pure and applied mathematics as well as mathematical physics Editorial Board: M. Brin (College Park, USA) F. R Cohen (Lexington, USA) V. Enss (Berlin, FRG) R Fintushel (East Lansing, USA) M. Fliess (Gif-sur-Yvette, France) M. Fukushima (Osaka, Japan) G. Gallavotti (Rome, Italy) R Gobel (Essen, FRG) K H. Hofmann (Darmstadt, FRG) J. Lindenstrauss (Jerusalem, Israel) D. H. Phong (New York, USA) D. Ramakrishnan (Ithaca, USA) A. Ranicki (Edinburgh, GB) P.-A. Raviart (Palaiseau, France) D. S. Scott (Pittsburgh, USA) D. Segal (Oxford, GB) B. Shiffman (Baltimore, USA) F. Skof (Torino, Italy) K Strambach (Erlangen, FRG) G. Talenti (Florence, Italy) H. Triebel (Jena, GDR) R B. Warfield (Seattle, USA)

Authors may submit original research articles for publication directly to one of the editors or to: "Fomm Mathematicum" Mathematisches Institut der Universitiit Bismarckstrasse 1 1h D-8520 Erlangen, Federal Republic of Germany l8J de Gmyter Berlin · New York de Gmyter Studies in Mathematics Editors: Heinz Bauer (University of Erlangen-Niimberg, West-Germany) Peter Gabriel (University of Zurich, Switzerland)

An international series of monographs and textbooks of high standard in pure and applied mathematics. Written of outstanding experts, the volumes within this series cover a wide spectrum of contemporary mathematics and will be of interest to active researchers in mathematics and related fields as well as to graduate students.

W. Klingenberg H. Strasser Riemannian Geometry Mathematical Theory of Statistics 1982. X, 396 pages. Cloth DM 128,-/US $ 59.95 Statistical Experiments and Asymptotic ISBN 3 11 008673 5 (Vol. 1) Decision Theory 1985. XII, 492 pages. Cloth DM 158,-/US $ 72.95 M. Metivier ISBN 3 11 010258 7 (Vol. 7) Semimartingales T. tom Dieck A Course on Stochastic Processes 1982. XII, 287 pages. Cloth DM 88,-/US $ 44.95 Transformation Groups ISBN 311 008674 3 (Vol. 2) 1987. X, 312 pages. Cloth DM 128,-/US $ 59.95 ISBN 311 0097451 (Vol. 8) L. Kaup/B. Kaup Holomorphic Functions of Several Variables An Introduction to the Fundamental Theory 1983. XVI, 350 pages. Cloth DM 112,-/US $ 54.95 ISBN 311 004150 2 (Vol. 3)

C. Constantinescu Spaces of Measures Forthcoming Titles 1988: 1984. 444 pages. Cloth DM 138,-/US $ 64.95 H. 0. Georgii ISBN 3 11 008784 7 (Vol. 4) Gibbs Measures and Phase G. Burde/H. Zieschang Transitions Knots Approx. 500 pages. Cloth approx. DM 150,-/ 1984. XII, 400 pageS. Cloth DM 138,-/US $ 64.95 approx. US $ 70.00 ISBN 3 11 010455 5 (Vol. 9) ISBN 3 11 0086751 (Vol. 5) M. Herve U. Krengel Analyticity in Infmite Ergodic Theorems Dimensional Spaces 1985. VIII, 357 pages. Cloth DM 128,-/US $ 59.95 Approx. 250 pages. Cloth approx. DM 130,-/ ISBN 3 11 008478 3 (Vol. 6) approx. US $ 60.00 ISBN 3 11 010995 6 (Vol. 10)

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THE UNIVERSITY SERIES IN CONTEMPORARY SOVIET MATHEMATICS MATHEMATICS Series Editor: Revaz Gamkrelidze Series Editor: Joseph J. Kohn Steklov Institute, Moscow USSR Princeton University ASYMPTOTICS OF OPERATOR AND TOPICS IN NUMBER THEORY PSEUDO-DIFFERENTIAL EQUATIONS by J. S. Chahal, Brigham Young University by V. P. Maslov and V. E. Nazaikinskii, This book presents a self-contained introduction to the Moscow Institute of Electronic Engineering, theory of numbers. Chapters cover basic properties of the USSR integers, algebraic methods, representation of integers by Beginning with an introduction to asymptotic problems for forms, algebraic number fields, algebraic curves, the linear equations, the authors go on to cover functional Mordell-Weil theorem, computation of the Mordell-Weil calculus, asymptotic solutions for pseudo-differential equa­ group, equations over finite fields, and the Weierstrass tions, and the quasi-inversion theorem for functions of a theory. tuple of noncommuting operators. Specific subjects include 0-306-42866-0/206 pp./ill./1988/$35.00 the functions of a single operator, the functions of several operators, the canonical operator on a Lagrangian sub­ manifold of a symplect!c manifold, equations with coeffi­ ELLIPTIC DIFFERENTIAL cients at infinity, and Poisson algebras and nonlinear com­ EQUATIONS AND OBSTACLE mutation relations. PROBLEMS 0-306-11014-8/320 pp./ill./1988/$79.50 by Giovanni Maria Troianiello, Universitd degli Studt di Roma, Italy DIFFERENTIAL GEOMETRY Professor Troianiello fills the need for a thorough analytic AND TOPOLOGY presentation of fundamental theory and techniques related by A. T. Fomenko, Moscow State University, to second-order elliptic operators in this book. Chapters cover function spaces, the variational theory of elliptic USSR boundary value problems, H'·'' and C ... theory, variational translated from Russian by D. A. Leites inequalities, and nonvariational obstacle problems. Exer­ A review of some geometric and differential methods used cises follow each chapter. widely in theoretical investigations and in various applica­ 0-306-42448-7/370 pp./ill./1987/$49.50 tions. This book incorporates several mathematical disciplines and interlinks various themes including com­ plexes, homology, bundles, topology of three-dimensional MATRIX THEORY manifolds, symplectic geometry, problems of integration of A Second Course mechanical systems, and more. by James M. Ortega, University of Virginia 0-306-10995-6/338 pp./ill./1987/$75.00 Matrix Theory bridges the gap between elementary linear algebra texts and the more advanced matrix theory books geared toward upper-level graduate students. Ortega Coming in 19891 presents both general broad matrix theory and more JOURNAL OF DYNAMICS AND specialized sections on topics including diagonalization of symmetric matrices, the Lyapunov criterion for stability, DIFFERENTIAL EQUATIONS nonnegative matrices, and eigenvalue problems. Exercises Editor-in-Chief: George R. Sell, University of are included in each section, as well as historical Minnesota, Minneapolis background. This new journal will publish original papers on the theory 0-306-42433-9/274 pp./ill./1987/$34.50 of the dynamics of differential equations-and their discrete analogues. Dynamical issues treated in the journal will Newt cover all the classical topics including attractors, bifurca­ tion theory, dichotomies, ergodic theory, finite and infinite JOURNAL OF THEORETICAL dimensional systems, invariant manifolds, Lyapunov ex­ PROBABILITY ponents, normal forms, scattering, stability theory, sym­ metries, topological methods, and transversality. Editor: Arunava Mukherjea, University of South Florida Write for a free examination copy of any Plenum An international forum for research in all aspects of joumaU theoretical probability, this journal offers peer-reviewed Come join the celebration at the Plenum booth-#3 original papers. Topics include probability theory on semigroups, groups, vector spaces, and other abstract structures as well as random matrices/random operators and applications of these to statistics, computer science, Pliifdli pattern theory, and other disciplines. PlJBliSHING CORPORATION Subscription: Volume 1, 1988 (4 issues) PLENUM PUBLISHING CORPORATION Institutional rate: $95.00 233 Spring Street Personal rate: $45.00 New York, NY 10013-1578 Telephone orders: 212-620-8000/1-800-221-9369 Mathematics Journals from Academic Press

EDITOR: Joseph F. Traub, Columbia University, New York City Original Research Containing Substantial Mathematical Results on Complexity as Broadly Conceived

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Focusing on topics from the theory of computation, Information and Computation is directed to computer scientists, mathematicians, and electrical engineers. In addition to papers that contribute new results to active theoretical research, expository surveys and reviews are occasionally published.

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Publications of the Anthony J. Tromba Vieweg Seminar on new Results in Nonlinear Max-Pianck-lnstitut Partial Differential Equations congratulates fiir Mathematik 1987. VI, 198 pp. Softcover US$ 27.00 the Bonn, Federal Republjc of Germany The papers by U. Abresch, J. L. Kazdan, S. Klai­ American Adviser: Friedrich Hirzebruch nerman, V. Moncrief, J. E. Marsden and 1 Ratiu, in the series Aspects of Mathematics W.-M. Ni, A. J. Tromba, and H. C. Wente give an Mathematical (edited by Klas Diederich) insight into important modern research activi­ Society ties in nonlinear partial differential equations and differential geometry. on its centennial Masaaki Yoshida Fuchsian Differential Equations With special Emphasis on the Gauss-Schwarz­ Theory. 1987. XIV, 215 pp. Softcover US$ 30.00 The aim of this book lies in the study of linear ordinary differential equations and systems of linear partial differential equations with finite dimensional solution spaces in the complex analytic category. This field has a long history and has attracted many famous mathemati­ cians, such as Euler, Gauss, Schwarz, Rie­ mann, Poincare, and Kummer. * Mathematics for the eye Gottfried Barthel, Friedrich Hirzebruch, Thomas Hofer Gerd Fischer (Ed.) Geradenkonfigurationen Mathematical Models und Algebraische Flachen From the Collection of Universities and 1987. XII, 308 pp. Softcover US $ 40.00 Museums. 1986. Photograph Volume: XII, (in German) 129 pp. with 132 Photographs; Commentary: The methods in. this· book provide new and VIII, 83 pp. with 90 Figures. Hardcover US interesting examples of algebr~ic surfaces, $ 74.00 using arrangement of lines in the complex pro­ jective plane and ramified coverings. The Available are four of the well known models: famous Miyaoka-Yau inequality for the Chern Graph of w = 1/z 12 x 12 x 12 em; To order these or other numbers of an algebraic surface relates to sur­ us$ 80.00 Vieweg titles, prising results in elementary geometry. Weierstrass ~-function, real part please contact: 16 x 16 x 16 em. US$ 124.00 GLP International Gerd Faltings, Gisbert WUstholz et al. Steiner's Roman surface 10 x 1 0 x 1 0 em. Vieweg-Dept. Rational Points us$ 62.00 560, Sylvan Avenue, 2nd edition 1986, VIII, 268 pp. Softcover Clebsch diagonal surface 17 x 15 x 24 em. Englewood Cliffs, NJ 07622 us$ 30.00 us $124.00 Telephone (201) 871-1010 This book consists of the notes from the semi­ telex (00230) 13-5368 nar Bonn/Wuppertal1983/84 about arithmetic Francois Apery algebraic geometry and contains a proof of the Models of the Real Projective Plane Vieweg Publishing Mordell-conjecture. This topics include moduli Computer Graphics of Steiner and Boy Sur­ P. 0. Box 5829 spaces, heights, finite group schemes, Tate faces. With a Preface by Egbert Brieskorn. D-6200 Wiesbaden, F. R. G. conjecture, Shafarevich conjecture, and arith­ 1987. XII, 156 pp. with 46 fig. and 64 Color telex 4186 928 metic surfaces. Plates. Softcover US $ 49.00 telefax 160229 A New Way to View Mathematics AMS-MAA Invited Addresses and ICM-86 Plenary Addresses now on videotape!======~ Today's eminent mathematicians... The AMS is now offering a selection of important mathematics lectures on videotape: Joint AMS-MAA Invited Addresses, presented at the January Joint Mathematics meetings, and Plenary Addresses presented at the International Congress of Mathematicians in August, 1986. The lecturers in these two series are among the world's most distinguished mathematicians. ... bring the subject to life Because these lectures are intended to be expository and to appeal to a wide audience, the videotapes make excellent teaching tools. This medium communicates the excitement and enthusiasm for mathematics in a way that written material cannot. Students, teachers, and researchers will all appreciate these videotapes not only for their important mathematical content, but also for the historical perspective and personal touches the speakers bring to them.

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PREPAYMENT REQUIRED. Order from American Mathematical Society, P.O. Box 1571, Annex Station, Provi­ dence, RI 02901-9930 USA or call (800) 556-7774 in the continental U.S. to charge on VISA or MasterCard. All prices are subject to change without notice. Please add shipping & handling: 1st video $2, each add'l $1, max. $25; by air, 1st video $5, each add'l $3, max. $100. Can you identify these prominent members of the mathematical community ? Learn their identities in I HAVE A PHOTOGRAPHIC MEMORY by Paul R. Halmos

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600,000+ Item Access For further information or to place an order through MathDoc, write to MathDoc has access to over 600,000 items at present MathDoc Document Delivery in mathematics and related subjects, including 70,000 Mathematical Reviews articles from Soviet journals. 416 Fourth Street P. 0. Box 8604 95% Success Rate in Two Ann Arbor, MI 48107-8604 USA Working Days OR telephone (313) 996-5268 and ask for And MathDoc has over a 95% success rate in quickly MathDoc. locating and supplying the articles in journals, collec­ tions, and conference proceedings listed in CMP, MR, OR send electronic mail (Internet) to and MathSci...usually within two worldng days. [email protected] All electronic mail will be acknowledged. Although MathDoc cannot ordinarily supply an entire book, it often can give you infonnation about the OR order through the DIALOG DIALORDER or book's availability through other sources. the BRS systems. AMERICAN MATHEMATICAL SOCIETY Expands Publication Program

In its Centennial year, the American Mathematical Several new book series and journals are planned to Society looks forward to Increased opportunity to begin publication in 1988 and 1989. Some of them­ serve the mathematical community through an ex­ such as the History of Mathematics Series - repre· panded publications program. One of the Society's sent new directions for the Society. Others- such central goals is to promote communication about as the translation of expository articles from the mathematical sciences research through a diverse Japanese mathematical journal Sugaku- reflect the and dynamic publications program. As one of the need for mathematicians to communicate more world's largest mathematical publishers, the AMS Is easily across International lines. All of them exem­ committed to providing the high quality, reasonably plify the Society's commitment to serving the mathe­ priced publications the mathematical community matical community In the next century and beyond. needs.

HISTORY OF MATHEMATICS SERIES ___ AMS REPRINTS------­ This series begins in 1988 with books about Ameri­ The Society receives many requests for help in can mathematics during the past century. The titles obtaining copies of a number of excellent books in this series will present historical perspectives on that have gone out of print. As a service to the individuals who have profoundly influenced the community, the AMS will publish the series AMS development of mathematics, as well as those who Reprints, consisting of important research mono­ have made great contributions to the mathematical graphs and graduate level textbooks that have community, or will trace the development of special been declared out of print by the original publishers. areas of research.

SUGAKU EXPOSITIONS------­ Sugaku, published by the Mathematical Society of UNIVERSITY LECTURE SERIES ____ Japan, is the Japanese counterpart of the Bulletin Lecture series provide an excellent forum for in­ of the AMS. Each issue of Sugaku contains several depth, and sometimes inspired, presentation of expository articles which provide highly informative mathematical topics, but often benefit few beyond accounts of a variety of current areas of mathe­ those attending them. To address this limitation, matical research. The AMS will publish these the AMS will publish books in the new University articles in a new journal called Sugaku Expositions. Lecture Series which will preserve important lecture series given at various institutions by outstanding mathematicians.

To submit a manuscript, contact To begin a STANDING ORDER for titles In the Director of Publication History of Mathematics or University Lecture American Mathematical Society Series or to begin a subscription to Sugaku P.O. Box 6248 Expositions, contact the Membership and Sales Providence, Rl 02940 USA Department at the AMS. (401) 272-9500 or (800) 556-77741n the continental U.S. •NEW• Journal of the American Mathematical Society

Editors The new Journal of the American Mathematical Society be­ Michael Artin gan publication with the January 1988 issue. This quarterly H. Blaine Lawson, Jr. journal contains research articles of the highest quality in all Richard Melrose areas of pure and applied mathematics. Selected articles ap­ Wilfried Schmid pearing in Volume 1· include: Homology of the zero set of a Robert E. Tarjan unipotent vector field on a flag manifold by C. De Concici, G. Lusztig, and C. Procesi; Reduced Hausdorff dimension and the concentration-cancellation law for 2-dimensional incom­ pressible flows by Ronald J. DiPerna and Andrew Majda; Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem by David Jerison and John M. Lee; p-adic Hodge theory by G. Faltings; Flip theorem and the existence of minimal models for 3-folds by Shigefumi Mori; Zero-one laws for sparse random matrices by Saharon Shelah and Joel Spencer; and Arithmeticity of holonomy groups of Lie foliations by Robert J. Zimmer

ISSN 0894-034 7 Quarterly, Volume 2, 1989

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".A~~~';/~· Oljl\TOF.D \flY' American Mathematical Society Centennial Celebration- August 8-12, 1988 Providence, Rhode Island American Mathematical Society v v JOURNALS OF DISTINCTION

D MATHEMATICAL REVIEWS Since 1940, Mathematical Reviews has been the recognized reviewing and ab­ stracting journal covering published mathematical research literature. Over 40,000 reviews or abstracts, by 12,000 international reviewers, are published each year. The reviews in each issue are arranged according to the 1980 Math­ ematics Subject Classification {1985 Revision). Subscriptions include annual author and subject indexes. Published monthly. 1989 subscription rate: Either paper or microfiche subscriptions: List 83510; Inst member S2808; Indiv member 8421*; MR Reviewer S281* Postage surcharge: India U53; other foreign 860 Combination paper and microfiche subscriptions: List 84668; Inst member 83'135 $200 late charge applies Optional delivery: First class $70; airmail $190 *Add $10 for subscriptions in the US; $60 for subscriptions outside the US. D CURRENT MATHEMATICAL PUBLICATIONS This current awareness journal, which is published every three weeks, is a sub­ ject index of recent and forthcoming mathematical publications which have been classified by the editors of Mathematical Reviews. (The classification scheme used is the 1980 Mathematics Subject Classification {1985 Revision), published in the most recent annual index of Mathematical Reviews.) Each issue contains an author and key index; author and key indexes covering a half year are in­ cluded in issues 9 and 17. Each issue contains a list of the serials represented in that issue and a separate listing of serial additions and changes, as well as a section containing the tables of contents of certain journals. Volume 21 is the 1989 volume. 1989 subscription rate: List 82'15; Inst member 8220; Indiv U65; MR Reviewer 8110 Postage surcharge: India 838; other foreign S25 Optional delivery: First class $30; airmail $60 10% late charge applies.

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Singularities and Groups in Bifurcation Stochastic Differential Systems, Stochastic Theory, Volume D Control Theory and Applications M. Golubitsky, I. Stewart, and D. Schaeffer W Fleming and P.L. Lions Bifurcation theory studies how the structure of solu­ This volume is based on the proceedings of a work­ tions to equations change as parameters are varied. The shop on stochastic control and related topics in applied nature of these changes depend both on the number of probability: The ch01:-e of topics was deliberately made parameters and on the symmetries of the equations. Four to obtain a mix of traditional areas of stochastic control broad topics are covered in this volume: group theory theory and topics arising in .·•ewer areas of application, and steady state bifurcation, equivariant singularity including queuing networks, simulated annealing, and theory, Hopf bifurcation theory with symmetry and mode stochastic scheduling. The papers emphasize various interactions. The opening chapter provides an introduc­ underlying mathematical theories, modelling issues and tion to these subjects and motivates the study of systems questions of computational implementation. This volume with symmetry: Three case studies are included which will interest several audiences in mathematics, elec­ illustrate how group theoretic methods can be used to tricaVcomputer engineering, and management science. analyze specific bifurcation problems arising in applica­ 19881609 pp./10 illus./8 tableslhardcover/$49.80 tions. The topics of these studies are Benard convection, The IMA Volumes in Mathematics and Its Applications, Vol. /0 ISBN 0-387-96641-2 traction for Mooney-Rivlin material and pattern forma­ tion in the Thylor-Couette systems. Groups and Symmetry 1988/553 pp./96 illus./80 tables/hardcover/$69.50 Armstrong Applied Mathematical Sciences, Vol. 69 M.A. ISBN 0-387-96652-8 This text provides a gentle introductic ' to the high­ lights of elementary group theory: Throughout the book Bolomorphic Functions and Moduli I emphasis is placed on concrete examples, many of them geometrical in nature so that finite rotation groups and Bolomorphic Functions and Moduli II the seventeen wallpape1 groups are treated in detail Edited by D. Drasin, C.]. Earle, F.W Gehring, I. Kra, and alongside theoretical results such as Lagrange's the­ A. Marden orem, the Sylow theorems, and the classification the­ These volumes, part of a continuing series of proceed­ orem for finitely generated abelian groups. A novel ings from MSRI workshops, cover a broad range of feature at this level is a proof of the Nielsen-Schreier themes from geometric function theory: There are 41 theorem, using group actions on trees. Contains over 300 contributions. The subject matter includes non-linear exercises and 50 illustrations to help develop the stu­ partial differential equations, value distribution theory dent's intuition. and univalent functions, as well as several papers on 1988/app. 192 pp./52 illus.lhardcover/$34.00 complex iteration. However, the main focus is on the Undergraduate Texts in Mathematics influence of Teichmiiller eypaces, quasi-conformal map­ ISBN 0-387-96675-7 pings, and Mobius groups on subjects which have as Ordering Information their genesis the classification of Riemann surfaces. This 'l'oozdel'~ Ol'othel'llpdagw-Vel'lagtltlee: stop by our booth during area also has had major interactions with dynamical the AMS meeting, or write to llpdagw-Vedag N- TOI'k, lac., Attn: B. systems and low-dimensional topologies. This collection Bay, 175 Fifth Ave., New York, NY 10010. Checks anrl money orders (plus $2.50 for shipping) are acceptable forms of payment. NY. NJ and CA is intended for research mathematicians and advanced residents, please add state sales tax. 1b order by credit card, call TOLL graduate students. FREE 1-800-526-7254 (in NJ, 20 1-348-4033).

Vol. I: 1988/246 pp./22 illus./hardcover/$29.50 The MathematJcal Sciences Research Institute Publications, Vol /0 ISBN 0-387-96766-4 Vol. D: 1988/290 pp./29 illus./hardcover/$32.80 The MathematJcal Sc1ences Research Institute Pub/icallons. Vol. JJ Springer-Verlag Newt York Berlin Heidelberg Vienna London Paris 1bkyo ISBN 0-387-96786-9