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International Congress of Mathematicians

The lecture hall at the Technical University of Berlin, site of the Plenary Lectures for ICM98 .

About 3,500 people attended the 1998 Interna- Mathematicians under the Nazi Regime tional Congress of Mathematicians (ICM98) in Berlin 1933–1945”. last August. The tremendous variety on the pro- Of all German cities, Berlin perhaps most po- gram ensured that there was always something in- tently symbolizes how much the country has teresting to do, leaving participants exhausted but changed in the past several decades. At times satisfied. The meeting was in many ways a marvel, Berlin feels like one gigantic construction site, with combining Germany’s legendary efficiency with new buildings going up everywhere. The western its deep cultural heritage. and eastern parts of the city have long since rein- tegrated, but there are many reminders of the Berlin’s Culture on Display Berlin Wall, from streets that dead-end in peculiar It has been ninety-four years since the ICM was last ways to memorials to those who perished trying held in Germany. The reason, of course, is that Ger- to cross to the other side. Still, the city retains a man mathematics nearly died out during World War great deal of old world elegance and grandeur, as II, when many of the country’s best mathematicians well as a high degree of cultural sophistication, fled the Nazi regime. ICM98 clearly meant a great which was amply showcased during the Congress. deal to German mathematicians, as it provided an It happened by chance that on the Saturday night opportunity to heal the wounds of the past and to during the ICM, Berlin held an event called “The show the world that their country has regained Long Night of Museums”, for which museums all some, if not all, of its mathematical strength. At over the city stayed open until 2 a.m. and offered the Congress, the chilling facts of Germany’s past special programs of music and dance. One of the were not glossed over, but confronted directly. At main events on the ICM social program was a per- the Opening Ceremonies, ICM Honorary President formance of The Magic Flute by the Deutsche Oper Friedrich Hirzebruch, who has been a central fig- Berlin, for which ICM participants could purchase ure in the rebuilding of mathematics in postwar specially priced tickets. Germany, devoted nearly his entire speech to the In celebration of the ICM, the Berlin-Brandenburg subject. In particular, he called attention to a spe- Academy of Science sponsored a public lecture by cial ICM event organized by the Deutsche Math- the noted German writer Hans-Magnus Enzens- ematiker Vereinigung (German Mathematical So- berger. The title of his lecture was “Zugbrücke ciety) called “Terror and Exile: Berlin außer Betrieb, oder die Mathematik im Jenseits der Kultur: Eine Außenansicht” (“Drawbridge Out Photographs used in this article are courtesy of Gerd Fis- of Order, or Mathematics Outside of Culture: A cher. View from the Outside”). Enzensberger pondered

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when it was realized they were too obvious. Gen- erally the press was favorable toward mathemat- ics, but there was at least one exception. Der Spiegel, one of the main nationwide news maga- zines in Germany, ran an unsigned article entitled “Nobelpreis für Quatsch” (“Nobel Prize for Non- sense”). The title referred to the work of Fields Medalist Richard Borcherds on the “moonshine” conjectures in the theory of finite simple groups. The story ridiculed the work of Borcherds and of (who received a special one-time award at the Congress) as lacking practical appli- cations. Even when the story grudgingly acknowl- edged that the work of Fields Medalist Maxim Kont- sevich was interesting, the jeering tone remained, as the story referred to Kontsevich as a “Milch- gesicht” (“babyface”). Fields Medalists William Tim- ICM Honorary President Friedrich Hirzebruch (left) and othy Gowers and Curtis McMullen fared better, as Andrew Wiles. they were not mentioned in the piece.1

the strange position of mathematics in today’s so- Efficiency Pays Off ciety, in which people proudly proclaim their ig- Many at the Congress remarked on its efficient or- norance of mathematics but would never take the ganization. This showed in many small ways— same attitude toward other parts of human culture, such as the fact that water, fruit juice, coffee, and such as music. The lecture, which presented a view tea were always available for free outside the lec- of mathematicians that was at once sympathetic ture rooms—and in many large ones too—such as and unsparing, drew rave reviews. “It was mar- velous,” said Hermann Karcher of the University the extensive use of e-mail and the Web to com- of Bonn. “It is amazing that someone outside of municate information about the Congress and to mathematics could have so much insight into the register participants. Another organizational feat field itself and its communication problems.” was getting two of the three proceedings volumes published in time for participants to pick them up Mathematics für Alles with their registration packets. The efficiency at- The Enzensberger lecture was part of an extensive tracted some jokes: The ICM “circular letters” sent program for the general public that took place in out in e-mail during the preceding year or so by the Urania, a public lecture institute in Berlin. The Martin Grötschel, president of the organizing com- stereotype of Germans as a dour, serious lot might mittee, were said to number in the thousands, but lead one to suspect that this part of the ICM pro- really totaled only thirty-four. In one of these let- gram would be filled with stuffy, instructive lec- ters Grötschel felt compelled to address com- tures. Quite the opposite was true. Fun and whimsy plaints that the ICM was being “overorganized”. But prevailed in the lobby of the Urania, where groups witnessing how smoothly the Congress ran, one of young and not-so-young people gathered around might conclude that too much organization was the many mathematical games and puzzles on dis- just enough. play. There was one contraption that makes enor- Consider the Opening Ceremonies, an especially mous soap bubbles: Pull a cord, and a hula hoop- complex event that was held in Berlin’s Interna- sized ring rises from a circular vat of soap solution, tional Congress Center. The musical interludes leaving a shimmering, tubular trail of a soap bub- were accompanied by a light show projected onto ble. There was a festival of mathematical videos, a screen on the stage. There was a succession of as well as public lectures on such topics as math- speakers, some of whose presentations were ac- ematics and sculpture, and financial mathematics. companied by slides or short video programs, and The ICM organizers worked hard in advance of all were flawlessly timed. (On the other hand, ef- the Congress to ensure that there was plenty of ficiency did not dictate every aspect: Two attrac- media coverage. Their work paid off in daily arti- tive young women with long blonde hair and very cles in the local German newspapers, as well as in short skirts ferried the Fields Medals around the a half-hour television broadcast about the Con- stage, calling to mind the presentation of trophies gress. To give reporters time to get a handle on the at an automobile race.) The most impressive dis- work of the Fields Medalists, the organizers gave play of efficiency came with the serving of a buf- newspapers information several months in ad- 1An article about the mathematical work of the Fields vance. To keep the names secret, each medalist was Medalists appears in this issue of the Notices. There was initially given a code name, like “Quantum” and also a shorter report in the November 1998 issue, pages “Moonshine”, but these were changed to numbers 1358-1361.

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fet lunch to the 3,000 people who attended the medals were an- Opening Ceremonies. The curtain on the stage nounced). These lec- went up, and a collective gasp rose from the crowd. tures provided an op- There in the enormous backstage area stood a portunity for dozen or so waiters and waitresses, carrying trays Congress partici- of drinks, and behind them was the buffet lunch, pants to get a better laid out on several tables. No one had to leave the understanding of the lecture hall to have lunch; they simply poured medalists’ work. All onto the stage. Although those who had sat in the of the medalists put far reaches of the auditorium had to wait in line in a lot of effort to quite a long time, everyone got fed. make their talks com- prehensible to a gen- Nevanlinna Prize winner Peter Shor, right, The Scientific Program eral mathematical au- with Ronald Graham of the University of The main part of the ICM scientific program con- dience. McMullen, California, who spoke on Shor’s work. sisted of twenty-one 1-hour Plenary Lectures, and known for being an more than one hundred fifty 45-minute Invited excellent expositor, Lectures; there were also poster sessions and short discussed the role communications of fifteen minutes’ duration. Some that topological rigid- participants commented that the Plenary Lectures ity plays in dynamical were presented at just the right level for a general systems. Borcherds mathematical audience. Among the highlights was provided an accessi- ble lecture about his the lecture by Nevanlinna Prize winner Peter Shor work on the “moon- of AT&T Labs. In the last few years, Shor has gained shine” conjectures in worldwide attention for his work on quantum com- finite group theory puting, which he described in his lecture. His most and the development famous result was to exhibit an algorithm that, if of the notion of ver- implemented on a quantum computer, could fac- tex algebras. Rather tor integers in polynomial time. Currently no such than talk about his algorithm is known to exist for conventional com- work in Banach puters. The simple and fundamental quality of the Fields Medalist Tim Gowers, right, with Béla spaces, which seems ideas in Shor’s lecture appealed to many ICM par- Bollobás of the University of Memphis, who to be the reason for ticipants. Some were so impressed that they said spoke on Gowers’ work. his getting a Fields his work seemed more exciting than that of the Medal, Gowers spoke Fields Medalists. on his work on a problem of Szeméredi Quantum computing also arose in a highly spec- in arithmetic number theory. Kontse- ulative Invited Lecture by of Mi- vich described some of his newest work crosoft Research, who spoke in the sec- concerning quantization of Poisson man- tion. In recent years, Freedman, a 1986 Fields ifolds using ideas involving motives from Medalist, has become interested in trying to use arithmetic geometry. topology to address some of the central questions in theoretical computer science. He began his lec- The talks at the ICM demonstrated ture by asking whether there is a “speed limit” on that classical questions in dynamical knowledge, akin to the limit on the speed of light: systems, such as questions about closed Is there inherent in the laws of nature an obstacle orbits, are alive and well. Examples in- to solving certain very hard problems within a rea- clude the Invited Lecture by Krystyna sonable amount of time? Here “reasonable” means Kuperberg of Auburn University, who an amount of time that grows only polynomially spoke on the real analytic counterex- with the size of the problem. After discussing the ample she produced to the so-called Fields Medalist Curtis idea of quantum computing, Freedman proposed Seifert Conjecture, a long-standing prob- McMullen. a new model called “quantum conformal field com- lem in this area, and the Plenary Lec- puting”, which attempts to exploit connections ture by Helmut Hofer of the Courant Institute, between the Jones polynomial for knots and con- which approached these questions from the view- formal field theory to attack hard computing prob- point of . Christopher lems that have not yielded to conventional algo- Deninger of the University of Münster presented rithms. an intriguing talk about an idea for a new kind of The ICM organizing committee made room on cohomology theory in number theory that has par- the program for the Fields Medalists to present 45- allels to the cohomology theory that already exists minute ad hoc lectures (Curtis McMullen was al- for dynamical systems associated with foliations. ready on the schedule to present an Invited Lec- If the theory works out as Deninger hopes, it would ture in the dynamical systems section before the produce a vast generalization of the Weil conjec-

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tures in number the- smaller rooms in this building as well as in the ory. Pierre Deligne mathematics building across the street. Some of proved the last of these smaller rooms were packed to the rafters, these conjectures, with people sitting or standing in the aisles. In known as the Rie- such cases it was clear that the talks were at- mann Hypothesis for tracting specialists and nonspecialists alike, mak- Finite Fields, and as a ing it hard for speakers to know which group to result received a address. As a result, there were some complaints Fields Medal in 1978. that the Invited Lectures were not as understand- Ideas from physics able to nonspecialists as were the Plenary Lec- arose in many of the tures. talks. In his Invited Fields Medalist , left, There were also complaints about scheduling Lecture, Clifford with Clifford Taubes of Harvard University, conflicts among the Invited Lectures, which were Taubes of Harvard who spoke on Kontsevich’s work. presented in parallel sessions. There were a num- University described ber of cases in which speakers in related areas his celebrated result were scheduled for the same time slot. Such con- showing that the flicts are unavoidable, but they seemed especially Seiberg-Witten invari- abundant at this Congress and afflicted several ants in gauge theory areas, from geometry to fluid dynamics. One es- and the Gromov in- pecially odd case concerned the talks of three variants in symplec- geometers who work in very closely aligned areas: tic geometry are one and , both of and the same. This Stanford University, and Clifford Taubes. Before the talk showed a sense Congress, Eliashberg complained to the organiz- in which Taubes was ing committee that Taubes and Donaldson had correct when, in his been scheduled to speak at the same time. Eliash- Plenary Lecture at the berg hoped to attend both his colleagues’ talks, but ICM in Zürich four it was not to be. The organizing committee elimi- years ago, he pre- nated one conflict but created another: It moved Fields Medalist Richard Borcherds, left, dicted that symplec- Eliashberg to Donaldson’s time slot, thereby putting with Peter Goddard of the University of tic geometry would Eliashberg at the same time as Taubes. After it be- Cambridge, who spoke on Borcherds’s one day be incorpo- came clear that there were many problems with work. rated into anti-self- conflicts between the talks in the geometry and dual geometry. The topology sections, the organizers attempted to influence of physics was also notable in the Ple- reschedule them in separate weeks of the Con- nary Lecture of Peter Sarnak of Princeton Univer- gress. However, the attempt came too late, as many sity. He discussed the close parallels between the of the speakers had already purchased nonre- statistics of the spacing of the zeroes of the Rie- fundable plane tickets. mann zeta function and the statistics of the dis- Although it was supposed to be a secret until tribution of eigenvalues of certain matrices, which the Closing Ceremonies, the fact that the next Con- physicists refer to as the Gaussian Unitary En- gress will take place in Beijing in 2002 was widely semble. The picture of a phase transition familiar known. (It had even been mentioned in the Notices from physics—which shows a graph that is hori- [August 1998, p. 864].) Glossy posters announcing zontal, suddenly dips sharply, and then becomes the event, which will take place in the Great Hall horizontal again—arose in surprising places, such of the People in Tianenman Square, were available as in the lectures of Freedman and McMullen. It also well before the end of the Berlin Congress. Some showed up in the Plenary Lecture by Persi Diaco- expressed discomfort about holding the ICM in a nis, in which he described a model for under- country where human rights abuses have been a standing the probabilities of distribution of cards continuing problem. It is also not clear whether after shuffling. It turns out that 6 shuffles tend to mathematicians from countries such as Taiwan leave the cards in almost the same order, but there would be allowed to attend. But perhaps Berlin will is a “phase transition” between 6 shuffles and 7, prove an exemplar for Beijing. ICM98 provided an after which the cards tend suddenly to become ran- opportunity for Germany to examine how the coun- domly mixed. Diaconis described how the card- try’s past abuses afflicted its mathematicians and shuffling model can be generalized to more com- to make a commitment to renewal. There could be plicated problems, such as random walks on no better precedent for the Beijing Congress. buildings. —Allyn Jackson The Plenary Lectures were held in a large audi- torium in one of the buildings of the Technical Uni- versity of Berlin, and the Invited Lectures were in

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