In 2006 Guillermo Curbera organized a comprehensive exhibition of historical materials for the International Congress of Mathematicians (ICM), held that year in Madrid. His extensive efforts are evident in this carefully researched text with its extraordinary collection of 400 illustrations, many of which have not appeared elsewhere, at least in recent times.

The text is a joy to read and the lavish layout is a delight to the eye. Even the cover is a hint of good things to come: a wrap-around group picture of the 1954 Congress in Amsterdam with a dust jacket echoing this cover with a similar group photograph from the 1950 congress at Harvard. The reproductions are so good the reader can identify individuals in the pictures.

The literature on the International Congresses of Mathematicians is small — up till now, two books in addition to the volumes of proceedings issued after each congress: (1) *International Mathematical Congresses: An Illustrated History 1893–1986* (Springer, 1986; reissued in 1987 and revised in 1990 in Japanese, and in 2002 in Chinese)) by D. Albers, G. Alexanderson, and C. Reid, and (2) Mathematics without Borders: A History of the International Mathematical Union (Springer, 1988) by O. Lehto. The first was published to coincide with the second Congress to be held in the United States (Berkeley, 1986). The second is a very scholarly work on the history of the International Mathematical Union (IMU), the current sponsor of the Congresses.

I eagerly awaited the appearance of this new volume. It has now appeared and a splendid work it is, profusely illustrated and elegantly presented. It is comprehensive, with lots of color pictures, not only of mathematicians and meeting venues, but also with much attention paid to ephemera: posters, invitations and tickets, logos, postage stamps, scenes of social events and excursions — and even the sheet music for a song by Tom Lehrer. It is almost as good as having attended the congresses!.

A Foreword by Lennart Carleson describes his experience attending his first Congress, appropriately that of 1962 in Stockholm. He says:

The congress was a great experience. I was amazed to encounter the richness of our field and how unimportant my own specialty was considered by many people. I made friends from different parts of the world and these contacts have lasted through the years. I saw icons of mathematics whose names I knew from theorems and listened to their lectures. I remember in particular Jacques Hadamard. The organizers of the congress had with great effort managed to get a visa for him for a few days, in spite of the risk for the security of the country to let an 85-year-old communist in. This was my first contact with the problem of how politics interferes with mathematics. Much more on this subject can be found in this book. At the congress I listened to lectures by not only Hadamard but also H. Cartan, K. Gödel, J. Leray,… and S. S. Chern, to just mention a few. I also remember the excitement of the Fields Medals — who would win? — and the discussions afterwards.

Here Carleson captures the excitement of attending one’s first congress, as well as subsequent ones. If one is not standing on the shoulders of giants, one is perhaps able, for a short time, to rub shoulders with them. Mainly we stand aside in awe—spotting Pontrjagin chatting with someone on a street corner, seeing together groups of Fields Medalists past and recent, catching a glimpse of Sierpiński or Bombieri, or… and hearing some great talks.

This is not a book about mathematics, but is primarily about the community of mathematicians. To be sure, by reading titles of plenary sessions, descriptions of major mathematical announcements, and reports on the work of Fields Medalists, the reader can get some idea of what kind of mathematics was going on at the time. But for details of the mathematics, a person really needs to consult the published proceedings.

Curbera catches the spirit of the congresses beautifully. Individually mathematicians, like others, can be difficult, remote, unwelcoming, but at these meetings they often appear to be very happy to see each other. Of course, as Lehto makes clear in his book, some of the politics of mathematics can be ugly. This was clear from all the difficulties after World War I that affected the organization of the postwar congresses in Strasbourg, Toronto and Bologna. Later there were political conflicts during the Cold War, evident from the delay of the Warsaw Congress in 1982. There were other problems during the Cold War, primarily in Vancouver and Helsinki where there were questions about whether Soviet mathematicians would be allowed to attend or how the delegates were being chosen. In Vancouver only 26 of the 42 Soviet delegates were allowed by their government to attend.

Curbera does not dwell too much on these problems and in his writing maintains a rather light touch, as one might expect from the book’s proletarian title. In the captions, in particular, one occasionally finds an unexpected remark, as when he describes the design of the postage stamp showing Jean Bernoulli, issued by Switzerland on the occasion of the 1994 Congress in Zürich. He says that “it could have been prettier.”

The narrative proceeds largely chronologically, from the international “congress” convened in Chicago on the occasion of the World’s Columbian Exposition in 1893 at which Felix Klein spoke. Papers by Hermite, Hilbert, Minkowski, Pincherle, and Pringsheim, who were not in attendance, were read by others. The first “official” congress was held in Zürich in 1897, with participants from 26 countries. This is in contrast to the 108 countries represented in 2006 in Madrid

Perhaps the most often cited congress was held three years later in Paris, again at the time of a World’s Fair, the 1900 Exposition Universelle. The main event was Hilbert’s famous list of 23 problems to set the mathematical agenda for the 20h century. Following successful congresses in Heidelberg (1904), Rome (1908), and Cambridge (England) (1912), held with little drama, there was a break in the series due to World War I. After the war the first Congress was in Strasbourg (1920), this time with a legacy from the recent conflict. The choice of location, Strasbourg was provocative. As a result of the war France had, under the provisions of the Treaty of Versailles, again taken control of Alsace, lost earlier to Germany in the Franco-Prussian War of 1871.

Further, invitations for Strasbourg had not been issued to mathematicians from the Central Powers (Germany, the Austro-Hungarian Empire, Bulgaria, and Turkey). To reflect this, the Congress was no longer called an International Congress of Mathematicians, but instead an International Congress of Mathematics. There’s a difference. The exclusion of German mathematicians and others continued into the Toronto Congress of 1924 with details too complex to go into here, but in Bologna in 1928, all nationalities were again welcome. The Toronto Congress did, however, contribute one innovation of note: the first group picture of the delegates was taken. In Bologna the president of the honorary committee was Benito Mussolini! But more important was the fact that the Congress took on its former name — it was back to being a meeting of mathematicians, not a congress of mathematics .

The Oslo congress in 1936 was the setting for the awarding of the first Fields Medals, but attendance was disappointing, probably due in part to the Great Depression but also to the ominous political events in Europe.

Throughout this chronological account the author intersperses chapters (“Interludes”) on: “Images of the ICM,” logos and memorabilia; “Awards of the ICM” (the Fields Medal, the Nevanlinna Prize, and the Gauss Prize); “Buildings of the ICM,” with pictures of the meeting sites from the Richelieu Amphitheatre in Paris (the size of a large classroom) to huge convention halls seating thousands; and “The Social Life at a Congress.”

The presence of this last aspect of congresses is important, and in keeping with the emphasis on their being meetings of mathematicians, as mentioned earlier. From the time of the 1897 Congress in Zürich, the value of enabling people to know each other was recognized. Of course, in early congresses one feature was to plan activities for “the ladies” while their husbands took care of the business of the Congress. Fortunately, if there are still teas and receptions today, they are no longer restricted to “the ladies.” The social events take various forms, and just as God in his wisdom, it has been remarked, always placed significant bodies of water next to large cities, someone sees to it that at ICMs, participants have an option of taking a cruise on the water: Zürich-See, (3 times!), Oslo Fjord, the Strait of Georgia, the Mediterranean off the Côte d’Azur, the Gulf of Finland and San Francisco Bay, to name a few. Of course there were exceptions. In Rome participants had to make do with a visit to Hadrian’s Villa (fountains only), and in Bologna 400 participants made a long journey to Ravenna to go wading in the Adriatic. A non-aquatic event for the Berkeley congress in 1986 was a rodeo and a Western barbecue. The excursion of record length was the 18-day rail trip from Toronto to Vancouver and back in 1924. The author notes wryly that after this trip, J. C. Fields (who had organized the Congress — and for whom the medals were named) suffered a sudden decline in his health. Small wonder!

As with many large mathematical meetings there have been concerts. The Cambridge Congress in 1950 seemed to set records: a concert by the Busch Srring Quartet, along with recitals by Helen Traubel, the reigning Wagnerian soprano of the day, and the folk singer, Richard Dyer-Bennett. (Can it be coincidence that Dyer-Bennett’s brother was a mathematician?) In Beijing there were performances of three Chinese operas and this book provides short versions of the plots, thus demonstrating that Verdi and Wagner did not exhaust the supply of silly opera scenarios.

If you enjoy reading about mathematicians, their foibles as well as their passion for this subject, you can be assured that this book has much to offer.

Earlier I mentioned that the lists of plenary speakers can provide a hint about where mathematics is going, but because mathematicians in this class often work in more than one field or indeed unify seemingly disparate fields, it’s hard to conclude very much. Early lists of speakers contain names of mathematicians who have withstood the test of time: in Bologna, Borel, Fréchet, Hadamard, von Kármán, Lusin, Veblen, Volterra, and Weyl, for example. Seventy years later, we have a list of plenary speakers who are not yet exactly “mathematical household” names, but with names like Eliashberg, Iwaniec, Stanley, Tao, … the hope is that this list too will, decades hence, be seen as the equivalents of those others. It’s an interesting game to play, to predict whose work will last.

Less difficult to assess is another trend. The first female plenary speaker was Emmy Noether (Zürich, 1932). The next one appeared 58 years later, Karen Uhlenbeck (Kyoto, 1990). Then in 1994 (Zürich) there were two, Ingrid Daubechies and Marina Ratner, but in Berlin (1998), there was only one (Dusa McDuff). Progress is not necessarily monotonic.

A clearer trend is in meeting sites. The first six congresses were in Western Europe, as have been almost all of them since. But four have been in North America (Toronto, Cambridge, Vancouver, Berkeley) and now two in Asia (Kyoto, Beijing), with another scheduled there in 2010.

Next year, in Hyderabad!

All images are from *Mathematicians of the World, Unite!* We thank A K Peters for supplying the sample images and giving us permission to use them.

Gerald L. Alexanderson (galexand@math.scu.edu) is Valeriote Professor of Science at Santa Clara University. He has served as Secretary and President of the MAA.