Book Reviews Holden and Piene (eds)

Reviewer: Ulf Persson This is the sequel to the first book on the Abel Prize winners, which covered the first five years and which was reviewed for the EMS a couple of years ago. The format of this book is the same but this time, even more ambitiously, the editors have produced a tome that dwarfs the preceding one in bulk. One wonders whether this will set a trend. The basic elements consist of autobiographical sketches followed by lengthy presentations of the mathematical accomplishments of the winners. In addition, there are complete lists of each winner’s publications and short, formal CVs. There is also some additional material, which we will consider now and return to later. The book starts out with an introduction, written by an historian Kim Helswig, presenting in documentary detail the history of how the prize came into being, a process in which the Abel biographer Arild Stubhaug played a key role both as an initiator and, perhaps more importantly, as a caretaker, making sure it came to fruition. There is also an account of the subsequent failure to connect it more closely with the Nobel Prize. In fact, the desired connection to the Nobel Prize is also made explicit by the title of the contribution. As was already discussed in the previous review, the Abel Prize is meant to be the Nobel Prize of mathematics, with the hope of also allowing mathematics to partake in the glamour and attention that comes to the natural sciences once a year. As was noted back then, it is one thing to set up a prize but quite another to make it famous. Financial generosity, as displayed by the Norwegian Government, may not hurt but it is far from being sufficient. Tradition is something of the past over which we have no control; it is different with the future for which we can always hold out hope. One can see these volumes as part of a sustained effort to establish the Abel Prize, at least in the world of mathematicians. The glamour of a prize never stems from the size of the prize itself, only from that of its recipients. The cursory reader may be expected to read through the introduction and sample the autobiographical sketches for their human interest. Such a reader may also have the ambition to read through the mathematical sections (at least at some later date) but may find themselves bogged down in technicalities. It is indeed in the mathematical sections that we find the explanation for the added bulk; when it comes to autobiographies, the efforts are bound to be very sketchy and I do not believe that this will change in the future, as it is obviously not a requirement with which most of the mathematicians will feel comfortable. Thompson appears most uncomfortable; I suspect his one page submission, including a lengthy quote from Stendhal, must have been an effort for the editors to extricate. In the case of Tits, the interview reports serve the purpose; we learn that he must have been something of a prodigy, which may not have been revealed had he written a biography himself. Mathematicians tend to be modest and when it comes to autobiographies, this is a definite disadvantage. Tate and Milnor give succinct and impeccable accounts of their lives, especially their mathematical childhood and youth, which of course is of the greatest interest. We learn that Tate originally intended to be a physics graduate student at Princeton (although his interest and ability were definitely superior in mathematics) because from Bell’s classic book ‘Men of Mathematics’ he had received the impression that you needed to be a genius to pursue mathematics (while, on the other hand, his father was a physicist). Milnor emphasises his shyness as a youngster and his consequent isolation, unsurprisingly taking advantage of any books he could lay his hands on, including the few mysterious mathematics books his father, an engineer, happened to own. When he came to Princeton at the age of 17, he was truly captivated and a subsequent stint in Zürich with Hopf as a graduate student capped it all off, not only mathematically. He also admits to a passing interest in game theory (after all, he was a fellow student of Nash) but decided that its main difficulties were not mathematical. The account of Szmeredi is not devoid of charm either but he plays it safe by concentrating on his early mathematical career. Gromov takes the most original approach by turning the task into a reflection on what it means to be a mathematician, in the process eschewing any systematic chronological account. The result is one of the gems in the volume. Scientists, like children, are good at non-understanding, he points out, in particular in their propensity for asking stupid questions, such as whether four elephants can beat a whale in a fight. One should never despise the trivial observation, he cautions, and refers to a chance remark in a lecture that made him realise that group theory was more than just slippery formalism, the consequences of which it would take him 20 years to work out. Mathematics is about asking the right questions and asking stupid questions is the way to start, he seems to imply. Gromov’s parents were pathologists and the breadth of his interests is legendary; one surmises that he keeps on asking questions regardless of the context he finds himself in. The meat of the book is to be found in the mathematical surveys – these differ widely. The most conventional, in a way, and this is not meant to be disparaging, is that The Abel Prize 2008–2012