When I was a junior professor at my university in the early 1990s, I had the pleasure of being advisor and major professor to a gifted young woman who was on the fast track to a very solid graduate school stint in pure mathematics. In her senior year she had pretty much exhausted our regular offerings and asked me for a reading course in analysis. I opted to have her go through part of Ahlfors’s classic text on complex analysis and it indeed proved quite a sporty business. Of course, given the level at which the book is pitched, it was clear from the outset that my student was in for the right kind of torture for a mathematician, and it certainly had the right impact.
In any case, Ahlfors was a living legend at that time (he passed away a few years later, in 1996, at 89), and although he was known to outsiders, i.e. non analysts, primarily through the effect of his famous text mentioned above, there was, and is, a lot more to be told about his fascinating life. We now have a compact, readable, and mathematically meaningful biography by Olli Lehto, another Finn, another analyst, another member of Nevanlinna’s circle, and a very close friend of Ahlfors. It doesn’t get any better than that.
Thus, in the pages of the book under review, we learn any number of marvelous things about this great scholar, an early Fields medalist who, early on, moved from Helsinki to Harvard, where he spent on the order of 30 years, and who became a major and influential figure on the international mathematical scene. We learn about his early breakthrough at age 21, proving Denjoy’s conjecture and including it in his doctor’s thesis, and about his relationship with his teachers Ernst Lindelöf and the aforementioned Rolf Nevanlinna. We learn about Ahlfors’s close friendship with Arne Beurling, who also solved the Denjoy problem but didn’t publish until some years later, the friends’ titanic capacity for strong drink, and the poignant episode of Ahlfors’s eulogy for Beurling (who died in 1986) starting off with the phrase “Arne Beurling was the best friend I ever had.”
Then there is the following passage (on p. 108) from a speech by Raoul Bott, giving a sketch of Ahlfors’s personality:
“In the [Harvard mathematics] department Lars was a quiet, reserved, and benevolent presence. At home he was the warmest of hosts, a delightful drinking companion, a devotee of beauty in every form, from literature, music, and painting, to the admiration of the opposite sex. Yet there was no impropriety in Lars’ gallantry. All his life he was enthralled with his wonderful Erna.
Indeed, his wife properly figures throughout this thoughtful biography, from their courtship, through the adventures of a long and full career, to the last year of Ahlfors’s life, when he was stricken with dementia. Lehto presents this personal dimension of the life of his friend and the subject of his biography with great sensitivity and poignancy. And, when appropriate, humor can’t help but filter through: there are, for instance, a number of tales about drinking (and sometimes even drinking and driving) in the account.
Lehto is a mathematician, as already mentioned, so that aspect of Ahlfors’s life is amply taken care of, and is done proper justice to. Compex analysis, conformality (and Ahlfors’s own quasi-conformality), and Riemann surfaces are everywhere, and rightly so. That said, the book under review has something to offer to every one of us, analyst or not: we all know of his famous text, at least, and we should all be keen to read about such an interesting mathematician’s life, played out on two continents, divided by a World War, and of course dominated by the grace of prolonged mathematical exploration and discovery at the highest level.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.