by Jonathan Kujawa
Several weeks ago Alexander Grothendieck passed away. It is hardly possible to overstate his influence on twentieth (and twenty-first!) century mathematics. With the help of others he rebuilt vast amounts of mathematics from the ground up. He had a vision that still seems futuristic many decades later [1]. I compare it to Braque, Picasso, and company blowing up the art world with their entirely new vision of what art could be. In Grothendieck's case you'll have it about right if you imagine him as a visiting scholar from an alien civilization whose mathematics is to ours as ours is to one of those Amazonian tribes who can only count to three.
Grothendieck's life was as interesting as his mathematics. It's bound to be turned into one of those movies made to win Oscars [2]. His parents were anarchist political activists and artists, he moved to France as a refugee of Germany in 1938, and for most of his life was legally stateless and traveled with a Nansen passport. Grothendieck was at the peak of his public mathematical life during the 50's and 60's, receiving the Fields medal in 1966. Starting in the 70's he withdrew from the mathematical community and, ultimately, his family and friends as well.
For the last two decades he lived in a village in the south of France and only a very few people knew where he was. Grothendieck issued a letter in 2010 insisting that his work not be published and any existing publications be withdrawn from libraries. He was so isolated that it wasn't immediately clear to many in the math community if he was still alive and, if so, if he was the one who had written the letter.
Since that request seems to no longer be in force we should now have the chance to learn what did with himself for the past thirty years. There are rumors of tens to hundreds of thousands of pages of mathematics and political and philosophical writings. I'm sure I'm not the only one who had idle fantasies of running into Grothendieck at a cafe in France and getting on like gangbusters over espresso while hearing all about what he'd been up to [3].
Grothendieck's tools, language, and point of view are now ubiquitous across a broad spectrum of contemporary mathematics. They are certainly part of the everyday lexicon and mode of thought in my area of research (representation theory). Grothendieck reconsidered such fundamental questions as what is a “point” (there's a lot more to say than you might think!). For an excellent overview of Grothendieck's work I recommend Steve Landsburg's recent essay. It was linked to here on 3QD, but you may have missed it in the shuffle.
