Friday, November 14, 2014

Grothendieck

Peter Woit reports that Alexander Grothendieck, one of the greatest mathematicians of the Twentieth Century, has died at age 86. He has links to a number of stories about Grothendieck - here is a fragment of a superb one by Grothendieck's friend and colleague Pierre Cartier:

Grothendieck’s journey? A childhood devastated by Nazism and its crimes, a father who was absent in his early years and then disappeared in the storm, a mother who kept him in her orbit and long disturbed his relationships with other women. He compensated for this with a frantic investment in mathematical abstraction until psychosis, kept at bay through this very involvement, caught up with him and swallowed him in morbid anguish.

Grothendieck is difficult to categorize. Like Carl Friedrich Gauss, Bernhard Riemann, and many other mathematicians, he was obsessed with the notion of space. But his originality lay in deepening of the concept of a geometric point.1 Such research may seem trifling, but the metaphysical stakes are considerable; the philosophical problems it engenders are still far from solved. In its ultimate form, this research, Grothendieck’s proudest, revolved around the concept of a motive, or pattern, viewed as a beacon illuminating all the incarnations of a given object through their various ephemeral cloaks. But this concept also represents the point at which his incomplete work opened to a void. Grothendieck’s idiosyncrasy prompted him fully to accept this flaw. Most scientists are somewhat keener to erase their footprints from the sand, silence their fantasies and dreams, and devote themselves to the statue within, as François Jacob puts it.

Cartier also has an introduction to some of Grothendieck's most important work, including some tantalizing hints of a connection between quantum mechanics and the deepest aspects of Grothendieck's conception of the mathematical point - unfortunately, most of the air up there is too thin for me to breathe. Another excerpt:

...The answer, inspired by Zariski’s work, was simple and elegant: the scheme of an algebraic variety is the collection of local rings of the sub-varieties found inside the rational function field. There is no need for an explicit topology, a point of distinction between Chevalley and Serre, who at roughly the same time introduced his algebraic varieties using Zariski topologies and sheaves. Each of the two approaches had advantages, but also limitations: Serre had an algebraically closed base field; Chevalley had to work only with irreducible varieties. In both cases, the two fundamental problems of products of varieties and base change could only be approached indirectly. All the same, Chevalley’s point of view was better suited to future extensions to arithmetic, as Nagata soon observed.

Évariste Galois was certainly the first to notice the polarity between equations and their solutions. One must distinguish between the domain in which coefficients of the algebraic equation are chosen and the domain in which solutions are sought. Grothendieck created a synthesis out of these ideas, based in essence on the conceptual presentation of Zariski-Chevalley-Nagata. Schemes are thus a way of encoding systems of equations as well as the transformations to which one may subject them.

For those who don't speak the language - and I myself only understand tiny fragments - this may seem like abstract nonsense, but in addition to probing how very real world mathematical problems can be solved, this work also speaks very deeply to ways the human mind can represent reality.