Alexander Grothendieck

Readers of this blog are probably already aware that Alexander Grothendieck died on Thursday. He is widely regarded as one of the most influential mathematicians in the twentieth century. Since his is not my field of study, I felt that now was a good time to learn a little about why he is so well regarded — I took the day to read a couple of articles from 10 years ago that provide an overview of his life, research, personality, and philosophies. I highly recommend the read: here and here.

(I say that his is not my field of study, and yet I have still seen his influence. For example, the Grothendieck inequality is a beautiful result in functional analysis that he proved as a graduate student before changing fields, and it has since found applications in hardness of approximation. Also, his development of Grothendieck groups provided a starting point for K-theory, which is the source of the best known lower bound for the 4M-4 conjecture.)

The rest of this entry briefly summarizes some of what I learned from the reading:

  • As a boy, he ran away from an internment camp with the intention of assassinating Hitler.
  • He rediscovered measure theory and the Lebesgue integral on his own.
  • He lived his life without nationality.
  • One time, he mistook 57 for being prime. (Please don’t hold me to a standard of never committing a brain fart!)
  • He was fond of giving names to new mathematical objects.
  • In the 1970s, he was seriously concerned for the fate of the human race due to environmental degradation and military conflict.
  • That same decade, he started a commune that eventually collapsed. He later tried again with similar results.
  • Instead of reading, he would talk to others to find out what was happening in the math community. By contrast, he wrote a lot.
  • He attempted to convince colleagues to join a campaign against Springer (perhaps a precursor of sorts to the recent anti-Elsevier campaign).
  • In the mid-1990s, he disappeared into the Pyrenees to live in complete isolation.

When he studied math, he was chiefly interested in solving problems that pointed to deeper hidden structures. This pursuit of truth and beauty matched his overall research philosophy, which was to build up a theory of mathematics as a series of small, natural steps. Apparently, he was not interested in publishing his proof of the Riemann–Roch theorem because it didn’t follow this philosophy; it used a “trick.” Similarly, he wasn’t satisfied with Deligne’s proof of the “most stubborn” of the Weil conjectures because it used a trick instead of going through the standard conjectures, which would have been more telling of underlying structures.

His approach to mathematics was very general, and he didn’t work much with examples. He had a knack for taking leaps of abstraction to prove a result, but he didn’t seek generality for the mere sake of generality — he knew which general things to think about.

As a leader in the field, he was charismatic. He was full of energy and ideas, and he consistently worked hard toward his mathematical agenda. He also knew how to match people with open problems. While he could be intense, he was a good communicator, an excellent teacher, patient, charming, and he loved to laugh.

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