Originally appeared on Quora.
1. Évariste Galois
The mathematician Évariste Galois died at age 20 after laying the foundations of much of modern algebra. Regarding the letter in which he laid out his ideas at age 19, Hermann Weyl said, "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind"—a rather favorable review.
He was heavily involved in the revolutionary politics of France, was arrested for threatening the king's life at a banquet, and was imprisoned for wearing a uniform without authorization, narrowly missing a conspiracy charge that other members of his unit were later charged with.
He died in a duel under mysterious circumstances with an unknown assailant, either as a planned martyrdom for the republican cause, over a jealous lover, or due to an agent provocateur for the royalists using the love affair as a cover.
2. John von Neumann
He is notable not so much for his personal life but for his influence on world affairs. Besides his technical work on the atomic bomb, von Neumann is also credited with the concept of mutual assured destruction, which more or less saved us from World War III at the expense of the Cold War. He was considered so valuable by the military that he was kept under armed guard to prevent him from divulging military secrets under delirium.
He also displayed many of the qualities TV associates with geniuses:
1) Photographic memory
2) The ability to do complex calculations instantly mentally
3) Effortlessly mastery of subjects outside his expertise.
Other Nobel Prize winners literally wondered if he was of "a species superior to that of man." Dr. Strangelove is a composite of him along with a few other people (he was not a Nazi).
3. Kurt Gödel
He grew up a rather strange, sickly child in Vienna. From an early age his parents took to referring to him as “Herr Varum”, Mr Why, for his insatiable curiosity. At the University of Vienna, Gödel first studied number theory, but soon turned his attention to mathematical logic, which was to consume him for most of the rest of his life.
In the mid 1930s, he suffered a series of mental breakdowns and spent some significant time in a sanatorium. Nevertheless, he threw himself into the same problem that had destroyed the mental well-being of Georg Cantor during the previous century, the continuum hypothesis. In fact, he made an important step in the resolution of that notoriously difficult problem (by proving that the the axiom of choice is independence from finite type theory), without which Paul Cohen would probably never have been able to come to his definitive solution. Like Cantor and others after him, though, Gödel too suffered a gradual deterioration in his mental and physical health.
He was only kept afloat at all by the love of his life, Adele Numbursky. Together, they witnessed the virtual destruction of the German and Austrian mathematics community by the Nazi regime. Eventually, along with many other eminent European mathematicians and scholars, Gödel fled the Nazis to the safety of Princeton in the USA, where he became a close friend of fellow exile Albert Einstein, contributing some demonstrations of paradoxical solutions to Einstein's field equations in general relativity (including his celebrated Gödel metric of 1949).
But, even in the USA, he was not able to escape his demons, and was dogged by depression and paranoia, suffering several more nervous breakdowns. Eventually, he would only eat food that had been tested by his wife Adele, and, when Adele herself was hospitalized in 1977, Gödel simply refused to eat and starved himself to death.
4. Paul Erdős
Paul was shown to be a mathematical prodigy very early in his life; he could multiply 3 digit numbers and had independently developed the idea of "negative numbers" by the age of three.
He did not cultivate human contact outside of his mathematical interactions, with the exception of his Mother, whom he loved dearly. He didn't have to cook, clean or accomplish any of the other range of daily tasks that might sap the time and energy that he could devote to mathematics. Instead, he had a cadre of people who looked after him, and saw to it that he had food, shelter and, incidentally, a visa for his next destination.
He was a habitual amphetamine user the last 20 years of his life, and a heavy consumer of other more mainstream stimulants his entire life.
Erdős's friends worried about his drug use, and in 1979 Graham bet Erdős $500 that hecouldn't stop taking amphetamines for a month. Erdős accepted, and went cold turkey fora complete month. Erdős's comment at the end of the month was "You've showed me I'm not an addict. But I didn't get any work done. I'd get up in the morning and stare at a blank piece of paper. I'd have no ideas, just like an ordinary person. You've set mathematics back a month." He then immediately started taking amphetamines again.
But beyond his skill at collaboration, his apparently unending energy, and his voracious appetite for discovery, Erdős was, incidentally, also a mathematical genius of his own right.
5. Grigori Perelman
He turned down a $1 million prize for the resolution of the Poincare conjecture despite being unemployed and living in poverty in Russia.
Why did Grigori Perelman refuse the Fields Medal?
The key to this question is to understand Perelman's motivation and its standards.
Motivation: to solve a very important problem that no one else could solve for more than a hundred years. He single-mindedly dedicated, practically in isolation, 7 years of his life to it. Absolutely nothing else mattered to him. He REALLY didn't care about money or fame (so we can conclude already that he is a very peculiar man).
Standards: Perelman is described as "impeccably honest" by people who have known him since he was a teenager. He also believed in the absolute righteousness of mathematicians about giving credit where credit is due. But in 2003 a more senior and very influential mathematician (a Fields Medallist himself) was involved in publishing a paper that clearly tried to take away from Perelman some of the credit.
Other people would have worked out the politics by making the right calls and explaining everything. Not Perelman. Throughout his adult life Perelman onlyfelt comfortable around few other mathematicians and now felt betrayed by the mathematical community. He shut out. He quit his job. He abandoned mathematics altogether in 2005.
So in May 2006, when approached in Russia by the president of the mathematical union (John Ball) he said he wouldn't go to the international congress and he wouldn't accept the prize.
'I don't want to be on display like an animal in a zoo. I'm not a hero of mathematics. I'm not even that successful; that is why I don't want to have everybody looking at me.'
- Grigori Perelman