When I first thought of this question, I wanted to ask: If you could give one mathematician his remaining life until an average age for the time he lived in, who would you choose? However, this question is a bit too subjective for SE. So I decided to modify it to: Which mathematicians died very young or in a tragic way?
Two obvious candidates are Galois and Abel. I included more detail about them in an answer. But I'd like to know whether there are more examples.
Maryam Mirzakhani, the first Iranian and first woman to win the Fields medal, died of breast cancer in July 2017. She was only 40 years of age.
Évariste Galois (1811 - 1832), aged 20, was killed in a duel. He is known for Galois theory and he wrote his most notable results down in the night before the duel. You can also find more information about him and why he was killed in this question.
Niels Henrik Abel (1802 - 1829), aged 26, died of tuberculosis. He is mainly known for proving the Abel-Ruffini theorem, that had been open for hunderds of years, but also for several other results. However, he was barely recognized during his lifetime and because of that he was very poor. He died two days before he got a letter that he was appointed as a professor at the Berlin university.
Srinivasa Ramanujan (1887 - 1920) died at the age of $32$, according to Wikipedia the cause was:
A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B. Young concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver widespread in Madras, where Ramanujan had spent time. He had two episodes of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis, a difficult disease to diagnose, but once diagnosed readily cured.
Bernhard Riemann (1826–1866) died at the age of 39. Paraphrasing the following from Wikipedia:
The son of a poor pastor, he enrolled at age 19 at the University of Göttingen to obtain a degree in Theology. There Gauss urged him to become a mathematician. When he was 31, there was an attempt to promote him to extraordinary professor status. When he was 33, Dirichlet died, and he became head of the department of mathematics.
He is known for Riemannian geometry (Riemann surfaces etc.), the Riemann integral (the first rigorous formulation of the integral), one of the most influential papers in analytic number theory, in which he stated the Riemann hypothesis, and a few other things.
In 1866, the Austro-Prussian War broke out. The armies of Prussia and Hanover clashed in Göttingen. Riemann fled the city. In Italy, he died of tuberculosis. (The war lasted just over two months, from June to August 1866.) He was 39. Much of his unpublished work was lost.
Eugenio Elia Levi (1883-1917), the one of Levi decomposition in Lie algebras, was killed in action during WWI to which he participated as a volunteer. It is often said that his early death played a role in the Italian school of geometry leavng largely unexplored the subject of Lie algebras and Lie groups.
Andreas Floer (1956-1991) who introduced what is nowadays called Floer homology, ICM plenary speaker in 1990, commited suicide at age of 34. He opened up a completely new subject (symplectic topology) and, in fact, the list of posthomous paper speaks in itself.
Frank Ramsey died before his 27th birthday. He started Ramsey theory, which is a very active field in mathematics.
Yutaka Taniyama (12 November 1927 – 17 November 1958) was a notable Japanese mathematician known for the Taniyama–Shimura or Taniyama–Shimura–Weil conjecture now referred to as the Modularity theorem which "states that elliptic curves over the field of rational numbers are related to modular forms."
Proof of a significant special case of the modularity theorem (for semistable elliptic curves) was an essential component of the proof of Fermat's last theorem.
From here:.
On 17 November 1958, Taniyama committed suicide. He left a note explaining how far he had gotten with his teaching duties, and apologizing to his colleagues for the trouble he was causing them. His mystifying suicide note read:
"Until yesterday I had no definite intention of killing myself. But more than a few must have noticed that lately I have been tired both physically and mentally. As to the cause of my suicide, I don't quite understand it myself, but it is not the result of a particular incident, nor of a specific matter. Merely may I say, I am in the frame of mind that I lost confidence in my future. There may be someone to whom my suicide will be troubling or a blow to a certain degree. I sincerely hope that this incident will cast no dark shadow over the future of that person. At any rate, I cannot deny that this is a kind of betrayal, but please excuse it as my last act in my own way, as I have been doing my own way all my life."
Mikhail Yakovlevich Suslin (1894-1919). Known for "Suslin sets" and the "Suslin hypothesis". He died of typhus following the Russian Revolution. He was 25. LINK
I recently read a very interesting book Naming Infinity about the Moscow school of mathematics.
Gotthold Eisenstein is a classic example. This prodigious young mathematician was held in the highest esteem by none other than Gauss himself, and he was also given encouragement and support by Jacobi and Kummer. He worked in number theory and analysis, publishing, for example, proofs on the laws of quadratic, cubic and quartic reciprocity. He always suffered from ill health, however, and died at the age of 29 of tuberculosis.
Pavel Urysohn, known for his eponymous Metrization Theorem and also eponymous Lemma. He died at the age of 26 in a drowning accident off the coast of Brittany, France. He did fundamental work in the then nascent field of point-set topology.
Felix Hausdorff, his wife, and his wife's sister committed suicide in 1942 after being ordered to a Nazi death camp. He was 74 at the time but his death is certainly tragic. Wikipedia
Joseph Liberman was killed in WW2 at the age of 23. He defended his thesis during a short vacation in Leningrad. The idea in his thesis is one of the most fruitful in comparison geometry, which was developed further by Alexandrov's school.
Jacques Herbrand (1908-1931) died et 23 in an accident during a mountain excursion. He made fundamental works in logic.
"This page features a collection of mathematicians who died under unfortunate or unfitting circumstances."
Some of those not already mentioned in previous answers are
Vladimir Markov, 1871-1897 (25, tuberculosis)
René Gâteaux, 1889-1914 (25, died in battle during WWI)
Stanisław Saks, 1897-1942 (44, murdered by the Gestapo)
Dénes Kőnig, 1884-1944 (60, suicide after Hungarian Nazi coup)
Dmitri Egorov, 1869-1931 (61, starvation while on hunger strike)
Archimedes, 287-212 (75, killed)
Let's add Hypatia and Hippasus (although we're straying into legend here).
In a footnote on page 4 of Mathematicians Fleeing from Nazi Germany, author Reinhard Siegmund-Schultze names over two dozen mathematicians murdered by the Nazis.
EDIT: I should also mention Walter Koppelman, murdered by a graduate student at Penn in 1970.
Oswald Teichmüller (1913–1943), was a gifted German mathematician who worked on Riemann surfaces, among other subjects. He was also a dedicated Nazi, who volunteered for combat on the Eastern Front and was killed in action in September 1943.
Raymond E. Paley (7 January 1907 – 7 April 1933) fulfills both conditions asked for in the question. He died young, with 26 years, just after having started his collaboration with Wiener (see Paley-Wiener integral), and in a tragic way. Namely, he was killed by an avalanche, while his death was witnessed by companions lower down the mountainside.
Source: Wikipedia
Gustav Roch, well-known for his Riemann-Roch theorem, died at the age of 26 of tuberculosis, in 1866, just a few months after Riemann's death.
Wolfgang Döblin, interested in probability theory, and had Maurice Fréchet as doctoral advisor, committed suicide at the age of 25.
Heinrich Kornblum proved an analogue to Dirichlet's theorem on arithmetic progressions in the setting of polynomial rings over finite fields for his PhD thesis. Sadly he was called to fight in WWI and died in battle. His advisor, Edmund Landau, was tasked with typing and publishing the work of his student. You can find a source in Number Theory in Function Fields by Rosen.
Lu Jiaxi (1935 - 1983) was a self-taught Chinese mathematician who solved a major problem in combinatorial design theory. He was born in a poor family and his father died when he was in junior middle school. Working in the factory for several years, he self-studied and got into university majoring in physics.
Fascinated by the Kirkman's schoolgirl problem while in university, he self-studied math and solved a generalized version of the problem in early 1960s, and he wrote a paper on it. However, being a physics teacher at a middle school in a remote city, his paper was rejected by Chinese math journals, and then the Cultural Revolution started. After the Cultural Revolution, he found out that the same problem was later solved and published in 1971 by Ray-Chaudhuri and R. M. Wilson.
He turned his attention to the problem of existence of large sets of disjoint Steiner triple systems, which was very much open at the time. As a school teacher, he could only worked in the evenings on the problem. His health was poor due to heavy work and poverty. He solved the problem and submitted it to Journal of Combinatorial Theory, Series A. The journal published first three of his papers in March 1983, it was then the Chinese mathematicians discovered him. Tragically, after attending the general conference of the Chinese Mathematical Society in October that year, he hurried back home, and in that very evening of his arrival he had a heart attack and died, leaving the seventh paper which was the last part of his proof unfinished. The President of the University of Toronto wrote to the principal of his school asking the principal to allow Lu Jiaxi to be transferred to a university, but it was too late.
Vladimir Voevodsky died on September 30, 2017 at the age of 51. He won the Fields medal in 2002 (due to the proof of the Milnor conjecture)
Ludwig Scheeffer (1859-1885) was a gifted mathematician. At 25 he became a Privatdozent in Munich. He died at only 26. In 1884, only 10 years after the invention of set theory and way before its breaktrough in 1891 he wrote in an Acta Mathematica paper about sets of first cardinality. Georg Cantor wrote an obituary which is reprinted in his collected papers pp. 368f.
Mukremin Nesheli.
He was a young Turkish mathematician. He had such strong mathematical abilities that the Canadian mathematician Langlands immediately recognized him and Langlands wanted to invite him to Yale University. But Mukremin was a leftist and was killed by a right-wing group in Turkey. This story always reminds me of Evariste Galois.
James Gregory also died very young. He discovered the basic ideas of Calc the same time Newton did...he died at 36 only 1 year after accepting a position at the University of Edinburgh
Oded Schramm died in a climbing accident at the age of 46. He is best known for his work on Schramm–Loewner evolution. He also contributed to Wikipedia.
Felix Hausdorff LINK
This is not "very young" but it is still tragic.
Hausdorff was a German Jewish mathematician. He is also known for his work in astronomy, philosohy, and even literature (under the pseudonym "Paul Mongré"). When the Nazis came to power in Germany, Hausdorff was retired, and he was allowed to remain in his house in Bonn. For a long time, the Nazi government ignored him, instead concentrating on purging active Jewish academics.
But finally in 1942 Hausdorff heard that they were coming to deport him to a camp. Felix, his wife, and his wife's sister committed suicide by taking barbiturates. He was 73.
https://www.realclearscience.com/blog/2015/02/mathematicians_die_in_horrible_ways.html refers to Kellen Myers's list, and exemplifies with:
Or what about Austrian American mathematician Kurt Gödel? Considered by some to be just as influential a logician and philosopher as Aristotle, he sadly succumbed to crippling paranoia later in life. In his sixties, he became convinced that his food was being poisoned, and would only trust the cooking of his wife Adele. When she was hospitalized for six months in 1977, Gödel refused to eat, and subsequently died of starvation.
https://www.usatoday.com/story/tech/2015/03/08/mathematician-deaths/24605901/ exemplifies with:
There's Alan Turing and Dénes Konig, both of whom killed themselves.
Wikipedia details Kőnig's rational suicide:
After the occupation of Hungary by the Nazis, he worked to help persecuted mathematicians. On October 15, 1944 the National Socialist Arrow Cross Party took over the country. Days later on October 19, 1944 he committed suicide to evade persecution from the Nazis being a Hungarian Jew.[1]
Roger Cotes, a contemporary of Newton and best known for the Newton-Cotes quadrature rules in numerical integration, died of a fever aged only 33.
Newton supposedly said of his death 'If he had lived we would have known something': this is extremely high praise given that Newton was generally not inclined to praise the work or intellect of others.
There are a few notable ones from Computer Science and Logic:
Alan Turing (famously) died at the age of 41, most likely due to suicide from mental issues cause by estrogen he was being forced to take
Gerhard Gentzen died at the age of 45, during WWII. His death isn't always viewed as a tragedy, since he was fighting for the Nazis. But he invented both Natural Deduction and Sequent Calculus, and was hugely fundamental in the study of logic.
Kurt Gödel wasn't young (71), but he suffered from extreme mental illness, so his death was tragic. He died when his wife was hospitalized and he refused to eat any food, for fear that others would poison him.
Archimedes of Syracuse (287 - 212 BC) died in a tragic episode of the Second Punic War between Rome and Carthage. At that time he was already 75 years old.
Syracuse was an important Greek colony on the island of Sicily with good relations to Rome. However, during the Second Punic War Syracuse ended up being under a long Roman siege. One reason for the length of the siege was the ingenuity of Archimedes.
Finally, the city fell and Archimedes was killed by an angry Roman soldier as he didn't want to interrupt his geometrical studies. His alleged last words were:
"Noli turbare circulos meos" ("Do not disturb my circles!")
C P Ramanujam, an Indian Mathematician, not to be confused with Srinivasa Ramanujan, one who worked with GH Hardy.
Ramanujam's achievements at High School had been outstanding and he had shown that he was extraordinarily gifted, so he entered Loyola College with great expectations. He continued his interest in chemistry but it was mathematics that he specialised in, taking Mathematics Honours after obtaining his Intermediate qualification. He was awarded a B.A. with Honours in Mathematics in 1957 but, strangely for such an outstanding student, he only obtained a second class degree. This may have been a result of starting his university education at so young an age before he was really ready, for the second class degree no way reflected his remarkable mathematical abilities. On the other hand it may have resulted from a lack of belief in himself which haunted Ramanujam throughout his life.
Ramanujam felt that he did not have what it takes to solve the big problems of mathematics, and he had no wish to solve small routine problems. Again, as in his undergraduate course, it would appear to be a psychological problem rather than a mathematical one but for Ramanujam, it was a very real problem and he became more and more frustrated.
Mumford writes
It was a stimulating experience to know and collaborate with C P Ramanujam. He loved mathematics and he was always ready to take up a new thread or pursue an old one with infectious enthusiasm. He was equally ready to discuss a problem with a first year student or a colleague, to work through an elementary point or puzzle over a deep problem. On the other hand he had high standards. He felt the spirit of mathematics demanded of him not merely routine developments but the right theorem an any given topic. He was sometimes tormented by these high standards, but, in retrospect, it is clear to us how often he succeeded in adding to our knowledge, results both new, beautiful and with a genuine original stamp.
Back in India after his year at the University of Warwick, Ramanujam asked for a Professorship at the Tata Institute ,but be based in Bangalore where a new branch dealing with applications of mathematics was being set up. This was agreed and he taught analysis in Bangalore but, again in the depths of depression caused by his illness, he tried again to leave the Institute and obtain a university teaching post. While waiting for an offer of such a post from the Indian Institute at Simla, he took his life with an overdose of barbiturates.
Sonya Kovalevskaya died at the age of 41, of the 'coronavirus' of her time. From Wikipedia:
In 1889, Kovalevskaya fell in love with Maxim Kovalevsky, a distant relation of her deceased husband, but insisted on not marrying him because she would not be able to settle down and live with him.
Kovalevskaya died of epidemic influenza complicated by pneumonia in 1891 at age forty-one, after returning from a vacation in Nice with Maxim...
Kovalevskaya's mathematical results, such as the Cauchy–Kowalevski theorem, and her pioneering role as a female mathematician in an almost exclusively male-dominated field...
Also her husband, paleontologist Vladimir Kovalevsky, had committed suicide 8 years earlier.
