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Gerd Faltings studied for his doctorate at the University of Münster, being awarded his Ph.D. in 1978. Following the award of his doctorate, Faltings went to the United States where he spent a year doing postdoctoral work as a research fellow at Harvard University in 197879.
In 1979 Faltings returned to Germany, taking up an appointment as professor of mathematics at the University of Wuppertal. In 1985 Faltings was appointed to the faculty at Princeton.
Faltings proved conjectures by Mordell, Shafarevich and Tate during 1983. In the same year he received the Danny Heinemen Prize from the Akademie der Wissenschaften, Göttingen.
In 1986 Faltings received the highest honour that a young mathematician can receive when he was awarded a Fields Medal at the International Congress of Mathematicians at Berkeley. At the Congress B Mazur gave an address describing the work by Faltings which had led to the award. He received the medal primarily for his proof of the Mordell Conjecture which he achieved using methods of arithmetic algebraic geometry.
Faltings has been closely linked with the work leading to the final proof of Fermat's Last Theorem by Andrew Wiles. In 1983 Faltings proved that for every n > 2 there are at most a finite number of coprime integers x, y, z with x + y = z. This was a major step but a proof that the finite number was 0 in all cases did not seem likely to follow by extending Falting's arguments. However, Faltings was the natural person that Wiles turned to when he wanted an opinion on the correctness of his repair of his proof of Fermat's Last Theorem in 1994.
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G Faltings won a Fields Medal in 1986. You can see a history of the Fields Medal and a list of winners.