James Burton Ax (1937–2006) was a mathematician who proved several results in algebra and number theory by using model theory. He shared the seventh Frank Nelson Cole Prize in Number Theory with Simon B. Kochen, which was awarded for a series of three joint papers on Diophantine problems.
James Ax earned his Ph.D. from the University of California, Berkeley in 1961 under the direction of Gerhard Hochschild, with a dissertation on The Intersection of Norm Groups. After one year at Stanford University, he joined the mathematics faculty at Cornell University. In 1969, he moved to the mathematics department at Stony Brook University and remained on the faculty until 1977. In the 1980s, he and James Simons founded a quantitative finance firm, Axcom Trading Advisors, which was later acquired by Renaissance Technologies and renamed the Medallion Fund. The latter fund was named after the Cole Prize won by James Ax and the Veblen Prize won by James Simons.
The Ax Library in the Department of Mathematics at the University of California, San Diego houses his mathematical books.
- James B. Ax and Simon B. Kochen Diophantine problems over local fields. I American Journal of Mathematics 87 (1965), pp. 605-630
- James B. Ax and Simon B. Kochen Diophantine problems over local fields. II American Journal of Mathematics 87 (1965), pp. 631-648
- James B. Ax and Simon B. Kochen Diophantine problems over local fields. III Annals of Mathematics, Ser. 2 83 (1966), pp. 437-456
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