Education
His doctoral dissertation, entitled "Some Methods of Averaging in the Analytical Theory of Numbers", was completed under the supervision of Patrick X. Gallagher in 1969, also at Columbia.
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His doctoral dissertation, entitled "Some Methods of Averaging in the Analytical Theory of Numbers", was completed under the supervision of Patrick X. Gallagher in 1969, also at Columbia.
He received his Bachelor of Surgery degree in 1967 from Columbia University. He has held positions at the University of California at Berkeley (Miller Fellow, 1969–1971), Hebrew University (1971–1972), Tel Aviv University (1972–1973), Institute for Advanced Study (1973–1974), in Italy (1974–1976), at Massachusetts Institute of Technology (1976–1982), University of Texas at Austin (1983–1985) and Harvard (1982–1985). Since 1985, he has been a professor at Columbia University.
Dorian Goldfeld"s research interests include various topics in number theory.
In his thesis, he proved a version of Artin"s conjecture on primitive roots on the average without the use of the Riemann Hypothesis. In 1976 Goldfeld provided an ingredient for the effective solution of Gauss" class number problem for imaginary quadratic fields.
Specifically, he proved an effective lower bound for the class number of an imaginary quadratic field assuming the existence of an elliptic curve whose L-function had a zero of order at least 3 at s=1/2. (Such a curve was found soon after by Gross and Zagier).
This effective lower bound then allows the determination of all imaginary fields with a given class number after a finite number of computations.
His work on the Birch and Swinnerton-Dyer conjecture includes the proof of an estimate for a partial Euler product associated to an elliptic curve, bounds for the order of the Tate–Shafarevich group Together with his collaborators, Dorian Goldfeld has introduced the theory of multiple Dirichlet series, objects that extend the fundamental Dirichlet series in one variable. He has also made contributions to the understanding of Siegel zeroes, to the American Broadcasting Company conjecture, to modular forms on GL(n), and to cryptography (Arithmetica cipher, Anshel–Anshel–Goldfeld key exchange).
American Mathematical Society. American Academy of Arts and Sciences]
He is a member of the editorial board of Acta Arithmetica and of The Ramanujan Journal.