Dorian M. Goldfeld
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Dorian Morris Goldfeld (born 21 January 1947, Marburg, Germany) is an American mathematician.
He received his B.S. degree in 1967 from Columbia University. His doctoral dissertation entitled "Some Methods of Averaging in the Analytical Theory of Numbers" was completed under the supervision of Patrick Gallagher in 1969, also at Columbia. 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 MIT (1976-1982), University of Texas at Austin (1983-1985) and Harvard (1982-1985). Since 1985, he has been a professor at Columbia University.
In 1987 he received the Frank Nelson Cole Prize in Number Theory, one of the most prestigious prizes in Number Theory, for his solution of Gauss' class number problem for imaginary quadratic fields. He has also held the prestigious Sloan Fellowship (1977-1979) and in 1985 he received the Vaughn prize.
He is a member of the editorial board of Acta Arithmetica and of The Ramanujan Journal.
He is a co-founder and Board member of SecureRF, a corporation that has developed the world’s first linear-based security solutions.
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[edit] Research interests
Dorian Goldfeld's research interests include various topics in Number Theory. In his thesis,[1] he proved a version of Artin's conjecture on primitive roots on the average without the use of Riemann Hypothesis.
In 1976 Goldfeld provided the key ingredient for the effective solution of Gauss' class number problem for imaginary quadratic fields.[2] Specifically, he proved an effective lower bound for the class number of an imaginary quadratic fields 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,[3] bounds for the order of the Tate-Shafarevich group[4]
Together with his collaborators, Dorian Goldfeld has introduced the theory of multiple Dirichlet series, important objects that extend the fundamental Dirichlet series in one variable.[5]
He has also made important contributions to the understanding of Siegel zeroes,[6] to the ABC conjecture,[7] to modular forms on GL(n),[8] to Cryptography (Arithmetica cipher, etc.)[9] etc.
Together with Iris Anshel and Michael Anshel, Dorian Goldfeld founded the field of Braid Group Cryptography [10] [1].
[edit] References
- ^ Goldfeld, Dorian, Artin's conjecture on the average, Mathematika, 15 1968
- ^ Goldfeld, Dorian, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 4
- ^ Goldfeld, Dorian, Sur les produits partiels eulériens attachés aux courbes elliptiques, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 14,
- ^ Goldfeld, Dorian; Szpiro, Lucien Bounds for the order of the Tate-Shafarevich group, Compositio Math. 97 (1995), no. 1-2, Goldfeld, Dorian; Lieman, Daniel Effective bounds on the size of the Tate-Shafarevich group. Math. Res. Lett. 3 (1996), no. 3; Goldfeld, Dorian, Special values of derivatives of L-functions. Number theory (Halifax, NS, 1994), 159--173, CMS Conf. Proc., 15, Amer. Math. Soc., Providence, RI, 1995.
- ^ Goldfeld, Dorian; Hoffstein, Jeffrey Eisenstein series of 1/2-integral weight and the mean value of real Dirichlet L-series. Invent. Math. 80 (1985), no. 2; Diaconu, Adrian; Goldfeld, Dorian; Hoffstein, Jeffrey Multiple Dirichlet series and moments of zeta and L-functions. Compositio Math. 139 (2003), no. 3
- ^ Goldfeld, Dorian, A simple proof of Siegel's theorem Proc. Nat. Acad. Sci. U.S.A. 71 (1974); Goldfeld, D. M.; Schinzel, A. On Siegel's zero. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4
- ^ Goldfeld, Dorian Modular elliptic curves and Diophantine problems. Number theory (Banff, AB, 1988), 157--175, de Gruyter, Berlin, 1990.
- ^ Bump, Daniel; Friedberg, Solomon; Goldfeld, Dorian Poincaré series and Kloosterman sums. The Selberg trace formula and related topics (Brunswick, Maine, 1984), 39--49, Contemp. Math., 53, Amer. Math. Soc., Providence, RI, 1986
- ^ Anshel, Iris; Anshel, Michael; Goldfeld, Dorian An algebraic method for public-key cryptography. Math. Res. Lett. 6 (1999), no. 3-4, Anshel, Michael; Goldfeld, Dorian Zeta functions, one-way functions, and pseudorandom number generators. Duke Math. J. 88 (1997), no. 2
- ^ Anshel, Iris; Anshel, Michael; Goldfeld, Dorian An algebraic method for public-key cryptography. Math. Res. Lett. 6 (1999), no. 3-4, Anshel, Michael
[edit] Books
- Editors: Gerritzen, Goldfeld, Kreuzer, Rosenberger and Shpilrain (2006). Algebraic Methods in Cryptography. ISBN 0-8218-4037-1.
- Goldfeld, Dorian (2006). Automorphic Forms and L-Functions for the Group GL(n,R). ISBN 0-521-83771-5.
- Anshel, Iris and Goldfeld, Dorian (1995). Calculus: a Computer Algebra Approach. ISBN 1-57146-038-1.
