Dorian M. Goldfeld

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Dorian M. Goldfeld
Dorian Goldfeld.jpg
Dorian Goldfeld at The Analytic Theory of Automorphic Forms workshop, Oberwolfach, Germany (2011)
Born (1947-01-21) January 21, 1947 (age 67)
Marburg, Germany
Residence Tenafly, New Jersey
Nationality American
Fields Mathematics
Institutions Columbia University
Massachusetts Institute of Technology
Alma mater Columbia University
Doctoral advisor Patrick X. Gallagher
Doctoral students Jeffrey Hoffstein
M. Ram Murty
Eric Stade
Known for Number Theory, Cryptography
Notable awards Frank Nelson Cole Prize in Number Theory (1987)
Sloan Fellowship (1977–1979)
Vaughan prize (1985)
Fellow of the American Academy of Arts and Sciences (April 2009)
Spouse Iris Lee Anshel
Children Dahlia (daughter)
Ada (daughter)

Dorian Morris Goldfeld (born January 21, 1947) is an American mathematician.

Professional career[edit]

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 X. 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.

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.[1]

Research interests[edit]

Dorian Goldfeld's research interests include various topics in number theory. In his thesis,[2] 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.[3] 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,[4] bounds for the order of the Tate–Shafarevich group[5]

Together with his collaborators, Dorian Goldfeld has introduced the theory of multiple Dirichlet series, objects that extend the fundamental Dirichlet series in one variable.[6]

He has also made contributions to the understanding of Siegel zeroes,[7] to the ABC conjecture,[8] to modular forms on GL(n),[9] and to cryptography (Arithmetica cipher, Anshel–Anshel–Goldfeld key exchange).[10]

Together with his wife, Dr. Iris Anshel,[11] and father-in-law, Dr. Michael Anshel,[12] both mathematicians, Dorian Goldfeld founded the field of Braid Group cryptography.[13][14]

Awards and honors[edit]

In 1987 he received the Frank Nelson Cole Prize in Number Theory, one of the prizes in Number Theory, for his solution of Gauss' class number problem for imaginary quadratic fields. He has also held the Sloan Fellowship (1977–1979) and in 1985 he received the Vaughan prize. In April 2009 he was elected a Fellow of the American Academy of Arts and Sciences. In 2012 he became a fellow of the American Mathematical Society.[15]

Selected works[edit]

  • 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. 

References[edit]

  1. ^ SecureRF Corporation, co-founded by Dorian Goldfeld
  2. ^ Goldfeld, Dorian, Artin's conjecture on the average, Mathematika, 15 1968
  3. ^ 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
  4. ^ Goldfeld, Dorian, Sur les produits partiels eulériens attachés aux courbes elliptiques, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 14,
  5. ^ Goldfeld, Dorian; Szpiro, Lucien Bounds for the order of the Tate–Shafarevich group, Compositio Mathematica 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.

  6. ^ 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 Mathematica 139 (2003), no. 3
  7. ^ 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
  8. ^ Goldfeld, Dorian Modular elliptic curves and Diophantine problems. Number theory (Banff, AB, 1988), 157–175, de Gruyter, Berlin, 1990.
  9. ^ 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
  10. ^ 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
  11. ^ Dr. Iris Anshel page at SecureRF Corporation
  12. ^ Dr. Michael Anshel page at City College of New York
  13. ^ Anshel, Iris; Anshel, Michael; Goldfeld, Dorian An algebraic method for public-key cryptography. Math. Res. Lett. 6 (1999), no. 3-4, Anshel, Michael
  14. ^ Braid Group Cryptography page
  15. ^ List of Fellows of the American Mathematical Society, retrieved 2013-01-19.

External links[edit]