Dorian Morris Goldfeld (born January 21, 1947) is an American mathematician working in
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
and
automorphic forms at
Columbia University.
Professional career
Goldfeld 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
Patrick Ximenes Gallagher (January 2, 1935 – March 30, 2019) was an American mathematician who pioneered large sieve theory and invented the larger sieve.
Biography Early life
Patrick Ximenes Gallagher was born on January 2, 1935, in Elizabet ...
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
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...
(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
''The Ramanujan Journal'' is a peer-reviewed scientific journal covering all areas of mathematics, especially those influenced by the Indian mathematician Srinivasa Ramanujan. The journal was established in 1997 and is published by Springer Science ...
''. On January 1, 2018 he became the Editor-in-Chief of the
Journal of Number Theory
The ''Journal of Number Theory'' (''JNT'') is a bimonthly peer-reviewed scientific journal covering all aspects of number theory. The journal was established in 1969 by R.P. Bambah, P. Roquette, A. Ross, A. Woods, and H. Zassenhaus (Ohio State Un ...
.
He is a co-founder and board member o
Veridify Security formerly SecureRF, a corporation that has developed the world's first linear-based security solutions.
Goldfeld advised several doctoral students including
M. Ram Murty. In 1986, he brought
Shou-Wu Zhang
Shou-Wu Zhang (; born October 9, 1962) is a Chinese-American mathematician known for his work in number theory and arithmetic geometry. He is currently a Professor of Mathematics at Princeton University.
Biography Early life
Shou-Wu Zhang was b ...
to the United States to study at Columbia.
Research interests
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 mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
.
In 1976, Goldfeld provided an ingredient for the effective solution of
Gauss'
class number problem for imaginary quadratic fields
In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each ''n'' ≥ 1 a complete list of imaginary quadratic fields \mathbb(\sqrt) (for negative integers ''d'') having ...
. 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
. (Such a curve was found soon after by
Gross and
Zagier
Don Bernard Zagier (born 29 June 1951) is an American-German mathematician whose main area of work is number theory. He is currently one of the directors of the Max-Planck-Institut für Mathematik, Max Planck Institute for Mathematics in Bonn, Ger ...
). 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 In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group that become trivial in all of the completions of ...
.
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
ABC conjecture, to
modular forms on
, and to cryptography (Arithmetica cipher,
Anshel–Anshel–Goldfeld key exchange Anshel–Anshel–Goldfeld protocol, also known as a commutator key exchange, is a key-exchange protocol using nonabelian groups. It was invented by Drs. Michael Anshel, Iris Anshel, and Dorian Goldfeld. Unlike other group-based protocols, it does n ...
).
Together with his wife, Dr.
Iris Anshel
Iris most often refers to:
*Iris (anatomy), part of the eye
* Iris (mythology), a Greek goddess
* ''Iris'' (plant), a genus of flowering plants
*Iris (color), an ambiguous color term
Iris or IRIS may also refer to:
Arts and media
Fictional ent ...
, and father-in-law, Dr.
Michael Anshel, both mathematicians, Dorian Goldfeld founded the field of
braid group cryptography.
Awards and honors
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
In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each ''n'' ≥ 1 a complete list of imaginary quadratic fields \mathbb(\sqrt) (for negative integers ''d'') having ...
. He has also held the
Sloan Fellowship (1977–1979) and in 1985 he received the Vaughan prize. In 1986 he was an
invited speaker at the International Congress of Mathematicians
This is a list of International Congresses of Mathematicians Plenary and Invited Speakers. Being invited to talk at an International Congress of Mathematicians has been called "the equivalent, in this community, of an induction to a hall of fame." ...
in Berkeley. 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.
List of Fellows of the American Mathematical Society
retrieved 2013-01-19.
Selected works
*
*
*
*
*
References
External links
*
Dorian Goldfeld's Home Page at Columbia University
{{DEFAULTSORT:Goldfeld, Dorian
20th-century American mathematicians
21st-century American mathematicians
Fellows of the American Academy of Arts and Sciences
Fellows of the American Mathematical Society
Number theorists
Columbia School of Engineering and Applied Science alumni
Columbia University faculty
University of California, Berkeley faculty
Academic staff of the Hebrew University of Jerusalem
Academic staff of Tel Aviv University
Institute for Advanced Study visiting scholars
University of Texas at Austin faculty
Harvard University faculty
1947 births
Living people
People from Marburg