Dorian M. Goldfeld
Encyclopedia
Dorian Morris Goldfeld (born January 21, 1947) 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 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
.
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 Vaughan prize. In April 2009 he was elected a Fellow of the American Academy of Arts and Sciences
.
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.
. 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 the key 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, important objects that extend the fundamental Dirichlet series in one variable.
He has also made important contributions to the understanding of Siegel zero
es, to the ABC conjecture
, to modular forms on GL(n), to Cryptography (Arithmetica cipher, Anshel–Anshel–Goldfeld key exchange, etc.) etc.
Together with Iris Anshel and Michael Anshel, Dorian Goldfeld founded the field of Braid Group
Cryptography.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician.
He received his B.S. degree in 1967 from Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
. 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
Tel Aviv University
Tel Aviv University is a public university located in Ramat Aviv, Tel Aviv, Israel. With nearly 30,000 students, TAU is Israel's largest university.-History:...
(1972–1973), Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
(1973–1974), in Italy (1974–1976), at MIT (1976–1982), University of Texas at Austin
University of Texas at Austin
The University of Texas at Austin is a state research university located in Austin, Texas, USA, and is the flagship institution of the The University of Texas System. Founded in 1883, its campus is located approximately from the Texas State Capitol in Austin...
(1983–1985) and Harvard
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
(1982–1985). Since 1985, he has been a professor at Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
.
In 1987 he received the Frank Nelson Cole Prize in Number Theory
Cole Prize
The Frank Nelson Cole Prize, or Cole Prize for short, is one of two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number theory. The prize is named after Frank Nelson Cole, who...
, one of the most prestigious prizes in Number Theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, for his solution of Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
' class number problem for imaginary quadratic fields. He has also held the prestigious Sloan Fellowship
Sloan Fellowship
The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This is distinct from the Sloan Fellows in business....
(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
American Academy of Arts and Sciences
The American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
.
He is a member of the editorial board of Acta Arithmetica
Acta Arithmetica
Acta Arithmetica is a scientific journal of mathematics publishing papers on number theory. It was established in 1935 by Salomon Lubelski and Arnold Walfisz. The journal is published by the Institute of Mathematics of the Polish Academy of Sciences....
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.
Research interests
Dorian Goldfeld's research interests include various topics in number theoryNumber theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. In his thesis, he proved a version of Artin's conjecture on primitive roots
Artin's conjecture on primitive roots
In number theory, Artin's conjecture on primitive roots states that a given integer a which is not a perfect square and not −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes...
on the average without the use of the Riemann Hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
.
In 1976 Goldfeld provided the key ingredient for the effective solution of Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
' class number problem for imaginary quadratic fields
Class number problem for imaginary quadratic fields
In mathematics, the Gauss class number problem , as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields with class number n. It is named after the great mathematician Carl Friedrich Gauss. It can also be stated in terms of discriminants...
. Specifically, he proved an effective lower bound for the class number of an imaginary quadratic field assuming the existence of an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
whose L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
had a zero of order at least 3 at s=1/2. (Such a curve was found soon after by Gross
Benedict Gross
Benedict Hyman Gross is an American mathematician, the George Vasmer Leverett Professor of Mathematics at Harvard University and former Dean of Harvard College....
and Zagier). This effective lower bound then allows the determination of all imaginary fields with a given class number
Ideal class group
In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
after a finite number of computations.
His work on the Birch and Swinnerton-Dyer conjecture
Birch and Swinnerton-Dyer conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. Its status as one of the most challenging mathematical questions has become widely recognized; the conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay...
includes the proof of an estimate for a partial Euler product
Euler product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.-Definition:...
associated to an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
, bounds for the
order of the Tate–Shafarevich group
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.
He has also made important contributions to the understanding of Siegel zero
Siegel zero
In mathematics, more specifically in the field of analytic number theory, a Siegel zero, named after Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeroes of Dirichlet L-function....
es, to the ABC conjecture
Abc conjecture
The abc conjecture is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser in 1985. The conjecture is stated in terms of three positive integers, a, b and c , which have no common factor and satisfy a + b = c...
, to modular forms on GL(n), to Cryptography (Arithmetica cipher, Anshel–Anshel–Goldfeld key exchange, etc.) etc.
Together with Iris Anshel and Michael Anshel, Dorian Goldfeld founded the field of Braid Group
Braid group
In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...
Cryptography.