Gerhard Frey
Encyclopedia
Gerhard Frey is a German
mathematician
, known for his work in number theory
. His Frey curve, a construction of an elliptic curve
from a purported solution to the Fermat equation
, was central to Wiles' proof of Fermat's Last Theorem
.
He studied mathematics
and physics
at the University of Tübingen
, graduating in 1967. He continued his postgraduate studies in Heidelberg
where he received the Ph.D.
degree in 1970 and his "Habilitation
" in 1973. He was assistant professor at the University of Heidelberg from 1969-1973, professor at the University of Erlangen
(1973-1975) and at the University of Saarbrücken
(1975-1990) and until 2009 held a chair for number theory
at the Institute for Experimental Mathematics
at the University of Duisburg-Essen
, campus Essen.
His research areas are number theory and arithmetical geometry as well as applications to coding theory and cryptography
. He was a visiting scientist at several universities and research institutions, including The Ohio State University
, Harvard University, University of California at Berkeley
, the Mathematical Sciences Research Institute (MSRI)
, the Institute for Advanced Studies at the Hebrew University of Jerusalem
, and IMPA in Rio de Janeiro
.
In 1985 Frey pointed out a connection between Fermat's last theorem
and the Taniyama conjecture, and this connection was made precise shortly thereafter by Kenneth Ribet, who proved that the Taniyama conjecture implies Fermat's last theorem
. This approach provided a framework for the subsequent successful attack on Fermat's last theorem
by Andrew Wiles
in 1990s.
Frey was co-editor of the Manuscripta Mathematica. He was awarded the Gauss medal of the Braunschweigische Wissenschaftliche Gesellschaft in 1996 for his work on Fermat's Last Theorem
. Since 1998 he has been a member of the Göttingen Academy of Sciences
.
In 1998, Frey proposed the idea of Weil Descent Attack for elliptic curves over finite fields with composite degree. As a result of this attack, cryptographers lost their interest in these curves.
In 2006 Frey received the Certicom ECC Visionary Award for his contributions to Elliptic Curve Cryptography.
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, known for his work 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...
. His Frey curve, a construction 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...
from a purported solution to the Fermat equation
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
, was central to Wiles' proof of Fermat's Last Theorem
Wiles' proof of Fermat's Last Theorem
Wiles's proof of Fermat's Last Theorem is a proof of the modularity theorem for semistable elliptic curves released by Andrew Wiles, which, together with Ribet's theorem, provides a proof for Fermat's Last Theorem. Wiles first announced his proof in June 1993 in a version that was soon recognized...
.
He studied mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
at the University of Tübingen
Tübingen
Tübingen is a traditional university town in central Baden-Württemberg, Germany. It is situated south of the state capital, Stuttgart, on a ridge between the Neckar and Ammer rivers.-Geography:...
, graduating in 1967. He continued his postgraduate studies in Heidelberg
Heidelberg
-Early history:Between 600,000 and 200,000 years ago, "Heidelberg Man" died at nearby Mauer. His jaw bone was discovered in 1907; with scientific dating, his remains were determined to be the earliest evidence of human life in Europe. In the 5th century BC, a Celtic fortress of refuge and place of...
where he received the Ph.D.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
degree in 1970 and his "Habilitation
Habilitation
Habilitation is the highest academic qualification a scholar can achieve by his or her own pursuit in several European and Asian countries. Earned after obtaining a research doctorate, such as a PhD, habilitation requires the candidate to write a professorial thesis based on independent...
" in 1973. He was assistant professor at the University of Heidelberg from 1969-1973, professor at the University of Erlangen
Erlangen
Erlangen is a Middle Franconian city in Bavaria, Germany. It is located at the confluence of the river Regnitz and its large tributary, the Untere Schwabach.Erlangen has more than 100,000 inhabitants....
(1973-1975) and at the University of Saarbrücken
Saarbrücken
Saarbrücken is the capital of the state of Saarland in Germany. The city is situated at the heart of a metropolitan area that borders on the west on Dillingen and to the north-east on Neunkirchen, where most of the people of the Saarland live....
(1975-1990) and until 2009 held a chair for 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...
at the Institute for Experimental Mathematics
Institute for Experimental Mathematics
The Institute for Experimental Mathematics was founded, with the support of the VolkswagenFoundation, as a central scientific facility of the former University of Essen, now University of Duisburg-Essen in 1989. With the addition of the Alfried Krupp von Bohlen und Halbach Foundation Chair on 1...
at the University of Duisburg-Essen
University of Duisburg-Essen
The University Duisburg-Essen is a public university in Duisburg and Essen, North Rhine-Westphalia, Germany and a member of the new founded University Alliance Metropolis Ruhr....
, campus Essen.
His research areas are number theory and arithmetical geometry as well as applications to coding theory and cryptography
Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...
. He was a visiting scientist at several universities and research institutions, including The Ohio State University
Ohio State University
The Ohio State University, commonly referred to as Ohio State, is a public research university located in Columbus, Ohio. It was originally founded in 1870 as a land-grant university and is currently the third largest university campus in the United States...
, Harvard University, University of California at Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
, the Mathematical Sciences Research Institute (MSRI)
Mathematical Sciences Research Institute
The Mathematical Sciences Research Institute , founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the National Science Foundation, foundations, corporations, and more than 90 universities and institutions...
, the Institute for Advanced Studies at the Hebrew University of Jerusalem
Hebrew University of Jerusalem
The Hebrew University of Jerusalem ; ; abbreviated HUJI) is Israel's second-oldest university, after the Technion – Israel Institute of Technology. The Hebrew University has three campuses in Jerusalem and one in Rehovot. The world's largest Jewish studies library is located on its Edmond J...
, and IMPA in Rio de Janeiro
Rio de Janeiro
Rio de Janeiro , commonly referred to simply as Rio, is the capital city of the State of Rio de Janeiro, the second largest city of Brazil, and the third largest metropolitan area and agglomeration in South America, boasting approximately 6.3 million people within the city proper, making it the 6th...
.
In 1985 Frey pointed out a connection between Fermat's last theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
and the Taniyama conjecture, and this connection was made precise shortly thereafter by Kenneth Ribet, who proved that the Taniyama conjecture implies Fermat's last theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
. This approach provided a framework for the subsequent successful attack on Fermat's last theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
by Andrew Wiles
Andrew Wiles
Sir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at Oxford University, specializing in number theory...
in 1990s.
Frey was co-editor of the Manuscripta Mathematica. He was awarded the Gauss medal of the Braunschweigische Wissenschaftliche Gesellschaft in 1996 for his work on Fermat's Last Theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
. Since 1998 he has been a member of the Göttingen Academy of Sciences
Göttingen Academy of Sciences
The Göttingen Academy of Sciences is the second oldest of the seven academies of sciences in Germany. It has the task of promoting research under its own auspices and in collaboration with academics in and outside Germany...
.
In 1998, Frey proposed the idea of Weil Descent Attack for elliptic curves over finite fields with composite degree. As a result of this attack, cryptographers lost their interest in these curves.
In 2006 Frey received the Certicom ECC Visionary Award for his contributions to Elliptic Curve Cryptography.
See also
- Fermat's last theoremFermat's Last TheoremIn number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
- Elliptic curveElliptic curveIn 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...
s - CryptographyCryptographyCryptography is the practice and study of techniques for secure communication in the presence of third parties...
- Elliptic curve cryptographyElliptic curve cryptographyElliptic curve cryptography is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S...