Dorian Goldfeld

Dorian Goldfeld

Dorian Morris Goldfeld (born January 21, 1947 in Marburg ) is an American mathematician who deals with number theory.

Career

Goldfeld graduated from Columbia University in New York in 1967 and received his doctorate there in 1969 with Patrick X. Gallagher with Some methods of averaging in analytical number theory , in which he wrote an averaged version of the Artin conjecture (on the distribution of prime numbers p , for which a given number a is a primitive root mod p) proved (Mathematika Vol. 15, 1968). From 1969 to 1971 he was a Miller Fellow in Berkeley , 1971/2 at the Hebrew University in Jerusalem and then 1971/2 at Tel Aviv University , 1973/4 at the Institute for Advanced Study inPrinceton , 1974–76 in Italy, 1976–1982 at MIT (1973–1974), 1983–1985 at the University of Texas at Austin and 1983–1985 at Harvard . Since 1985 he has been a professor at Columbia University.

In 1986 he was invited speaker at the International Congress of Mathematicians in Berkeley (Kloosterman Zetafunctions for GL (n, Z)). He is a fellow of the American Mathematical Society .

Field of activity

Goldfeld is best known for his work on the class number problem of imaginary quadratic number fields . Gauss had listed 9 class bodies with class number 1, and it was proven by Harold Stark in the 1960s that this list was complete (an older proof by Kurt Heegner had also been rehabilitated shortly afterwards and an alternative proof was given by Alan Baker ) . Goldfeld showed in a paper from 1976 (Ann. Sc. Norm. Sup. Pisa Vol. 3, No. 4) a possibility to tackle the general case of any number of classes by connecting to properties of L-functions created elliptical curves. Benedict Gross and Don Zagier completed this method of effectively determining the class field with a given class number in the 1980s.

He continued to work z. B. via the abc conjecture (a generalization of the Fermat conjecture), size estimates of the Tate-Shafarevich groups (important in connection with the detailed version of the conjecture by Birch and Swinnerton-Dyer ), modular forms for the general linear group GL (n) (in connection with the Langlands program ) and examined multiple Dirichlet series.

Goldfeld also dealt with cryptography . Together with Iris and Michael Anshel, he founded encryption methods based on the braid group . He is also a co-founder of the company SecureRF, which offers fast authentication procedures for RFID .

Goldfeld received the Cole Prize in number theory in 1987 , was a Sloan Research Fellow from 1977 to 1979, and received the Vaughn Prize in 1985.

He has been a member of the American Academy of Arts and Sciences since 2009 .

His students include Jeffrey Hoffstein , M. Ram Murty , Ilan Vardi .

literature

• Goldfeld, Iris Anshel: Calculus: a Computer Algebra Approach , Boston International Press, 1995, ISBN 1-57146-038-1 .
• Gerritzen, Goldfeld, Kreuzer, Rosenberger, Shpilrain (Eds.): Algebraic Methods in Cryptography , 2006, ISBN 0-8218-4037-1 .
• Goldfeld: Automorphic Forms and L-Functions for the Group GL (n, R) , Cambridge University Press, 2006, ISBN 0-521-83771-5 .
• Goldfeld Automorphic Representations and L-Functions for the general linear group , 2 volumes, Cambridge University Press 2011

Web links

• Homepage with an introduction to the class number problem
• THE GAUSS CLASS NUMBER PROBLEM FOR IMAGINARY QUADRATIC FIELDS (PDF; 145 kB)
• Links to Braid Group Cryptography ( Memento from March 22, 2008 in the Internet Archive )
• Goldfeld: The class number of quadratic fields and the conjecture of Birch and Swinnerton-Dyer , Ann. Sc. Standard. Pisa 1976
• Goldfeld, Szpiro: Bounds for the order of the Tate-Shafarevich group , Compositio Math. 1995