Jeffrey Ezra Hoffstein | |
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| Born | November 28, 1953 New York City |
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Jeffrey Ezra Hoffstein (born September 28, 1953 in New York City) [1] is an American mathematician, specializing in number theory, automorphic forms, and cryptography. [2]
Hoffstein graduated with a bachelor's degree in 1974 from Cornell University. He received his Ph.D. in 1978 from Massachusetts Institute of Technology with thesis Class numbers of totally complex quadratic extensions of totally real fields under the supervision of Harold Stark. [3] Hoffstein was a postdoc at the Institute for Advanced Study and then at the University of Cambridge. From 1980 to 1982 he was an assistant professor at Brown University. From 1982 he was an assistant professor and then an associate professor at the University of Rochester. Since 1989 he is a full professor at Brown University and he was from 2009 to 2013 the chair of the mathematics department there. [4]
His research uses analytic and algebraic methods to investigate L-series of automorphic forms over GL(n) and number fields. With co-workers he has developed new techniques for Dirichlet series in several complex variables. He was several times a visiting scholar at the Institute for Advanced Study (1978/79, 1985, 1986/87). [5] At MSRI, in the academic year 1994/95 he initiated seminars on automorphic functions. In 1984 he was a Fulbright Fellow. He was a visiting professor at the University of Texas at Austin in the spring of 1984 and at the University of Göttingen in the fall of 1986. [4] He became a Fellow of the American Mathematical Society in the class of 2019.
In 1996, Hoffstein, along with Jill Pipher, Joseph Silverman, and Daniel Liemann (Hoffstein's former doctoral student), founded NTRU Cryptosystems, Inc. to market their cryptographic algorithms, NTRUEncrypt and NTRUSign. NTRU Cryptosystems was acquired by Security Innovation in 2009. [6]
In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands, it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as "a kind of grand unified theory of mathematics."
Gorō Shimura was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture which ultimately led to the proof of Fermat's Last Theorem.
NTRU is an open source public-key cryptosystem that uses lattice-based cryptography to encrypt and decrypt data. It consists of two algorithms: NTRUEncrypt, which is used for encryption, and NTRUSign, which is used for digital signatures. Unlike other popular public-key cryptosystems, it is resistant to attacks using Shor's algorithm. NTRUEncrypt was patented, but it was placed in the public domain in 2017. NTRUSign is patented, but it can be used by software under the GPL.
In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan (1916, p.176), states that Ramanujan's tau function given by the Fourier coefficients τ(n) of the cusp form Δ(z) of weight 12
Neal I. Koblitz is a Professor of Mathematics at the University of Washington. He is also an adjunct professor with the Centre for Applied Cryptographic Research at the University of Waterloo. He is the creator of hyperelliptic curve cryptography and the independent co-creator of elliptic curve cryptography.
Joseph Hillel Silverman is a professor of mathematics at Brown University working in arithmetic geometry, arithmetic dynamics, and cryptography.
Dorian Morris Goldfeld is an American mathematician working in analytic number theory and automorphic forms at Columbia University.
In mathematics, the local Langlands conjectures, introduced by Robert Langlands, are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group G over a local field F, and representations of the Langlands group of F into the L-group of G. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups.
Lattice-based cryptography is the generic term for constructions of cryptographic primitives that involve lattices, either in the construction itself or in the security proof. Lattice-based constructions are currently important candidates for post-quantum cryptography. Unlike more widely used and known public-key schemes such as the RSA, Diffie-Hellman or elliptic-curve cryptosystems, which could, theoretically, be easily attacked by a quantum computer, some lattice-based constructions appear to be resistant to attack by both classical and quantum computers. Furthermore, many lattice-based constructions are considered to be secure under the assumption that certain well-studied computational lattice problems cannot be solved efficiently.
In the mathematical theory of automorphic representations, a multiplicity-one theorem is a result about the representation theory of an adelic reductive algebraic group. The multiplicity in question is the number of times a given abstract group representation is realised in a certain space, of square-integrable functions, given in a concrete way.
In mathematics, the Rankin–Selberg method, introduced by (Rankin 1939) and Selberg (1940), also known as the theory of integral representations of L-functions, is a technique for directly constructing and analytically continuing several important examples of automorphic L-functions. Some authors reserve the term for a special type of integral representation, namely those that involve an Eisenstein series. It has been one of the most powerful techniques for studying the Langlands program.
Freydoon Shahidi is an Iranian American mathematician who is a Distinguished Professor of Mathematics at Purdue University in the U.S. He is known for a method of automorphic L-functions which is now known as the Langlands–Shahidi method.
Jill Catherine Pipher is the president of the American Mathematical Society. She began a two-year term in 2019. She is the past-president of the Association of Women in Mathematics, and she was the first director of the Institute for Computational and Experimental Research in Mathematics, an NSF-funded mathematics institute based in Providence, Rhode Island.
In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic representation π of a reductive group G over a global field and a finite-dimensional complex representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a Dirichlet character and the Mellin transform of a modular form. They were introduced by Langlands.
Kohji Matsumoto is a mathematician, Doctor of Mathematics, and professor of mathematics at Nagoya University in Nagoya, Japan. His specializations include number theory, zeta theory, and mathematical analysis. He is mostly recognized for the Matsumoto zeta function, a zeta function named after him. His academic papers have been published in several scientific journals. He co-edited Analytic Number Theory, a tome about prime numbers, divisor problems, Diophantine equations, and other topics related to analytic number theory, including Diophantine approximations, and the theory of zeta and L-functions. His other book, Algebraic And Analytic Aspects Of Zeta Functions And L-Functions, a compilation of lectures at the French-Japanese Winter School, was published in 2010.
Daniel Willis Bump is a mathematician who is a professor at Stanford University. He is a fellow of the American Mathematical Society since 2015, for "contributions to number theory, representation theory, combinatorics, and random matrix theory, as well as mathematical exposition".
Günter Harder is a German mathematician, specializing in arithmetic geometry and number theory.
David Drasin is an American mathematician, specializing in function theory.
Samuel James Patterson is a Northern Irish mathematician specializing in analytic number theory. He has been a professor at the University of Göttingen since 1981.
James Wesley Cogdell is an American mathematician.
