Harold Edwards (mathematician)
Encyclopedia
Harold Mortimer Edwards, Jr. (born August 6, 1936 in Champaign, Illinois
) is an American mathematician working in number theory
, algebra
, and the history and philosophy of mathematics.
He was (with Bruce Chandler) founding editor of The Mathematical Intelligencer.
He is the author of expository books on the Riemann zeta function, on Galois theory
, and on Fermat's Last Theorem
. He wrote a book on Leopold Kronecker
's work on divisor theory providing a systematic exposition of that work—a task that Kronecker never completed. He has written textbooks on linear algebra
, advanced calculus, and number theory. He also wrote a book of essays on constructive mathematics.
Edwards received his Ph.D. in 1961 from Harvard University
, under the supervision of Raoul Bott
.
He has taught at Harvard and Columbia University
; he joined the faculty at New York University
in 1966, and has been an emeritus professor since 2002.
In 1980, Edwards won the Leroy P. Steele Prize for Mathematical Exposition of the American Mathematical Society
, for his books on the Riemann zeta function and Fermat's Last Theorem. For his contribution in the field of the history of mathematics he was awarded the Albert Leon Whiteman Memorial Prize
by the AMS in 2005.
Edwards is married to Betty Rollin
, a former NBC News
correspondent, author, and breast cancer
survivor
.
course, but follows a constructivist
viewpoint in focusing on algorithm
s for solving problems rather than allowing purely existential solutions. However, unlike much other work in algorithmic number theory, there is no analysis of how efficient these algorithms are in terms of their running time
.
Essays in Constructive Mathematics (2005).:Although motivated in part by the history and philosophy of mathematics, the main goal of this book is to show that advanced mathematics such as the fundamental theorem of algebra
, the theory of binary quadratic form
s, and the Riemann–Roch theorem
can be handled in a constructivist framework.
Linear Algebra, Birkhäuser, (1995).:
Divisor Theory (1990).:Algebraic divisors
were introduced by Kronecker as an alternative to the theory of ideals
. According to the citation for Edwards' Whiteman Prize, this book completes the work of Kronecker by providing "the sort of systematic and coherent exposition of divisor theory that Kronecker himself was never able to achieve."
Galois Theory (1984).:Galois theory
is the study of the solutions of polynomial equations
using abstract symmetry group
s. This book puts the origins of the theory into their proper historical perspective, and carefully explains the mathematics in Évariste Galois
' original manuscript (reproduced in translation). Mathematician Peter M. Neumann
won the Lester R. Ford Award of the Mathematical Association of America
in 1987 for his review of this book.
Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (1977).:As the word "genetic" in the title implies, this book on Fermat's Last Theorem
is organized in terms of the origins and historical development of the subject. It was written some years prior to Wiles' proof
of the theorem, and covers research related to the theorem only up to the work of Ernst Kummer
, who used p-adic number
s and ideal theory
to prove the theorem for a large class of exponents, the regular primes.
Riemann's Zeta Function (1974).:This book concerns the Riemann zeta function and the Riemann hypothesis
on the location of the zeros of this function. It includes a translation of Riemann's original paper on these subjects, and analyzes this paper in depth; it also covers methods of computing the function such as Euler–MacLaurin summation and the Riemann–Siegel formula
. However, it omits related research on other zeta functions with analogous properties to Riemann's function, as well as more recent work on the large sieve and density estimates.
Advanced Calculus: A Differential Forms Approach (1969).:This textbook uses differential form
s as a unifying approach to multivariate calculus. Most chapters are self-contained. As an aid to learning the material, several important tools such as the implicit function theorem
are described first in the simplified setting of affine maps before being extended to differentiable maps.
Champaign, Illinois
Champaign is a city in Champaign County, Illinois, in the United States. The city is located south of Chicago, west of Indianapolis, Indiana, and 178 miles northeast of St. Louis, Missouri. Though surrounded by farm communities, Champaign is notable for sharing the campus of the University of...
) is an American mathematician working 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...
, algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, and the history and philosophy of mathematics.
He was (with Bruce Chandler) founding editor of The Mathematical Intelligencer.
He is the author of expository books on the Riemann zeta function, on Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
, and 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....
. He wrote a book on Leopold Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
's work on divisor theory providing a systematic exposition of that work—a task that Kronecker never completed. He has written textbooks on linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, advanced calculus, and number theory. He also wrote a book of essays on constructive mathematics.
Edwards received his Ph.D. in 1961 from Harvard University
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...
, under the supervision of Raoul Bott
Raoul Bott
Raoul Bott, FRS was a Hungarian mathematician known for numerous basic contributions to geometry in its broad sense...
.
He has taught at Harvard and 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...
; he joined the faculty at New York University
New York University
New York University is a private, nonsectarian research university based in New York City. NYU's main campus is situated in the Greenwich Village section of Manhattan...
in 1966, and has been an emeritus professor since 2002.
In 1980, Edwards won the Leroy P. Steele Prize for Mathematical Exposition of the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
, for his books on the Riemann zeta function and Fermat's Last Theorem. For his contribution in the field of the history of mathematics he was awarded the Albert Leon Whiteman Memorial Prize
Albert Leon Whiteman Memorial Prize
The Albert Leon Whiteman Memorial Prize is awarded by the American Mathematical Society for notable exposition and exceptional scholarship in the history of mathematics....
by the AMS in 2005.
Edwards is married to Betty Rollin
Betty Rollin
Betty Rollin , has been an NBC News correspondent and author.Rollin's reports have won both the DuPont and Emmy awards. She now contributes reports for PBS's Religion and Ethics News Weekly....
, a former NBC News
NBC News
NBC News is the news division of American television network NBC. It first started broadcasting in February 21, 1940. NBC Nightly News has aired from Studio 3B, located on floors 3 of the NBC Studios is the headquarters of the GE Building forms the centerpiece of 30th Rockefeller Center it is...
correspondent, author, and breast cancer
Breast cancer
Breast cancer is cancer originating from breast tissue, most commonly from the inner lining of milk ducts or the lobules that supply the ducts with milk. Cancers originating from ducts are known as ductal carcinomas; those originating from lobules are known as lobular carcinomas...
survivor
Cancer survivor
A cancer survivor is an individual with cancer of any type, current or past, who is still living. About 11 million Americans alive today—one in 30 people–are either currently undergoing treatment for cancer or have done so in the past." Currently nearly 65% of adults diagnosed with cancer in the...
.
Books
Higher Arithmetic: An Algorithmic Introduction to Number Theory (2008).:An extension of Edwards' work in Essays in Constructive Mathematics, this textbook covers the material of a typical undergraduate 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...
course, but follows a constructivist
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
viewpoint in focusing on algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
s for solving problems rather than allowing purely existential solutions. However, unlike much other work in algorithmic number theory, there is no analysis of how efficient these algorithms are in terms of their running time
Running Time
Running Time may refer to:* Running Time * see Analysis of algorithms...
.
Essays in Constructive Mathematics (2005).:Although motivated in part by the history and philosophy of mathematics, the main goal of this book is to show that advanced mathematics such as the fundamental theorem of algebra
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
, the theory of binary quadratic form
Binary quadratic form
In mathematics, a binary quadratic form is a quadratic form in two variables. More concretely, it is a homogeneous polynomial of degree 2 in two variableswhere a, b, c are the coefficients...
s, and the Riemann–Roch theorem
Riemann–Roch theorem
The Riemann–Roch theorem is an important tool in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles...
can be handled in a constructivist framework.
Linear Algebra, Birkhäuser, (1995).:
Divisor Theory (1990).:Algebraic divisors
Divisor (algebraic geometry)
In algebraic geometry, divisors are a generalization of codimension one subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors...
were introduced by Kronecker as an alternative to the theory of ideals
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
. According to the citation for Edwards' Whiteman Prize, this book completes the work of Kronecker by providing "the sort of systematic and coherent exposition of divisor theory that Kronecker himself was never able to achieve."
Galois Theory (1984).:Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
is the study of the solutions of polynomial equations
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
using abstract symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
s. This book puts the origins of the theory into their proper historical perspective, and carefully explains the mathematics in Évariste Galois
Évariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
' original manuscript (reproduced in translation). Mathematician Peter M. Neumann
Peter M. Neumann
Peter Michael Neumann OBE is a British mathematician. He is the son of the mathematicians Bernhard Neumann and Hanna Neumann and, after gaining a B.A. from The Queen's College, Oxford in 1963, obtained his D.Phil from Oxford University in 1966...
won the Lester R. Ford Award of the Mathematical Association of America
Mathematical Association of America
The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
in 1987 for his review of this book.
Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (1977).:As the word "genetic" in the title implies, this book 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....
is organized in terms of the origins and historical development of the subject. It was written some years prior to Wiles' proof
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...
of the theorem, and covers research related to the theorem only up to the work of Ernst Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
, who used p-adic number
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
s and ideal theory
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
to prove the theorem for a large class of exponents, the regular primes.
Riemann's Zeta Function (1974).:This book concerns the Riemann zeta function and 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...
on the location of the zeros of this function. It includes a translation of Riemann's original paper on these subjects, and analyzes this paper in depth; it also covers methods of computing the function such as Euler–MacLaurin summation and the Riemann–Siegel formula
Riemann–Siegel formula
In mathematics, the Riemann–Siegel formula is an asymptotic formula for the error of the approximate functional equation of the Riemann zeta function, an approximation of the zeta function by a sum of two finite Dirichlet series. It was found by in unpublished manuscripts of Bernhard Riemann...
. However, it omits related research on other zeta functions with analogous properties to Riemann's function, as well as more recent work on the large sieve and density estimates.
Advanced Calculus: A Differential Forms Approach (1969).:This textbook uses differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s as a unifying approach to multivariate calculus. Most chapters are self-contained. As an aid to learning the material, several important tools such as the implicit function theorem
Implicit function theorem
In multivariable calculus, the implicit function theorem is a tool which allows relations to be converted to functions. It does this by representing the relation as the graph of a function. There may not be a single function whose graph is the entire relation, but there may be such a function on...
are described first in the simplified setting of affine maps before being extended to differentiable maps.