Harold Mortimer Edwards, Jr. (August 6, 1936 – November 10, 2020) was an American mathematician working in
number theory,
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, and the history and philosophy of mathematics.
He was one of the co-founding editors, with Bruce Chandler, of ''
The Mathematical Intelligencer
''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released quar ...
''.
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
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
providing a systematic exposition of that work—a task that Kronecker never completed. He wrote textbooks on
linear algebra,
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, and number theory. He also wrote a book of essays on
constructive mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...
.
Edwards graduated from the
University of Wisconsin–Madison
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, the ...
in 1956, received a
Master of Arts from
Columbia University in 1957, and a Ph.D from
Harvard University in 1961, under the supervision of
Raoul Bott.
He taught at Harvard and
Columbia University; he joined the faculty at
New York University in 1966, and was an
emeritus professor starting in 2002.
In 1980, Edwards won the
Leroy P. Steele Prize for Mathematical Exposition of the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, 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.
[.] In 2012 he became a fellow of the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
.
Edwards was married to
Betty Rollin
Betty Rollin (born January 3, 1936, in New York City) 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.
Rollin is a gr ...
, a former
NBC News correspondent, author, and
breast cancer
Breast cancer is cancer that develops from breast tissue. Signs of breast cancer may include a lump in the breast, a change in breast shape, dimpling of the skin, milk rejection, fluid coming from the nipple, a newly inverted nipple, or a r ...
survivor. Edwards died on November 10, 2020 of colon cancer.
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 theory course,
[Review by ]Samuel S. Wagstaff, Jr.
Samuel Standfield Wagstaff Jr. (born 21 February 1945) is an American mathematician and computer scientist, whose research interests are in the areas of cryptography, parallel computation, and analysis of algorithms, especially number theoretic ...
(2009), '' Mathematical Reviews'', . but follows a
constructivist viewpoint in focusing on
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s for solving problems rather than allowing purely existential solutions.
The constructions are intended to be simple and straightforward, rather than efficient, so, unlike works on
algorithmic number theory, there is no analysis of how efficient they are in terms of their
running time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...
.
[Review](_blank)
by Luiz Henrique de Figueiredo, Mathematical Association of America
The Mathematical Association of America (MAA) 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 ...
, April 26, 2008.
* ''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
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables
: q(x,y)=ax^2+bxy+cy^2, \,
where ''a'', ''b'', ''c'' are the coefficients. When the coefficients can be arbitrary complex numbers, most results a ...
s, and the
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It re ...
can be handled in a constructivist framework. The second edition (2022) adds a new set of essays that reflect and expand upon the first. This was Edwards' final book, finished shortly before his death.
* ''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
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambie ...
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
The Mathematical Association of America (MAA) 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 ...
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
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) 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 h ...
, who used
p-adic numbers 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 forms 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 map
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
s before being extended to
differentiable maps.
[Review by R. S. Booth (1982), '' Mathematical Reviews'', .]
See also
*
Edwards curve
In mathematics, the Edwards curves are a family of elliptic curves studied by Harold Edwards in 2007. The concept of elliptic curves over finite fields is widely used in elliptic curve cryptography. Applications of Edwards curves to cryptograp ...
and
Twisted Edwards curve
In algebraic geometry, the twisted Edwards curves are plane models of elliptic curves, a generalisation of Edwards curves introduced by Bernstein, Birkner, Joye, Lange and Peters in 2008. The curve set is named after mathematician Harold M. Ed ...
References
External links
Web page at New York University
{{DEFAULTSORT:Edwards, Harold M.
1936 births
2020 deaths
20th-century American mathematicians
21st-century American mathematicians
Number theorists
Harvard University alumni
Columbia University faculty
Harvard University faculty
New York University faculty
American historians of mathematics
Fellows of the American Mathematical Society
People from Champaign, Illinois
Mathematicians from Illinois