Harold Widom
Encyclopedia
Harold Widom is an American mathematician well known for his contributions to operator theory
and random matrices. He was appointed to the Department of Mathematics at the University of California, Santa Cruz
in 1968 and became professor emeritus in 1994.
. He studied at Stuyvesant High School
, graduating in 1949, and was a member of the school's math team along with his brother Benjamin Widom
(1944, 1948). Widom attended City College of New York
until 1951, during which he was one of the winners of the
William Lowell Putnam Mathematical Competition
(1951). At the University of Chicago
he obtained an M.S.
(1952) and Ph.D.
, the latter on a thesis Embedding of AW*-algebras advised by Irving Kaplansky
(1955).
He taught mathematics at Cornell University
(1955-68) where he started his work on Toeplitz
and Wiener-Hopf
operators, partly inspired by Mark Kac
.
Widom was appointed in the Department of Mathematics at the University of California, Santa Cruz
and became professor emeritus in 1994. His research areas were in integral equations and operator theory
, in particular the determination of the spectra of a semi-infinite Toeplitz matrix
and Wiener-Hopf
operators, and the asymptotic behavior of the spectra of various classes of operators. The latter was looked at from the point of view of pseudodifferential operators (which generalize both integral and partial differential operators) on manifolds.
More recently, his mathematical contributions with his long term collaborator Craig Tracy
have been recognized through the award of several prizes for their joint work on Tracy–Widom distribution
functions for random matrices. They used integral operators to obtain explicit representations, in terms of Painlevé transcendents, of the limiting distributions of the largest and smallest eigenvalues in many models of random matrices (see Fredholm determinant
s). These same distributions have since been shown to arise in numerous other physical models, in random growth models, and in asymptotic combinatorics.
He has been the author of two books and more than 120 journal articles, and is an associate editor of Asymptotic Analysis, Journal of Integral Equations and Applications and
Mathematical Physics, Analysis, and Geometry. He is an honorary editor of Integral Equations and Operator Theory
.
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
and random matrices. He was appointed to the Department of Mathematics at the University of California, Santa Cruz
University of California, Santa Cruz
The University of California, Santa Cruz, also known as UC Santa Cruz or UCSC, is a public, collegiate university; one of ten campuses in the University of California...
in 1968 and became professor emeritus in 1994.
Education and research
Widom was born in New York CityNew York City
New York is the most populous city in the United States and the center of the New York Metropolitan Area, one of the most populous metropolitan areas in the world. New York exerts a significant impact upon global commerce, finance, media, art, fashion, research, technology, education, and...
. He studied at Stuyvesant High School
Stuyvesant High School
Stuyvesant High School , commonly referred to as Stuy , is a New York City public high school that specializes in mathematics and science. The school opened in 1904 on Manhattan's East Side and moved to a new building in Battery Park City in 1992. Stuyvesant is noted for its strong academic...
, graduating in 1949, and was a member of the school's math team along with his brother Benjamin Widom
Benjamin Widom
Benjamin Widom is the Goldwin Smith Professor of Chemistry at Cornell University. His research interests include physical chemistry and statistical mechanics...
(1944, 1948). Widom attended City College of New York
City College of New York
The City College of the City University of New York is a senior college of the City University of New York , in New York City. It is also the oldest of the City University's twenty-three institutions of higher learning...
until 1951, during which he was one of the winners of the
William Lowell Putnam Mathematical Competition
William Lowell Putnam Mathematical Competition
The William Lowell Putnam Mathematical Competition, often abbreviated to the Putnam Competition, is an annual mathematics competition for undergraduate college students of the United States and Canada, awarding scholarships and cash prizes ranging from $250 to $2,500 for the top students and $5,000...
(1951). At the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
he obtained an M.S.
Master of Science
A Master of Science is a postgraduate academic master's degree awarded by universities in many countries. The degree is typically studied for in the sciences including the social sciences.-Brazil, Argentina and Uruguay:...
(1952) and Ph.D.
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
, the latter on a thesis Embedding of AW*-algebras advised by Irving Kaplansky
Irving Kaplansky
Irving Kaplansky was a Canadian mathematician.-Biography:He was born in Toronto, Ontario, Canada, after his parents emigrated from Poland and attended the University of Toronto as an undergraduate. After receiving his Ph.D...
(1955).
He taught mathematics at Cornell University
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
(1955-68) where he started his work on Toeplitz
Toeplitz matrix
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant...
and Wiener-Hopf
Wiener-Hopf
The Wiener–Hopf method is a mathematical technique widely used in applied mathematics. It was initially developed by Norbert Wiener and Eberhard Hopf as a method to solve systems of integral equations, but has found wider use in solving two-dimensional partial differential equations with mixed...
operators, partly inspired by Mark Kac
Mark Kac
Mark Kac was a Polish mathematician. His main interest was probability theory. His question, "Can one hear the shape of a drum?" set off research into spectral theory, with the idea of understanding the extent to which the spectrum allows one to read back the geometry. Kac completed his Ph.D...
.
Widom was appointed in the Department of Mathematics at the University of California, Santa Cruz
University of California, Santa Cruz
The University of California, Santa Cruz, also known as UC Santa Cruz or UCSC, is a public, collegiate university; one of ten campuses in the University of California...
and became professor emeritus in 1994. His research areas were in integral equations and operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
, in particular the determination of the spectra of a semi-infinite Toeplitz matrix
Toeplitz matrix
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant...
and Wiener-Hopf
Wiener-Hopf
The Wiener–Hopf method is a mathematical technique widely used in applied mathematics. It was initially developed by Norbert Wiener and Eberhard Hopf as a method to solve systems of integral equations, but has found wider use in solving two-dimensional partial differential equations with mixed...
operators, and the asymptotic behavior of the spectra of various classes of operators. The latter was looked at from the point of view of pseudodifferential operators (which generalize both integral and partial differential operators) on manifolds.
More recently, his mathematical contributions with his long term collaborator Craig Tracy
Craig Tracy
Craig Arnold Tracy is an American mathematician, known for his contributions to mathematical physics and probability theory.Born in United Kingdom, he moved as infant to Missouri where he grew up and...
have been recognized through the award of several prizes for their joint work on Tracy–Widom distribution
Tracy–Widom distribution
The Tracy–Widom distribution, introduced by , is the probability distribution of the largest eigenvalue of a random hermitian matrix in the edge scaling limit. It also appears in the distribution of the length of the longest increasing subsequence of random permutations and in current fluctuations...
functions for random matrices. They used integral operators to obtain explicit representations, in terms of Painlevé transcendents, of the limiting distributions of the largest and smallest eigenvalues in many models of random matrices (see Fredholm determinant
Fredholm determinant
In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a matrix. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class operator...
s). These same distributions have since been shown to arise in numerous other physical models, in random growth models, and in asymptotic combinatorics.
He has been the author of two books and more than 120 journal articles, and is an associate editor of Asymptotic Analysis, Journal of Integral Equations and Applications and
Mathematical Physics, Analysis, and Geometry. He is an honorary editor of Integral Equations and Operator Theory
Integral Equations and Operator Theory
Integral Equations and Operator Theory is a journal dedicated to operator theory and its applications to engineering and other mathematical sciences. As some approaches to the study of integral equations constitute a subfield of operator theory, the journal also deals with the theory of integral...
.
Awards
- Norbert Wiener Prize in Applied MathematicsNorbert Wiener Prize in Applied MathematicsThe Norbert Wiener Prize in Applied Mathematics is a $5000 prize awarded every three years to for an outstanding contribution to "applied mathematics in the highest and broadest sense." It was endowed in 1967 in honor of Norbert Wiener by MIT's mathematics department and is provided jointly by the...
2006, shared with Craig TracyCraig TracyCraig Arnold Tracy is an American mathematician, known for his contributions to mathematical physics and probability theory.Born in United Kingdom, he moved as infant to Missouri where he grew up and... - American Academy of Arts and SciencesAmerican Academy of Arts and SciencesThe American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
elected, 2006 - George Polya Prize 2002, shared with Craig TracyCraig TracyCraig Arnold Tracy is an American mathematician, known for his contributions to mathematical physics and probability theory.Born in United Kingdom, he moved as infant to Missouri where he grew up and...
, for their work on random matrices - Guggenheim Fellow 1967 and 1972
- Sloan FellowshipSloan FellowshipThe Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This is distinct from the Sloan Fellows in business....
1964–65 - National Science FoundationNational Science FoundationThe National Science Foundation is a United States government agency that supports fundamental research and education in all the non-medical fields of science and engineering. Its medical counterpart is the National Institutes of Health...
Postdoctoral Fellowship, 1959–60