Melvin Hochster
Encyclopedia
Melvin Hochster is an eminent American
mathematician
, regarded as one of the leading commutative algebra
ists active today. He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the University of Michigan
.
Hochster attended Stuyvesant High School
, where he was captain of the Math Team, and received a B.A.
from Harvard University
. While at Harvard, he was a Putnam Fellow
in 1960. He earned his Ph.D.
in 1967 from Princeton University
, where he wrote a dissertation under Goro Shimura
characterizing the prime spectra
of commutative ring
s. He held positions at the University of Minnesota
and Purdue University
before joining the faculty at Michigan in 1977. Hochster shared the 1980 Cole Prize
with Michael Aschbacher
, received a Guggenheim Fellowship
in 1981, and has been a member of both the National Academy of Sciences
and the American Academy of Arts and Sciences
since 1992. In 2008, on the occasion of his 65th birthday, he was honored with a conference in Ann Arbor and with a special volume of the Michigan Mathematical Journal
.
Hochster's work is primarily in commutative algebra
, especially the study of module
s over local ring
s. He has established classic theorems concerning Cohen-Macaulay ring
s, invariant theory
and homological algebra
. For example, the Hochster-Roberts theorem states that the invariant
ring of a linearly reductive group
acting on a regular ring
is Cohen-Macaulay. His best-known work is on the homological conjectures, many of which he established for local rings containing a field, thanks to his proof of the existence of big Cohen-Macaulay modules and his technique of reduction to prime characteristic. His most recent work on tight closure
, introduced in 1986 with Craig Huneke, has found unexpected applications throughout commutative algebra and algebraic geometry
.
Hochster has been recognized for his efforts in mentoring and popularizing mathematics through lectures and articles. He has had more than 30 doctoral students, and the Association for Women in Mathematics
has pointed out his outstanding role in mentoring women students pursuing a career in mathematics
. He is currently (since 2009) serving as the chair of the department of Mathematics, University of Michigan, Ann Arbor.
Hochster's hobbies are cryptic crosswords and bridge (game).
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, regarded as one of the leading commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
ists active today. He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the University of Michigan
University of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
.
Hochster attended 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...
, where he was captain of the Math Team, and received a B.A.
Bachelor of Arts
A Bachelor of Arts , from the Latin artium baccalaureus, is a bachelor's degree awarded for an undergraduate course or program in either the liberal arts, the sciences, or both...
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...
. While at Harvard, he was a Putnam Fellow
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...
in 1960. He earned his 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...
in 1967 from Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, where he wrote a dissertation under Goro Shimura
Goro Shimura
is a Japanese mathematician, and currently a professor emeritus of mathematics at Princeton University.Shimura was a colleague and a friend of Yutaka Taniyama...
characterizing the prime spectra
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
of commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s. He held positions at the University of Minnesota
University of Minnesota
The University of Minnesota, Twin Cities is a public research university located in Minneapolis and St. Paul, Minnesota, United States. It is the oldest and largest part of the University of Minnesota system and has the fourth-largest main campus student body in the United States, with 52,557...
and Purdue University
Purdue University
Purdue University, located in West Lafayette, Indiana, U.S., is the flagship university of the six-campus Purdue University system. Purdue was founded on May 6, 1869, as a land-grant university when the Indiana General Assembly, taking advantage of the Morrill Act, accepted a donation of land and...
before joining the faculty at Michigan in 1977. Hochster shared the 1980 Cole Prize
Cole Prize
The Frank Nelson Cole Prize, or Cole Prize for short, is one of two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number theory. The prize is named after Frank Nelson Cole, who...
with Michael Aschbacher
Michael Aschbacher
Michael George Aschbacher is an American mathematician best known for his work on finite groups. He was a leading figure in the completion of the classification of finite simple groups in the 1970s and 1980s. It later turned out that the classification was incomplete, because the case of quasithin...
, received a Guggenheim Fellowship
Guggenheim Fellowship
Guggenheim Fellowships are American grants that have been awarded annually since 1925 by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the arts." Each year, the foundation makes...
in 1981, and has been a member of both the National Academy of Sciences
United States National Academy of Sciences
The National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
and the American Academy of Arts and Sciences
American Academy of Arts and Sciences
The 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...
since 1992. In 2008, on the occasion of his 65th birthday, he was honored with a conference in Ann Arbor and with a special volume of the Michigan Mathematical Journal
Michigan Mathematical Journal
Michigan Mathematical Journal is published by the mathematics department atUniversity of Michigan.Its initial editor was George Piranian....
.
Hochster's work is primarily in commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, especially the study of module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
s over local ring
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
s. He has established classic theorems concerning Cohen-Macaulay ring
Cohen-Macaulay ring
In mathematics, a Cohen–Macaulay ring is a particular type of commutative ring, possessing some of the algebraic-geometric properties of a nonsingular variety, such as local equidimensionality....
s, invariant theory
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
and homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
. For example, the Hochster-Roberts theorem states that the invariant
Invariant polynomial
In mathematics, an invariant polynomial is a polynomial P that is invariant under a group \Gamma acting on a vector space V. Therefore P is a \Gamma-invariant polynomial ifP = Pfor all \gamma \in \Gamma and x \in V....
ring of a linearly reductive group
Reductive group
In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...
acting on a regular ring
Regular ring
In commutative algebra, a regular ring is a commutative noetherian ring, such that the localization at every prime ideal is a regular local ring: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.Jean-Pierre...
is Cohen-Macaulay. His best-known work is on the homological conjectures, many of which he established for local rings containing a field, thanks to his proof of the existence of big Cohen-Macaulay modules and his technique of reduction to prime characteristic. His most recent work on tight closure
Tight closure
In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by Mel Hochster and Craig Huneke in the 1980s....
, introduced in 1986 with Craig Huneke, has found unexpected applications throughout commutative algebra and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
.
Hochster has been recognized for his efforts in mentoring and popularizing mathematics through lectures and articles. He has had more than 30 doctoral students, and the Association for Women in Mathematics
Association for Women in Mathematics
The Association for Women in Mathematics is a professional society whose mission is to encourage women and girls to study and to have active careers in the mathematical sciences. Equal opportunity and the equal treatment of women and girls in the mathematical sciences are promoted. The AWM was...
has pointed out his outstanding role in mentoring women students pursuing a career in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. He is currently (since 2009) serving as the chair of the department of Mathematics, University of Michigan, Ann Arbor.
Hochster's hobbies are cryptic crosswords and bridge (game).