Melvin Hochster (born August 2, 1943) is an

American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, pe ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

working in

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...

. He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the

University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...

Education

Hochster attended

Stuyvesant High School Stuyvesant High School (pronounced ), commonly referred to among its students as Stuy (pronounced ), is a State school, public university-preparatory school, college-preparatory, Specialized high schools in New York City, specialized high school ...

, where he was captain of the Math Team, and received a

B.A. Bachelor of arts (BA or AB; from the Latin ', ', or ') is a bachelor's degree awarded for an undergraduate program in the arts, or, in some cases, other disciplines. A Bachelor of Arts degree course is generally completed in three or four years ...

from

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...

. While at Harvard, he was a

Putnam Fellow The William Lowell Putnam Mathematical Competition, often abbreviated to Putnam Competition, is an annual mathematics competition for undergraduate college students enrolled at institutions of higher learning in the United States and Canada (regard ...

in 1960. He earned his

Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is a ...

in 1967 from

Princeton University Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...

, where he wrote a dissertation under

Goro 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 multipli ...

characterizing the prime spectra of

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

Career

He held positions at the

University of Minnesota The University of Minnesota, formally the University of Minnesota, Twin Cities, (UMN Twin Cities, the U of M, or Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Tw ...

and

Purdue University Purdue University is a public land-grant research university in West Lafayette, Indiana, and the flagship campus of the Purdue University system. The university was founded in 1869 after Lafayette businessman John Purdue donated land and money ...

before joining the faculty at Michigan in 1977. Hochster's work is primarily in

, especially the study of

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...

s over

local ring In abstract algebra, more specifically 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 num ...

s. He has established classic theorems concerning

Cohen–Macaulay ring In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a fini ...

invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...

and

homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

. For example, the ''Hochster–Roberts'' theorem states that the

invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

of a linearly

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...

acting on a

regular ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ide ...

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 In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by . Let R be a commutative noetherian ring containing a field of characteristic p > 0. Hence p is ...

, introduced in 1986 with

Craig Huneke Craig Lee Huneke (born August 27, 1951) is an American mathematician specializing in commutative algebra. He is a professor at the University of Virginia. Huneke graduated from Oberlin College with a bachelor's degree in 1973 and in 1978 earned ...

, has found unexpected applications throughout commutative algebra and

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

. He has had more than 40 doctoral students, and the

Association for Women in Mathematics The Association for Women in Mathematics (AWM) is a professional society whose mission is to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity for and the equal treatment o ...

has pointed out his outstanding role in mentoring women students pursuing a career in mathematics. He served as the chair of the department of Mathematics at the University of Michigan from 2008 to 2017.

Awards

Hochster shared the 1980

Cole Prize The Frank Nelson Cole Prize, or Cole Prize for short, is one of twenty-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 ...

with

Michael Aschbacher Michael George Aschbacher (born April 8, 1944) 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 turne ...

, received a

Guggenheim Fellowship Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the ar ...

in 1981, and has been a member of both the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...

and the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and ...

since 1992. In 2008, on the occasion of his 65th birthday, he was honored with a conference in

Ann Arbor Anne, alternatively spelled Ann, is a form of the Latin female given name Anna (name), Anna. This in turn is a representation of the Hebrew Hannah (given name), Hannah, which means 'favour' or 'grace'. Related names include Annie (given name), ...

and with a special volume of the

Michigan Mathematical Journal The ''Michigan Mathematical Journal'' (established 1952) is published by the mathematics department at the University of Michigan. An important early editor for the Journal was George Piranian. Historically, the Journal has been published a small ...

References

* * *

External links

*
Hochster's home page at MichiganHochster's blurb at the National AcademiesThe University Record (October 8, 2004)
{{DEFAULTSORT:Hochster, Melvin Members of the United States National Academy of Sciences 20th-century American mathematicians 21st-century American mathematicians Algebraists Stuyvesant High School alumni Harvard University alumni Princeton University alumni University of Michigan faculty Purdue University faculty University of Minnesota faculty Putnam Fellows Living people 1943 births Scientists from New York City Mathematicians from New York (state)

Education

Career

Awards

See also

References

External links