Oscar Zariski
Encyclopedia
Oscar Zariski was a Russian mathematician
and one of the most influential algebraic geometers
of the 20th century.
where he became a disciple of the Italian school of algebraic geometry
, studying with Guido Castelnuovo
, Federigo Enriques
and Francesco Severi
.
Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory
. At the time of his dissertation publication, he changed his name for professional purposes to Oscar Zariski.
in 1927 supported by Solomon Lefschetz
. He had a position at Johns Hopkins University
where he became professor in 1937. During this period, he wrote Algebraic Surfaces as a summation of the work of the Italian school.
The book was published in 1935 and reissued 36 years later, with detailed notes by Zariski's students that illustrated how the field of algebraic geometry had changed. It is still an important reference.
It seems to have been this work that set the seal of Zariski's discontent with the approach of the Italians to birational geometry
. The question of rigour he addressed by recourse to commutative algebra
. The Zariski topology
, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions. That theory is too limited for algebraic surfaces, and even for curves with singular points. A rational map is to a regular map as a rational function
is to a polynomial: it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, dense set of a given variety. The description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions. This introduces a need, in the surface case, to use also valuation theory
to describe the phenomena such as blowing up
(balloon-style, rather than explosively).
in 1947 where he remained until his retirement in 1969. In 1945, he fruitfully discussed foundational matters for algebraic geometry with André Weil
. Weil's interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski's interests. The two sets of foundations weren't reconciled at that point.
At Harvard, Zariski's students included Shreeram Abhyankar, Heisuke Hironaka
, David Mumford
, Michael Artin
and Steven Kleiman
— thus spanning the main areas of advance in singularity theory
, moduli theory and cohomology
in the next generation. Zariski himself worked on equisingularity theory. Some of his major results, Zariski's main theorem
and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck
that ultimately unified algebraic geometry.
Zariski proposed the first example of a Zariski surface
in 1958.
with Lars Ahlfors. He wrote also Commutative Algebra in two volumes, with Pierre Samuel
. His papers have been published by MIT Press
, in four volumes.
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....
and one of the most influential algebraic geometers
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...
of the 20th century.
Education
Zariski was born Oscher (also transliterated as Ascher or Osher) Zaritsky to a Jewish family (his parents were Bezalel Zaritsky and Hanna Tennenbaum) and in 1918 studied at the University of Kiev. He left Kiev in 1920 to study in RomeRome
Rome is the capital of Italy and the country's largest and most populated city and comune, with over 2.7 million residents in . The city is located in the central-western portion of the Italian Peninsula, on the Tiber River within the Lazio region of Italy.Rome's history spans two and a half...
where he became a disciple of the Italian school of algebraic geometry
Italian school of algebraic geometry
In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major...
, studying with Guido Castelnuovo
Guido Castelnuovo
Guido Castelnuovo was an Italian mathematician. His father, Enrico Castelnuovo, was a novelist and campaigner for the unification of Italy...
, Federigo Enriques
Federigo Enriques
Federigo Enriques was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry....
and Francesco Severi
Francesco Severi
Francesco Severi was an Italian mathematician.Severi was born in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry...
.
Zariski wrote a doctoral dissertation in 1924 on a topic in 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...
. At the time of his dissertation publication, he changed his name for professional purposes to Oscar Zariski.
Johns Hopkins University years
Zariski emigrated to the United StatesUnited States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
in 1927 supported by Solomon Lefschetz
Solomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...
. He had a position at Johns Hopkins University
Johns Hopkins University
The Johns Hopkins University, commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States...
where he became professor in 1937. During this period, he wrote Algebraic Surfaces as a summation of the work of the Italian school.
The book was published in 1935 and reissued 36 years later, with detailed notes by Zariski's students that illustrated how the field of algebraic geometry had changed. It is still an important reference.
It seems to have been this work that set the seal of Zariski's discontent with the approach of the Italians to birational geometry
Birational geometry
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...
. The question of rigour he addressed by recourse to 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...
. The Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions. That theory is too limited for algebraic surfaces, and even for curves with singular points. A rational map is to a regular map as a rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
is to a polynomial: it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, dense set of a given variety. The description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions. This introduces a need, in the surface case, to use also valuation theory
Valuation (mathematics)
In algebra , a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field...
to describe the phenomena such as blowing up
Blowing up
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point...
(balloon-style, rather than explosively).
Harvard University years
After spending a year 1946–1947 at the University of Illinois, Zariski became professor at Harvard UniversityHarvard 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...
in 1947 where he remained until his retirement in 1969. In 1945, he fruitfully discussed foundational matters for algebraic geometry with André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
. Weil's interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski's interests. The two sets of foundations weren't reconciled at that point.
At Harvard, Zariski's students included Shreeram Abhyankar, Heisuke Hironaka
Heisuke Hironaka
is a Japanese mathematician. After completing his undergraduate studies at Kyoto University, he received his Ph.D. from Harvard while under the direction of Oscar Zariski. He won the Fields Medal in 1970....
, David Mumford
David Mumford
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
, Michael Artin
Michael Artin
Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry. and also generally recognized as one of the outstanding professors in his field.Artin was born in Hamburg,...
and Steven Kleiman
Steven Kleiman
Steven Lawrence Kleiman is an American mathematician.He is a Professor of Mathematics at the Massachusetts Institute of Technology. Born in Boston, he did his undergraduate studies at the MIT. He received his Ph.D. from Harvard University in 1965, after studying there with Oscar Zariski and David...
— thus spanning the main areas of advance in singularity theory
Singularity theory
-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
, moduli theory and cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
in the next generation. Zariski himself worked on equisingularity theory. Some of his major results, Zariski's main theorem
Zariski's main theorem
In algebraic geometry, a field in mathematics, Zariski's main theorem is a theorem proved by that states that there is only one branch at any point of a normal variety...
and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
that ultimately unified algebraic geometry.
Zariski proposed the first example of a Zariski surface
Zariski surface
In algebraic geometry, a branch of mathematics, a Zariski surface is a surface over a field of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane to the surface...
in 1958.
Awards and recognition
Zariski was awarded the Steele Prize in 1981, and in the same year the Wolf Prize in MathematicsWolf Prize in Mathematics
The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts...
with Lars Ahlfors. He wrote also Commutative Algebra in two volumes, with Pierre Samuel
Pierre Samuel
Pierre Samuel was a French mathematician, known for his work in commutative algebra and its applications to algebraic geometry. The two-volume work Commutative Algebra that he wrote with Oscar Zariski is a classic. Other books of his covered projective geometry and algebraic number theory...
. His papers have been published by MIT Press
MIT Press
The MIT Press is a university press affiliated with the Massachusetts Institute of Technology in Cambridge, Massachusetts .-History:...
, in four volumes.
See also
- Zariski ringZariski ringIn commutative algebra, a Zariski ring, introduced by , is a commutative Noetherian topological ring A whose topology is defined by an ideal m contained in the Jacobson radical, the intersection of all maximal ideals...
- Zariski tangent spaceZariski tangent spaceIn algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V...
- Zariski surfaceZariski surfaceIn algebraic geometry, a branch of mathematics, a Zariski surface is a surface over a field of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane to the surface...
- Zariski topologyZariski topologyIn algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
- Zariski's theorem on holomorphic functions
- Zariski–Riemann surface
- Zariski spaceZariski spaceIn algebraic geometry, a Zariski space, named for Oscar Zariski, has several different meanings:*A topological space that is Noetherian...
- Zariski's main theoremZariski's main theoremIn algebraic geometry, a field in mathematics, Zariski's main theorem is a theorem proved by that states that there is only one branch at any point of a normal variety...
External links
- Biography from United States Naval AcademyUnited States Naval AcademyThe United States Naval Academy is a four-year coeducational federal service academy located in Annapolis, Maryland, United States...
.