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He was a student at the University of Kiev in 1918, moving to Rome to study in 1920. He became a disciple of the Italian school of algebraic geometry, studying with Guido Castelnuovo , Federigo Enriques and Francesco Severi . He wrote a doctoral dissertation in 1924, on a topic in Galois theory. It was when it came to be published that he accepted a suggestion to change his name for professional purposes.
He emigrated to the USA in 1927, supported by Solomon LefschetzSolomon Lefschetz ( 3 September 1884- 5 October 1972) was a US mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. He was born in Moscow into a. He had a position at Johns Hopkins UniversityThe Johns Hopkins University is a prestigious private institution of higher learning located in Baltimore, Maryland. Hopkins holds many "firsts" in American education: it was the first university in the United States to put an emphasis on research, founde, where he became professor in 1937.
It was this period that he wrote the celebrated book Algebraic Surfaces, intended as a summation of the work of the Italian school, but in effect its swansong, too. It was published in 1935. It was reissued many years later, with copious notes showing how much the field of algebraic geometry had changed, not only foundationally but in emphasis. 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 geometryGeometry 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 2, the birational geometry of algebrai. The question of rigour he addressed by recourse to commutative algebraIn abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. It is foundational both for algebraic geometry and for algebraic number theory. The most prominent example for commutative rings are p. The Zariski topologyIn mathematics, the Zariski topology is a structure basic to algebraic geometry, especially since 1950. It is named after its originator, Oscar Zariski. The Zariski topology is defined by defining the closed sets to be the sets consisting of the mutual ze, 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 functionIn mathematics, a rational function is a ratio of polynomials. For a single variable x a typical rational function is therefore P ''x Q ''x where P and Q are polynomials in x as indeterminate, and Q isn't the zero polynomial. Any non-zero polynomial Q is is to a polynomial: it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, denseTopology General topology In mathematics, the term dense has at least two different meanings. A subset A of a topological space X is said to be dense if the only closed subset of X containing A is X itself. This can also be expressed by saying that the cl 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 theoryModel Theory In logic and model theory, a valuation is a map from the set of variables of a first-order language to the universe of some interpretation of that language. Informally, it is an assignment of particular values to the variables in a mathematic to describe the phenomena such as 'blowing up' (balloon-style, rather than explosively).
Zariski became professor at Harvard University in 1947, retiring in 1969. In 1945 he fruitfully discussed foundational matters for algebraic geometry with André Weil; in fact 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 and Michael Artin - 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.
He was awarded the Steele Prize in 1981. He wrote also Commutative Algebra in two volumes, with Pierre Samuel . His papers have been published by MIT Press, in four volumes. The Unreal Life of Oscar Zariski (1991) is a biography by Carol Ann Parikh.
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