Oscar Zariski

Oscar Zariski
Oscar Zariski (1899–1986)
Oscar Zariski
Born	Ascher Zaritsky April 24, 1899
Birthplace	Kobrin, Russian Empire
Died	July 4, 1986 Brookline, Massachusetts, United States
Nationality	American
Occupation	Mathematician
Employer	Harvard University
Known for	Contributions to algebraic geometry
Alma mater	University of Rome
Awards	Cole Prize in Algebra (1944), National Medal of Science (1965), Wolf Prize (1981)

Oscar Zariski (born Ascher Zaritsky; April 24, 1899 – July 4, 1986) was a Russian-born American mathematician who became one of the most influential figures in algebraic geometry during the twentieth century. Born in the small town of Kobrin in the Russian Empire, Zariski emigrated first to Italy and then to the United States, where he spent the bulk of his career reshaping the foundations of his field. As the Dwight Parker Robinson Professor of Mathematics at Harvard University, he brought the rigorous methods of commutative algebra to bear on classical problems of algebraic geometry, transforming the discipline and training a generation of mathematicians who would carry his work forward. His contributions earned him numerous honors, including the Cole Prize in Algebra, the National Medal of Science, the Wolf Prize in Mathematics, and the Steele Prize from the American Mathematical Society. His work on the resolution of singularities, algebraic surfaces, and the foundations of algebraic geometry laid the groundwork for much of modern mathematics, and problems bearing his name continue to inspire research decades after his death.

Early Life

Oscar Zariski was born Ascher Zaritsky on April 24, 1899, in Kobrin, a town then part of the Russian Empire (now in Belarus). He grew up in a Jewish family during a period of considerable upheaval in the Russian Empire. The region experienced significant political turmoil during Zariski's youth, including the disruptions of World War I and the Russian Revolution. These events had a profound impact on his early education and his eventual decision to leave his homeland.

Zariski displayed an early aptitude for mathematics. He began his university studies at the University of Kyiv, where he first encountered advanced mathematical ideas. However, the political instability in the newly formed Soviet Union and the limited opportunities available to him there led the young mathematician to seek his education elsewhere. He made the consequential decision to move to Italy, a country that at the time was home to one of the world's leading schools of algebraic geometry.

Upon arriving in Italy, Zariski enrolled at the University of Rome, where he came under the influence of the Italian school of algebraic geometry. This school, which had been producing significant results since the late nineteenth century, was known for its geometric intuition and its sometimes informal approach to mathematical proof. The experience in Rome would prove to be formative for Zariski, both in exposing him to the beauty of algebraic geometry and in making him aware of the need for more rigorous foundations in the subject.

Education

At the University of Rome, Zariski studied under Guido Castelnuovo, one of the leading figures of the Italian school of algebraic geometry. Castelnuovo, along with his colleagues Federigo Enriques and Francesco Severi, had developed an extensive theory of algebraic surfaces using methods that relied heavily on geometric intuition. Zariski completed his doctoral dissertation in 1926, titled Sopra una classe di equazioni algebriche contenenti linearmente un parametro e risolubili per radicali (On a class of algebraic equations containing a parameter linearly and solvable by radicals).

The training Zariski received in Rome gave him a deep appreciation for the classical problems of algebraic geometry, particularly those involving algebraic curves and surfaces. At the same time, he became increasingly aware that many of the results claimed by the Italian school lacked the level of rigor that he considered necessary for mathematics. This realization would later drive his efforts to place algebraic geometry on firmer algebraic foundations, a project that would define much of his subsequent career.

Career

Move to the United States and Early Career

After completing his doctorate in Rome, Zariski emigrated to the United States in 1927. He joined the faculty of Johns Hopkins University in Baltimore, Maryland, where he would spend the first major phase of his American career. At Johns Hopkins, Zariski began the intellectual transformation that would make him one of the central figures in twentieth-century mathematics. While he had been trained in the geometric methods of the Italian school, he became convinced that the future of algebraic geometry lay in the application of the newly developing tools of abstract algebra.

During his years at Johns Hopkins, Zariski immersed himself in the algebraic methods being developed by mathematicians such as Emmy Noether and Wolfgang Krull. He studied commutative algebra intensively and began to see how its concepts—ideals, local rings, valuations—could be used to provide rigorous proofs of results in algebraic geometry that had previously relied on geometric intuition alone. This period of study and transformation was crucial to the development of his mathematical program.

Zariski published his first major book, Algebraic Surfaces, in 1935, which appeared as a volume in the Ergebnisse der Mathematik series. The book provided a systematic account of the theory of algebraic surfaces as developed by the Italian school, but it also identified gaps and unproven assertions in the existing literature. In a sense, the book served as both a monument to the Italian school's achievements and a blueprint for the work that needed to be done to put those achievements on solid footing. A review published in the Bulletin of the American Mathematical Society noted the significance of this contribution to the field.

Resolution of Singularities and Foundational Work

One of the central problems in algebraic geometry during the twentieth century was the resolution of singularities—the question of whether every algebraic variety can be transformed into a smooth variety by a sequence of algebraic operations. Zariski made fundamental contributions to this problem. He proved the resolution of singularities for algebraic varieties of dimension up to three over fields of characteristic zero, building on earlier work and using the algebraic methods he had developed.

Zariski's approach to the resolution of singularities was revolutionary in its use of valuation theory and local algebra. Rather than relying on the geometric techniques favored by the Italian school, he employed the theory of local rings to study the behavior of algebraic varieties near their singular points. This approach allowed him to achieve results of a generality and rigor that had not previously been possible.

In recognition of this work, Zariski was awarded the Frank Nelson Cole Prize in Algebra by the American Mathematical Society in 1944. The Cole Prize, one of the most prestigious awards in algebra, recognized the fundamental importance of his contributions to the algebraic foundations of geometry.

Harvard University

Zariski moved to Harvard University in 1947, where he would spend the remainder of his career. He also held a visiting position at the University of Illinois during parts of his career. At Harvard, he held the position of Dwight Parker Robinson Professor of Mathematics, one of the university's most distinguished chairs.

At Harvard, Zariski built one of the most important mathematical research groups of the mid-twentieth century. His seminar became a focal point for algebraic geometry, attracting students and visitors from around the world. He trained a remarkable number of doctoral students, many of whom went on to become leading mathematicians in their own right. His students included Shreeram Abhyankar, Heisuke Hironaka, David Mumford, Michael Artin, and Steven Kleiman, among many others. Heisuke Hironaka would later prove the resolution of singularities in all dimensions in characteristic zero, completing a program that Zariski had initiated, and received the Fields Medal for this work. David Mumford also received the Fields Medal for his contributions to algebraic geometry, work deeply influenced by Zariski's teaching.

Zariski's teaching style was known for its depth and intensity. He communicated not only the technical content of algebraic geometry but also a sense of the subject's beauty and the importance of rigorous foundations. The intellectual environment he created at Harvard was instrumental in making the university a world center for algebraic geometry, a status it maintained for decades.

Major Mathematical Contributions

Zariski's contributions to algebraic geometry were both broad and deep. His work fundamentally changed the way mathematicians think about algebraic varieties and their properties. Among his most important contributions were:

Zariski topology: Zariski introduced a topology on algebraic varieties that is now named after him. The Zariski topology, defined using the closed sets given by the zero loci of polynomial equations, became a fundamental tool in algebraic geometry. While coarser than the usual topology on complex varieties, it proved to be exactly the right notion for studying algebraic properties and became a cornerstone of the modern theory of schemes developed later by Alexander Grothendieck.

Theory of holomorphic functions on algebraic varieties: Zariski developed a theory of holomorphic functions in the algebraic setting, using the concept of formal power series and completion of local rings. His "Main Theorem" (now known as Zariski's Main Theorem) established fundamental results about the structure of birational morphisms between algebraic varieties.

Equisingularity theory: Later in his career, Zariski developed a theory of equisingularity, which provides a systematic way of studying families of singular algebraic varieties. This work aimed to classify and understand the different ways in which singularities can arise and deform.

Algebraic foundations: Perhaps most importantly, Zariski's program of rebuilding algebraic geometry on the foundations of commutative algebra set the stage for the vast generalizations achieved by Alexander Grothendieck and his school in the 1960s. Grothendieck's theory of schemes, which revolutionized algebraic geometry, was in many ways a natural extension of the algebraic approach that Zariski had pioneered.

Zariski also published extensively. His two-volume work Commutative Algebra, co-authored with Pierre Samuel, became a standard reference in the field and was used by generations of mathematicians. A second edition of Algebraic Surfaces appeared in 1971 with appendices by several of his former students, updating the original work and filling in the gaps that Zariski had identified decades earlier.

The Zariski Cancellation Problem

One of the problems bearing Zariski's name that has attracted significant attention is the Zariski Cancellation Problem. This problem asks, in essence, whether an algebraic variety is uniquely determined by certain algebraic data, or whether non-isomorphic varieties can become isomorphic after taking a product with an affine line. The problem remained open for decades and attracted the attention of many mathematicians.

In 2014, Neena Gupta of the Indian Statistical Institute in Kolkata provided a solution to the Zariski Cancellation Problem in positive characteristic for affine spaces of dimension three and higher. This achievement was recognized when Gupta received the Ramanujan Prize for Young Mathematicians, awarded in particular for her solution to this long-standing problem. The continued study of this problem decades after Zariski's death illustrates the lasting influence of his mathematical ideas.

Personal Life

Oscar Zariski spent the later decades of his life in Brookline, Massachusetts, near the Harvard campus where he had taught for so many years. He became a naturalized United States citizen after his emigration from Europe. His original name was Ascher Zaritsky, and he adopted the name Oscar Zariski after moving to Italy.

Zariski died on July 4, 1986, at his home in Brookline, at the age of 86. His death was reported by both The New York Times and The Harvard Crimson, reflecting the esteem in which he was held in the mathematical community and beyond.

A biography of Zariski, A Unreal Life of Oscar Zariski, was written by Carol Parikh and provides a detailed account of his personal and mathematical journey from Kobrin to Harvard.

Recognition

Oscar Zariski received numerous awards and honors throughout his career, reflecting the fundamental importance of his contributions to mathematics.

In 1944, he was awarded the Frank Nelson Cole Prize in Algebra by the American Mathematical Society, one of the society's most distinguished awards, for his work on the algebraic foundations of algebraic geometry and the resolution of singularities.

In 1965, Zariski received the National Medal of Science, the highest scientific honor bestowed by the United States government. President Lyndon B. Johnson announced the award, noting that the medal served as "a symbol of the Nation's desire to recognize outstanding achievement."

In 1981, Zariski was awarded the Wolf Prize in Mathematics, one of the most prestigious international awards in the field. The prize recognized his lifetime of contributions to algebraic geometry.

Also in 1981, the American Mathematical Society awarded Zariski the Leroy P. Steele Prize for his cumulative influence on mathematics. The Steele Prize recognized not only his own research contributions but also his extraordinary impact as a teacher and mentor.

Zariski was a member of the National Academy of Sciences and was recognized internationally as one of the foremost mathematicians of his era. A 1959 article in the Bulletin of the American Mathematical Society surveyed his scientific work, testifying to the significance attributed to his contributions by his contemporaries.

Legacy

Oscar Zariski's legacy in mathematics is multifaceted and enduring. His most lasting contribution was the transformation of algebraic geometry from a field based largely on geometric intuition into one grounded in the rigorous methods of commutative algebra. This transformation, which Zariski initiated and which was carried further by his students and by Alexander Grothendieck, fundamentally changed the character of the subject and opened up vast new areas of research.

The Zariski topology, which he introduced, remains a fundamental concept in algebraic geometry and commutative algebra. It is taught in every introductory course on algebraic geometry and is one of the basic tools used in the modern theory of schemes. The concept exemplifies Zariski's ability to find exactly the right algebraic formulation for geometric ideas.

Zariski's influence as a teacher and mentor was extraordinary. The list of his doctoral students reads as a roster of some of the most important algebraic geometers of the second half of the twentieth century. Through his students and their students, Zariski's mathematical vision has been transmitted to subsequent generations, ensuring that his approach to algebraic geometry continues to shape the field.

The problems and concepts that bear Zariski's name—the Zariski topology, Zariski's Main Theorem, the Zariski Cancellation Problem, Zariski surfaces, and others—are testament to the breadth of his contributions. The fact that the Zariski Cancellation Problem attracted a solution by Neena Gupta nearly three decades after his death demonstrates that his mathematical questions continue to generate important research.

The United States Naval Academy has noted that Zariski "did work of fundamental importance in the area of algebraic geometry," a field "which deals with the solution sets of polynomial" equations. This assessment reflects the consensus of the mathematical community that Zariski was one of the most important mathematicians of his century, a figure whose work reshaped an entire branch of mathematics and whose influence continues to be felt in research conducted around the world.