# Paul Cohen

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** Paul Joseph Cohen ** (April 2, 1934 – March 23, 2007) was an American mathematician.

## Paul_Cohen’s Biography

### First years

Paul Joseph Cohen was born on April 2, 1934, in Long Branch, New Jersey. His parents were Jewish immigrants from Poland, they made a hard living, his mother was a seamstress and his father brought money home as he could, arranging furniture, messages, everything that was presented to him. Paul, was the youngest of the four brothers and was raised in Brooklyn (New York). He was educated by his mother, when at nine years old, the parents separated. Since his childhood he was enthusiastic about mathematics, so he started studying advanced mathematics at an early age. In fact, her sister had to take the books from the library, because the librarians refused to give it to her, assuring that even some mathematics teachers did not understand those calculations.

In his adolescence he was declared a prodigy of mathematics, everyone was impressed by his ability to participate in math competitions. He attended the “Stuyvesant High School” in New York, graduating in 1950 at the young age of 16. He was a student at “Brooklyn College” from 1950 to 1953, but without getting his degree because he had been accepted at the University of Chicago after having gone to discuss their options to enter. So he took his master’s degree in Chicago, taking courses to fit his research of number theory and thus be guided. His knowledge of number theory before he came to Chicago was about some classic books he had read on his own at school. He started working on her being supervised by André Weil. He got the master’s degree in 1954, but he began to be more interested in the fact that certain results in the theory were more indescribable than the theory itself, even so, this is a fact that has marked him throughout his career. In the faculty, he used to ask his classmates which were the most difficult problems to solve in his field, since those were the problems he wanted to solve.

He continued studying in Chicago for his doctorate under the supervision of Antoni Zygmund. He obtained his doctorate in 1958 for his doctoral thesis of Topics in the Theory of Uniqueness of Trigonometric Series. In his student years he made several important friendships. John Thompson was one of them, because of their relationship, he began to take an interest in logic. The two of them, and other friends, treated that topic a lot, even Cohen stayed at Tennenbaum’s house for a while.

In 1957, before the granting of his doctorate, Cohen was appointed instructor at the University of Rochester for one year. In 1958 he was at the Massachusetts Institute of Technology, from 1959 to 1961 he went to the Institute of Advanced Studies at Princeton University. In 1961 Cohen was appointed to the faculty at Stanford University, being promoted to full professor there in 1964. In 1966 he was awarded the Fields Medal at the International Congress of Mathematicians in Moscow for his fundamental work on the foundations of the theory. determined. These were the years in which he made a significant number of mathematical advances.

### Professional jobs

In “Factorization in group algebras” (1959), he showed that any integrable function in a locally compact group is the convolution of two of these functions, solving a problem posed by Walter Rudin. In “On a conjecture of Littlewood and idempotent measures” (1960), Cohen made a great discovery by solving “Littlewood Conjecture”. Before that, he had already written to Harold Davenport about the result.

In 1961, Cohen was accepted at Stanford University as an assistant professor of mathematics. He was promoted to associate professor of mathematics in the following year and, also, in 1962, he was awarded an Alfred P Sloan research grant. In August 1962, he participated in the “International Congress of Mathematicians”, in Stockholm. He was invited to talk about “Idempotent measures and homomorphisms of group algebras”. On a cruise from Stockholm to Leningrad, he met Christina Karls, from Malung, in Sweden. They married on October 10, 1963 and had three children, the twins Eric and Steven and Charles.

He was promoted to full professor at “Stanford University”, in 1964, solving, at the same time one of the most challenging problems in mathematics, for this, he used a technique called “forcing”, to prove independence in the set of axiom choice theories and the generalized “continuum hypothesis”. Cohen explained that the idea of ”forcing” came from reading a book by Kurt Gödel, specifically “The Consistency of the Continuum Hypothesis”, a book based on notes taken in a course given at the “Institute for Advanced Study” in 1938 -1939. The “continuum hypothesis problem” was the first of the 23 famous problems of David Hilbert exposed at the “Second International Congress of Mathematicians” in Paris, in 1900.

He began to work on the independence of “continuum hypothesis” starting in 1962. In 1966, he published the monograph “Set theory and the continuum hypothesis”, based on a course he received at Harvard in the spring of 1965. In the same year, he was awarded “Fields Medal” for his fundamental work in his theories. He received it from President Lyndon B Johnson, at the White House on February 13, 1968. In addition, he was elected by “National Academy of Sciences” and “the American Academy of Arts and Sciences” as an honorary member of the “London Mathematical Society.”

To complete his set of theories he worked on “differential equation and harmonic analysis”. Paul Joseph Cohen was named “Marjorie Mhoon Fair Professor in Quantitative Science” at Standford, in 1972, being the first bearer of this position. He formally retired in 2004, but continued to teach at Stanford until a date near his death. He died of a strange lung disease, at “Stanford Hospital” in Palo Alto.

Apart from mathematics, Cohen also had a great interest in dance. He played piano and violin and sang in the Stanford choir.

## More Facts about Paul Cohen

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