Paul Cohen Kevine Boutcheko 11/4/14

Paul Cohen is an American mathematician whose work left a footprint in the world of mathematics and science. He was born on April 2, 1934 in Long Branch, New Jersey, the youngest child of Jewish immigrants Abraham and Minnie Cohen. Paul grew up in Brooklyn, NY, but at the age of nine, his parents separated and he stayed with his mother. It was around this time that he began to show interest in mathematics, and began to study advanced mathematics. He gained some popularity as a teenager when he began participating in math competitions.

Paul was accepted into the prestigious Stuyveysant High School in New York City and graduated when he was only 16 years old in 1950. He went on to attend Brooklyn College and while attending the school, he visited the University of Chicago to talk about the diﬀerent research opportuni- ties the school had to oﬀer. The young mathematician eventually left after three years , whithout graduation, when he was accepted into the Univer- sity of Chicago’s graduate program to study mathematics. Cohen was very interested in the Number Theory and worked on his research under the su- pervision of Andre Weil. He received his Master’s degree in 1954 but wanted to continue in his pursuit of solving the most important problems in the ﬁeld of mathematics at the time. Paul Cohen pursued his doctorate at the Uni- versity of Chicago under the supervision of Antoni Zygmund and wrote a thesis on the ”Topics in the Theory of Uniqueness of Trigonometic Series” which enabled him to receive his PhD in 1958.

Even before accepting his PhD, Cohen had been selected for a position as instructor in mathematics at the University of Rochester where his spent only one year. He then worked at the Massachusetts Institute of Technology right after achieving his doctorate from 1958-1959 and spent the next two years as a fellow at the Institute for Advanced Study at Princeton University. During these years, Cohen made several advances in math before attaining the level at which he is most famous for. In 1959, he solved a problem posed by Walter Rudin pertaining to ”Factorization in Group Algebras” and also solved the Littlewood Conjecture in 1960.

As he was quickly getting promoted to higher positions as a professor of mathematics at Stanford, Cohen began working on the indepenence of the

1 continuum hypothesis around the end of 1962. The continuum hypothesis problem was the ﬁrst on David Hilbert’s famous list of 23 problems. David Hilbert had introduced these problems to the world at the Second Inter- national Congress of Mathematicians in 1900. Paul Cohen gave credit to gaining the idea of ”forcing” to solve the problem, to Kurt Godel’s ”The Consistency of the Continuum Hypothesis”. A bit over a year after Cohen had started the problem, he believed he had ﬁnished it and sent his proof to Godel himself on May 9, 1963. Paul Cohen ﬁnally introduced his proof on the independence of the axiom of choice and the continuum hypothesis from the axioms of Zermelo-Fraenkel set theory in a lecture at the international symposium on ”Theory Models” at Berkeley July 3, 1963. In 1966, Cohen was awarded the Fields Medal for his work on the foundations of set the- ory at the age of 32 years old, the second youngest mathematician to have received the award at the time.

The Fields Medal is just one of the many prestigious awards and honors Paul Cohen received during his career. In 1964, he was given the Bocher Memorial Prize from the American Mathematical Society. In 1968, President Lyndon B. Johnson honored him with the National Medal of Science. The brilliant mathematician was elected to the National Academy of Sciences, The American Academy of Arts and Sciences, foreign honorary member of the London Mathematical Society and Marjorie Mhoon Fair Professor in Quantitationce Science at Stanford in 1972. Paul Cohen continued to teach mathematics at Stanford until after he retired in 2004. He became ill with a rare lung disease a short time after retiring and died March 23, 2007 in Palo Alto, CA.

References

[1] http : //www − history.mcs.st − andrews.ac.uk/Biographies/Cohen.html

[2] http : //www.storyofmathematics.com/20thcohen.html

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