Elliptic curves hit the front page of the New York Times in June, 1993 when Andrew Wiles announced a proof of Fermat’s Last Theorem. In the proof stitched together by Wiles, elliptic curves played a starring role. Amazingly, elliptic curves have been prominent in mathematics literally for millennia – and they continue to post simple-looking problems whose solutions continue to elude us. Perhaps most notoriously, a general conjecture of Birch and Swinnerton-Dyer, for which the Clay Math Institute has offered a $1,000,000 reward, remains unsolved even for elliptic curves. A proof of this conjecture in this case would settle the Congruent Number Problem, a problem about triangles that has been studied intensively at least since the tenth century.
In my talk, I will discuss some of the ways that elliptic curves have intervened in the field of cryptography. Although I am by no means an expert in this subject, I will give a course to Berkeley undergraduates on cryptography next spring and expect to be lecturing about elliptic curves at various points during the semester.
Kenneth Ribet studied at Brown University and Harvard University. He received his PhD in 1973 from Harvard, where his advisor was John Tate. After three years of teaching in Princeton and two years of research in Paris, Ribet joined the Berkeley faculty in 1978. He received his department's Distinguished Teaching Award in 1985.
Prof. Ribet is known for his work in number theory and algebraic geometry. He played a prominent role in the proof of Fermat's Last Theorem by showing that this statement was a logical consequence of a conjecture about elliptic curves. (Andrew Wiles proved this conjecture in 1995, thereby obtaining Fermat's Last Theorem as a corollary.) Prof. Ribet is a member of the scientific advisory board of the Institute for Pure & Applied Mathematics at UCLA and a member of the editorial boards of the following three Springer book series: Graduate Texts in Mathematics, Universitext, Undergraduate Texts in Mathematics. He serves also on the editorial boards of Mathematische Annalen, the Annales de l'Institut Fourier, the Journal of Number Theory and Mathematics Research Letters.
Prof. Ribet was elected to the American Academy of Arts and Sciences in 1997 and the National Academy of Sciences in 2000. He was awarded the Fermat Prize in 1989 and received an honorary PhD from Brown University in 1998. In 1988 he was inducted as a Vigneron d'honneur by the Jurade de Saint Emilion.
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