| Douglas C. Ravenel | |
|---|---|
| Born | 1947 |
| Nationality | United States |
| Fields | Mathematics |
| Institutions | University of Rochester |
| Alma mater | Brandeis University |
| Doctoral advisor | Edgar H. Brown, Jr. |
| Doctoral students | Andrew Salch |
| Known for | Ravenel Conjectures Work on Adams–Novikov spectral sequence |
Douglas Conner Ravenel (born 1947) is an American mathematician known for work in algebraic topology.
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He received his Ph.D. from Brandeis University in 1972 under the direction of Edgar H. Brown, Jr. with a thesis on exotic characteristic classes of spherical fibrations. From 1971 to 1973 he was instructor at the MIT and 1974/75 he was visiting the Institute for Advanced Study. He became assistant professor at the Columbia University in 1973 and at the University of Washington in Seattle in 1976, where he became associate professor in 1978 and professor in 1981. From 1977 to 1979 he was Sloan Fellow. Since 1988 he is professor at the University of Rochester. He was invited speaker at the International Congress of Mathematicians in Helsinki, 1978, and is an editor of the New York Journal of Mathematics since 1994.
Ravenel's main area of work is stable homotopy theory. Two of his most famous papers are Periodic phenomena in the Adams–Novikov spectral sequence, which he wrote together with H. R. Miller and W. S. Wilson, (Annals of Mathematics, 106 (1977), 469–516) and Localization with respect to certain periodic homology theories (Amer. J. Math., 106 (1984), 351–414).
In first of these two papers, the authors explore the stable homotopy groups of spheres by analyzing the E2-term of the Adams–Novikov spectral sequence. The authors set up the so-called chromatic spectral sequence relating this E2-term to the cohomology of the Morava stabilizer group, which exhibits certain periodic phenomena in the Adams–Novikov spectral sequence and can be seen as the beginning of chromatic homotopy theory. Applying this, the authors compute the second line of the Adams–Novikov spectral sequence and establish the non-triviality of a certain family in the stable homotopy groups of spheres. In all of this, the authors use work by Morava and themselves on Brown–Peterson cohomology and Morava K-theory.
In the second paper, Ravenel expands these phenomena to a global picture of stable homotopy theory leading to the Ravenel conjectures. In this picture, complex cobordism and Morava K-theory control many qualitative phenomena, which were understood before only in special cases. Here Ravenel uses localization in the sense of Bousfield in a crucial way. All but one of the Ravenel conjectures were proved by Ethan Devinatz, Mike Hopkins and Jeff Smith not long after the article got published. Frank Adams said on that occasion:
At one time it seemed as if homotopy theory was utterly without system; now it is almost proved that systematic effects predominate.[1]
In further work, Ravenel calculates the Morava-K theories of several spaces and proves important theorems in chromatic homotopy theory together with Hopkins. He was also one of the founders of elliptic cohomology. In 2009, he solved together with Hill and Hopkins the Kervaire invariant 1 problem for large dimensions.
Ravenel has written two books, the first on the calculation of the stable homotopy groups of spheres and the second on the Ravenel conjectures.