Douglas Ravenel

Douglas Ravenel in Seattle 1978

Douglas Conner Ravenel (born February 17, 1947 in Alexandria , Virginia ) is an American mathematician who deals with algebraic topology , in particular with homotopy theory .

Life

Ravenel graduated from Oberlin College with a bachelor's degree in 1969 and received his PhD from Brandeis University in 1972 . From 1971 to 1973 he was an instructor at MIT and 1974/75 at the Institute for Advanced Study . From 1973 he was an assistant professor at Columbia University and from 1976 at the University of Washington in Seattle , where he became an associate professor in 1978 and a professor in 1981. From 1988 he was a professor at the University of Rochester , from 1999 as Fayerweather Professor. From 1996 to 2005 he was chairman of the mathematics faculty there. Among other things, he was visiting scholar at MSRI (1989), in Cambridge (Isaac Newton Institute 2002), Oxford , Harvard , Bonn , Paris and at Johns Hopkins University . From 1977 to 1979 he was a Sloan Research Fellow .

In 1978 he was invited speaker at the International Congress of Mathematicians in Helsinki ( Complex cobordism and its application to homotopy theory ). He is a fellow of the American Mathematical Society .

Mathematical work

Ravenel's main field of work is stable homotopy theory . This deals with the properties of spaces , which remain the same when suspended , and with homology theories . Two of his most important works in this area are

• with HR Miller and WS Wilson: Periodic phenomena in the Adams-Novikov spectral sequence. In: Annals of Mathematics. 106, 1977, pp. 469-516.
• Localization with respect to certain periodic homology theories. In: Amer. J. Math. 106, 1984, pp. 351-414.

The first article is about an in-depth study of the stable homotopy groups of spheres. Drawing on their earlier work on Brown-Peterson cohomology and Morava-K theory (two cohomology theories closely related to complex cobordism ), the authors were able to set up the so-called chromatic spectral sequence, which calculates the initial term of the Adams-Novikov spectral sequence and profound periodic phenomena discovered in the Adams-Novikov spectral sequence and thus also in the stable homotopy groups of spheres. This enabled them to find, among other things, a new non-trivial infinite family of elements in these homotopy groups.

The second work mentioned extends the periodic phenomena mentioned above to a global picture of the stable homotopy theory, which culminates in the Ravenel conjectures. In this picture, complex cobordism and Morava-K theory control many qualitative phenomena that were previously only understood in special cases. All but one of these Ravenel conjectures were proven by Ethan Devinatz , Mike Hopkins, and Jeffrey H. Smith shortly after the article appeared. On the occasion of this, John Frank Adams said:

"At one time it seemed as if homotopy theory was utterly without system; now it is almost proved that systematic effects predominate. "

For the remaining, unproven telescope conjecture , Ravenel published a refutation in 1990, which later turned out to be incorrect. She is open.

Other work deals, among other things, with the computation of the Morava K theory of different classes of spaces. Ravenel wrote two books, the first on the computation of the stable homotopy groups of spheres, particularly with the Adams-Novikov spectral sequence, the second on the evidence of the Ravenel conjectures.

In 1986 he introduced elliptical cohomology with Peter Landweber and Robert Stong .

In 2009, with Michael A. Hill and Michael J. Hopkins, he succeeded in an almost complete solution to the Kervaire invariant problem (after Michel Kervaire ), which is closely related to exotic differentiable structures on spheres.

Fonts

• Complex cobordism and the stable homotopy groups of spheres. Academic Press 1986, 2nd edition: AMS 2003. ( online )
• Nilpotency and Periodicity in stable homotopy theory. In: Annals of Mathematical Studies. Princeton 1992.

Web links

• Douglas Ravenel on the Rochester University website
• Douglas Ravenel in the Mathematics Genealogy Project (English)

Footnotes

1. ^ JF Adams: The work of MJ Hopkins. In: JP May and CB Thomas (Eds.): The selected works of John Frank Adams. Vol. II. Cambridge University Press, Cambridge 1992, pp. 525-529.
2. ^ Ravenel, Hill, Hopkins, On the nonexistence of elements of Kervaire invariant one, Annals of Mathematics, Volume 184, 2016, pp. 1–262. Arxiv 2009 .
3. ↑ Douglas Ravenel on the Rochester University website ( Memento of the original from May 11, 2009 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.