Douglas Ravenel

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

Life

Ravenel studied at Oberlin College ( BA 1969) and received his doctorate in 1972 at Brandeis University. From 1971 to 1973 he was instructor at MIT and 1974/75 at the Institute for Advanced Study. From 1973 he was an Assistant Professor at Columbia University and in 1976 at the University of Washington in Seattle, where he became Associate Professor in 1978 and Professor in 1981. From 1988 he was professor at the University of Rochester, from 1999 as Fayerweather Professor. 1996 to 2005 he was chairman of the mathematics department there. He has been a visiting scientist at the MSRI (1989 ), Cambridge ( Isaac Newton Institute 2002), Oxford, Harvard, Bonn, Paris, and at Johns Hopkins University. From 1977 to 1979 he was Sloan 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

Ravenels main area of ​​work is the stable homotopy theory. This deals with properties of spaces that remain the same under device to attach, and homology theories. Two of his most important works in this field are

  • 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.

In the first article is about a profound investigation of the stable homotopy groups of spheres. Drawing on her earlier work on the Brown -Peterson cohomology and Morava K-theory (two cohomology theories that are closely related to complex cobordism ), the authors were able to establish the so-called chromatic spectral sequence that computes the initial term of the Adams - Novikov spectral sequence and profound periodic phenomena in the Adams - Novikov spectral sequence and thus discovered in the stable homotopy groups of spheres. This allowed them to see other things, a new non -trivial infinite family of elements in these homotopy groups.

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

Other work is concerned among other things with the computation of the Morava K-theory of various classes of spaces. Ravenel wrote two books, the first on the calculation of the stable homotopy groups of spheres, especially with the Adams - Novikov spectral sequence, the second about the evidence of Ravenel conjectures. In 2009 Ravenel has dissolved together with Michael Hill and Mike Hopkins along the Kervaire invariant 1 problem which is closely related to spheres with exotic differentiable structures.

In 1986, he led with Peter Landweber and Robert Stong an elliptic cohomology.

In 2009 he succeeded with Michael A. Hill and Michael J. Hopkins an almost complete solution of the Kervaire - Invariantenproblems (after Michel Kervaire ).

Writings

  • 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.
Alexandria (Virginia) Harvard University Topological space Haynes Miller
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