Douglas Ravenel
Encyclopedia
Douglas Conner Ravenel is an American
mathematician
known for work in algebraic topology
.
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.
. 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:
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.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
known for work in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
.
Life
He received his Ph.D. from Brandeis UniversityBrandeis University
Brandeis University is an American private research university with a liberal arts focus. It is located in the southwestern corner of Waltham, Massachusetts, nine miles west of Boston. The University has an enrollment of approximately 3,200 undergraduate and 2,100 graduate students. In 2011, it...
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
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
. He became assistant professor at the Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
in 1973 and at the University of Washington
University of Washington
University of Washington is a public research university, founded in 1861 in Seattle, Washington, United States. The UW is the largest university in the Northwest and the oldest public university on the West Coast. The university has three campuses, with its largest campus in the University...
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
University of Rochester
The University of Rochester is a private, nonsectarian, research university in Rochester, New York, United States. The university grants undergraduate and graduate degrees, including doctoral and professional degrees. The university has six schools and various interdisciplinary programs.The...
. He was invited speaker at the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Helsinki, 1978, and is an editor of the New York Journal of Mathematics
The New York Journal of Mathematics
The New York Journal of Mathematics is a peer-reviewed, scholarly journal. It has an editorial board which , consists of 18 university-affiliated scholars in addition to the Editor-in-chief. It is, however, published entirely electronically...
since 1994.
Work
Ravenel's main area of work is stable homotopy theoryStable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
. 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
Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting...
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
Morava K-theory
In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s...
.
In the second paper, Ravenel expands these phenomena to a global picture of stable homotopy theory leading to the Ravenel conjectures
Ravenel conjectures
The Ravenel conjectures are a set of mathematical conjectures in the field of stable homotopy theory posed by Douglas Ravenel at the end of a paper published in 1984. It was earlier circulated in preprint. The problems involved have largely been resolved, with all but the "telescope conjecture"...
. In this picture, complex cobordism
Complex cobordism
In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories...
and Morava K-theory
Morava K-theory
In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s...
control many qualitative phenomena, which were understood before only in special cases. Here Ravenel uses localization
Localization of a category
In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before...
in the sense of Bousfield in a crucial way. All but one of the Ravenel conjectures were proved by Ethan Devinatz, Mike Hopkins
Michael J. Hopkins
Michael Jerome Hopkins is an American mathematician known for work in algebraic topology.-Life:He received his Ph.D. from Northwestern University in 1984 under the direction of Mark Mahowald. In 1984 he also received his D.Phil...
and Jeff Smith not long after the article got published. Frank Adams
Frank Adams
John Frank Adams FRS was a British mathematician, one of the founders of homotopy theory.-Life:He was born in Woolwich, a suburb in south-east London. He began research as a student of Abram Besicovitch, but soon switched to algebraic topology. He received his Ph.D. from the University of...
said on that occasion:
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
Elliptic cohomology
In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.-History and motivation:Historically, elliptic cohomology arose from the study of elliptic genera...
. 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.