Douglas Ravenel
   HOME

TheInfoList



OR:

Douglas Conner Ravenel (born 1947) is an American
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
known for work in
algebraic topology Algebraic topology is a branch of mathematics that 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 classif ...
.


Life

Ravenel received his PhD from
Brandeis University Brandeis University is a Private university, private research university in Waltham, Massachusetts. Founded in 1948 as a nonsectarian, non-sectarian, coeducational institution sponsored by the Jews, Jewish community, Brandeis was established on t ...
in 1972 under the direction of
Edgar H. Brown, Jr. Edgar Henry Brown, Jr. (December 27, 1926 – December 22, 2021) was an American mathematician specializing in algebraic topology, and for many years a professor at Brandeis University. Life Brown was born in Oak Park, Illinois, Oak Park, Illin ...
with a thesis on exotic characteristic classes of spherical fibrations. From 1971 to 1973 he was a C. L. E. Moore instructor at the
Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private Land-grant university, land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern t ...
, and in 1974/75 he visited the
Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...
. He became an assistant professor at
Columbia University Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manha ...
in 1973 and at the
University of Washington The University of Washington (UW, simply Washington, or informally U-Dub) is a public research university in Seattle, Washington. Founded in 1861, Washington is one of the oldest universities on the West Coast; it was established in Seat ...
in
Seattle Seattle ( ) is a port, seaport city on the West Coast of the United States. It is the county seat, seat of King County, Washington, King County, Washington (state), Washington. With a 2020 population of 737,015, it is the largest city in bo ...
in 1976, where he was promoted to associate professor in 1978 and professor in 1981. From 1977 to 1979 he was a
Sloan Fellow The Sloan Fellows program is the world's first mid-career and senior career master's degree in general management and leadership. It was initially supported by a grant from Alfred P. Sloan, the late CEO of General Motors, to his alma mater, MI ...
. Since 1988 he has been a professor at the
University of Rochester The University of Rochester (U of R, UR, or U of Rochester) is a private university, private research university in Rochester, New York. The university grants Undergraduate education, undergraduate and graduate degrees, including Doctorate, do ...
. He was an
invited speaker at the International Congress of Mathematicians This is a list of International Congresses of Mathematicians Plenary and Invited Speakers. Being invited to talk at an International Congress of Mathematicians has been called "the equivalent, in this community, of an induction to a hall of fame." ...
in
Helsinki Helsinki ( or ; ; sv, Helsingfors, ) is the Capital city, capital, primate city, primate, and List of cities and towns in Finland, most populous city of Finland. Located on the shore of the Gulf of Finland, it is the seat of the region of U ...
, 1978, and is an editor of
The New York Journal of Mathematics The ''New York Journal of Mathematics'' is a peer-reviewed journal focusing on algebra, analysis, geometry and topology. Its editorial board, , consists of 17 university-affiliated scholars in addition to the Editor-in-chief. Articles in the ''New ...
since 1994. In 2012 he became a fellow of the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...
. In 2022 he received the Oswald Veblen Prize in Geometry.


Work

Ravenel's main area of work is
stable homotopy theory In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the ...
. Two of his most famous papers are ''Periodic phenomena in the Adams–Novikov spectral sequence'', which he wrote together with Haynes R. Miller and
W. Stephen Wilson W. Stephen Wilson is a mathematician based in Johns Hopkins University specializing in homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a top ...
(
Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
106 (1977), 469–516) and ''Localization with respect to certain periodic homology theories'' (
American Journal of Mathematics The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United ...
106 (1984), 351–414). In the first of these two papers, the authors explore the stable
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 ...
by analyzing the E^2-term of the
Adams–Novikov spectral sequence In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is no ...
. The authors set up the so-called chromatic spectral sequence relating this E^2-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 In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. ...
. 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
Jack Morava Jack Johnson Morava is an American homotopy theorist at Johns Hopkins University. Education Of Czech and Appalachian descent, he was raised in Texas' lower Rio Grande valley. An early interest in topology was strongly encouraged by his paren ...
and themselves on Brown–Peterson cohomology and
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. For every prime number ''p'' (which is su ...
. In the second paper, Ravenel expands these phenomena to a global picture of stable homotopy theory leading to the
Ravenel conjectures In mathematics, the Ravenel conjectures are a set of mathematical conjectures in the field of stable homotopy theory posed by Douglas Ravenel Douglas Conner Ravenel (born 1947) is an American mathematician known for work in algebraic topology. ...
. In this picture,
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 i ...
and Morava K-theory control many qualitative phenomena, which were understood before only in special cases. Here Ravenel uses localization in the sense of
Aldridge K. Bousfield Aldridge Knight Bousfield (April 5, 1941 – October 4, 2020), known as "Pete", was an American mathematician working in algebraic topology, known for the concept of Bousfield localization. Work and life Bousfield obtained both his undergrad ...
in a crucial way. All but one of the Ravenel conjectures were proved by Ethan Devinatz,
Michael J. Hopkins Michael Jerome Hopkins (born April 18, 1958) is an American mathematician known for work in algebraic topology. Life He received his PhD from Northwestern University in 1984 under the direction of Mark Mahowald, with thesis ''Stable Decomposit ...
and Jeff Smith not long after the article got published.
Frank Adams John Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory. Life He was born in Woolwich, a suburb in south-east London, and attended Bedford School. He began research ...
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 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 Michael Hill and Michael Hopkins the
Kervaire invariant In mathematics, the Kervaire invariant is an invariant of a framed (4k+2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a spher ...
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, colloquially known among topologists respectively as the green and orange books (though the former is no longer green, but burgundy, in its current edition).


Selected work


''Complex cobordism and the stable homotopy groups of spheres''
Academic Press 1986, 2nd edition, AMS 2003, onlin



Princeton, Annals of Mathematical Studies 1992


External links

* at the University of Rochester *


References

{{DEFAULTSORT:Ravenel, Douglas 1947 births 20th-century American mathematicians 21st-century American mathematicians Living people Fellows of the American Mathematical Society Topologists Brandeis University alumni University of Rochester faculty Columbia University faculty University of Washington faculty Sloan Research Fellows Institute for Advanced Study people Massachusetts Institute of Technology School of Science faculty