In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Ravenel conjectures are a set of mathematical conjectures in the field of
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 F ...
posed by
Douglas Ravenel
Douglas Conner Ravenel (born 1947) is an American mathematician known for work in algebraic topology.
Life
Ravenel received his PhD from Brandeis University in 1972 under the direction of Edgar H. Brown, Jr. with a thesis on exotic characterist ...
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" being proved in later papers by others. The telescope conjecture is now generally believed not to be true, though there are some conflicting claims concerning it in the published literature, and is taken to be an open problem.
[ Ravenel's conjectures exerted influence on the field through the founding of the approach of chromatic homotopy theory.
The first of the seven conjectures, then the ''nilpotence conjecture'', was proved in 1988 and is now known as the nilpotence theorem.
The telescope conjecture, which was #4 on the original list, remains of substantial interest because of its connection with the convergence of an Adams–Novikov spectral sequence. While opinion has been generally against the truth of the original statement, investigations of associated phenomena (for a triangulated category in general) have become a research area in its own right.
On June 6, 2023, Robert Burklund, Jeremy Hahn, Ishan Levy, and Tomer Schlank announced a disproof of the telescope conjecture. Their preprint is submitted on October 26, 2023.]
See also
* Homotopy groups of spheres
References
{{Reflist
Homotopy theory
Conjectures