mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 ...

posed by

Douglas Ravenel Douglas Conner Ravenel (born February 17, 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 ...

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. Ravenel's conjectures exerted influence on the field through the founding of the approach of

chromatic homotopy theory In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theory, complex-oriented cohomology theories from the "chromatic" point of view, which is based on Daniel Quillen, Quillen's ...

. The first of the seven conjectures, then the ''nilpotence conjecture'', was proved in 1988 and is now known as the

nilpotence theorem In algebraic topology, the nilpotence theorem gives a condition for an element in the homotopy groups of a ring spectrum to be nilpotent, in terms of the complex cobordism spectrum \mathrm. More precisely, it states that for any ring spectrum R, th ...

. The telescope conjecture, which was fourth on the original list, remains of substantial interest because of its connection with the convergence of an

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 now c ...

. While opinion has been generally against the truth of the original statement, investigations of associated phenomena (for a

triangulated category In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy ca ...

in general) have become a research area in its own right. On June 6, 2023, Robert Burklund, Jeremy Hahn, Ishan Levy, and

Tomer Schlank Tomer Moshe Schlank (; born 1982) is an Israeli mathematician and a professor at The University of Chicago. Previously, he was a professor at Hebrew University of Jerusalem. He primarily works in homotopy theory, algebraic geometry, and number ...

announced a disproof of the telescope conjecture. Their preprint was submitted to the

arXiv arXiv (pronounced as "archive"—the X represents the Chi (letter), Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not Scholarly pee ...

on October 26, 2023.

References

{{Reflist Homotopy theory Conjectures

See also

References