In mathematics, more specifically in
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 redshift conjecture states, roughly, that
algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sens ...
has chromatic level one higher than that of a complex-oriented
ring spectrum
In stable homotopy theory, a ring spectrum is a spectrum ''E'' together with a multiplication map
:''μ'': ''E'' ∧ ''E'' → ''E''
and a unit map
: ''η'': ''S'' → ''E'',
where ''S'' is the sphere spectrum. These maps have to satisfy as ...
''R''.
It was formulated by John Rognes in a lecture at
Schloss Ringberg
Schloss Ringberg (Ringberg Castle) is located in the Bavarian Alps, 50 km south of Munich, on a foothill overlooking the Tegernsee. Not open to the general public, it is a property of the Max Planck Society and used for conferences.
Histor ...
, Germany, in January 1999, and made more precise by him in a lecture at Mathematische Forschungsinstitut Oberwolfach, Germany, in September 2000. In July 2022, Robert Burklund,
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 ...
and Allen Yuan announced a solution of a version of the redshift conjecture for arbitrary
-ring spectra, after Hahn and Wilson did so earlier in the case of the truncated Brown-Peterson spectra
.
[Burklund, Schlank, Yuan (2022). ''The Chromatic Nullstellensatz'']
References
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Further reading
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External links
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Algebraic topology
Homotopy theory
Conjectures
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