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Conjecture

Spectral Gap Conjecture

Analysis

The spectral gap conjecture, in ergodic theory, is a statement of Alexander Lubotzky, Ralph S. Phillips, and Peter Sarnak on the spectral gaps of certain actions of a free group on the two-dimensional sphere. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Statement
For any integer n greater than 2, if n isometries are chosen uniformly at random, then the operator formed by summing each isometry and its inverse has a nontrivial spectral gap with probability 1. 1
Progress Toward Resolution
It is known that the answer is all or nothing, either a nontrivial spectral gap holds with probability 1 or the spectral gap is trivial with probability 1, and the full conjecture remains open. 1
Partially Attested
Proposed Year
1987 2
1987 is the year of the Lubotzky, Phillips and Sarnak paper the article cites for the conjecture; the article has no sentence stating in words the year the conjecture was first stated.
Classification
Resolution Status
Partially Resolved 1
Prize Status
Prize Status (category)
No Prize Offered 2
Connections

In Branch

Ergodic Theory, Branches of Mathematics
Source Spectral Gap Conjecture (Wikipedia)
Sources
1. Spectral Gap Conjecture (Wikipedia)
  • Formal statement section
    For any integer n greater than 2, if n isometries are chosen uniformly at random, then the operator formed by summing each isometry and its inverse has a nontrivial spectral gap with probability 1.
  • Current status section
    It is known that either there is a nontrivial spectral gap with probability 1 or that the spectral gap is trivial with probability 1.
View the Source
2. Spectral Gap Conjecture (Wikipedia)
Wikimedia Foundation
  • Lead section
    the spectral gap conjecture of Alexander Lubotzky, Ralph S. Phillips, and Peter Sarnak is a statement on the spectral gaps of certain actions of a free group on the sphere
  • References section, Lubotzky, Phillips and Sarnak 1987 entry
    Lubotzky, A.; Phillips, R.; Sarnak, P. (1987)
  • In Branch: Ergodic Theory, Lead sentence
    In ergodic theory, a branch of mathematics, the spectral gap conjecture of Alexander Lubotzky, Ralph S.
View the Source
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