Friday, February 1, 2013

Finding invariant subspaces

NewScientist reports:
Would a basketball spinning on a fingertip behave the same way in an infinite number of dimensions? The question has flummoxed mathematicians for 80 years, but now it looks as if the answer is yes – a find that could have implications for quantum theory.

The invariant subspace problem was studied in the 1930s by John von Neumann, a pioneer of operator theory, the mathematics behind quantum mechanics. The problem asks whether carrying out certain changes, or operations, will always leave part of an object unaltered, or invariant. In the case of the basketball, the operation is rotation. In three dimensions, the sphere's rotational axis remains unchanged, but you can't take that for granted in infinite dimensions.

On 25 January, a solution was unveiled at a meeting of the Royal Spanish Mathematical Society in La Coruña. Eva Gallardo of the Complutense University of Madrid in Spain and Carl Cowen of Purdue University in West Lafayette, Indiana, said they have proved that part of the object will always be unaltered.

No great prize rests on the proof, but, if it is correct, the method of solving it should enable innovations in operator theory, says Miguel Lacruz of the University of Seville. "Operator theory is the language of quantum mechanics," he adds.
No, this Invariant subspace problem has almost nothing to do with basketball or quantum mechanics.

This is a hard problem, so the proof might be wrong.

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