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• The paper is just 13 pages long and its list of references contains only a single entry.
• The paper purports to contain the final piece of a jigsaw puzzle that mathematicians have been picking away at for more than half a century: the invariant subspace problem.

Per Enflo: mathematics, music, and a live goose

• By solving the approximation problem, Enflo cracked an equivalent puzzle called Mazur’s goose problem.
• Polish mathematician Stanisław Mazur had in 1936 promised a live goose to anyone who solved his problem – and in 1972 he kept his word, presenting the goose to Enflo.

What’s an invariant subspace?

• But what about the invariant subspace problem itself?
• Another way to think about this is to say that the matrix transforms the eigenvectors (and any lines parallel to them) back onto themselves: these lines are invariant for this matrix.
• Taken together, we call these lines invariant subspaces of the matrix.

What about spaces with an infinite number of dimensions?

• The invariant subspace problem is a little more complicated: it is about spaces with an infinite number of dimensions, and it asks whether every linear operator (the equivalent of a matrix) in those spaces must have an invariant subspace.
• Mathematicians working on this problem narrowed their focus by restricting the problem to particular classes of spaces and operators.
• He answered the problem in the negative, by constructing an operator on a Banach space without a non-trivial invariant subspace.

What’s new about this new proposed solution?

• Resolving the invariant subspace problem for operators on Hilbert spaces has been stubbornly difficult, and it is this which Enflo claims to have achieved.
• This time Enflo answers in the affirmative: his paper argues that every bounded linear operator on a Hilbert space does have an invariant subspace.

Expert review is still to come

• Peer review of Enflo’s earlier proof, for Banach spaces in general, took several years.
• However, that paper ran to more than 100 pages, so a review of the 13 pages of the new paper should be much speedier.

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• "On behalf of everyone at Aspirion, we welcome Amy to the team and offer our thanks to interim-CEO, Linden Operating Partner, and board member Miki Kapoor."