Per Enflo on September 28 in the middle of the next step on the daily walk with his wife Lena in Östervåla Uppland, under inspection of plants on the ground in the spirit of Linné and with inspiration from the sky finally closing his constructive proof of the Invariant Subspace Problem for Hilbert Spaces, took a last breath sending a shock wave to family, friends and mathematical community. Per Enflo is certainly the most famous Swedish mathematician all times by having solved named major problems, also concert pianist expressing the true meaning of the music of Mozart, Beethoven, Schubert and Chopin.
I met first Per during my post doc years 1974-76 at the math department of the University of Chicago, when Per visited as the new shining star of Functional Analysis, with offers from all the big universities, after having solved one of the key open problems in that area formulated by its founder Stefan Banach in the 1930s, as a 9 page Counterexample to the Approximation Problem in Banach Spaces published in Acta Math in 1973, which took the math community with storm. Watch the documentary movie about Banach with Per in the main role (and me with little side role) and Per's home page.
Per then followed up in 1981 with an 101 page counterexample to the Invariant Subspace Problem in Banach Spaces, published in Acta Math after 6 years of refereeing, to return 40 years later to the case of Hilbert spaces with an explicit construction of an invariant subspace.
It took 43 years before we met again, in Stockholm in 2008 when Per had rejoined with his love from youth Lena and returned to Sweden after 25 years in the US, and Per welcomed me and my wife Ingrid to his piano trio concert at the Mazer Musical Society. We found each other on the spot into a 20 year long friendship along with our wifes, with music, math and love. It is very sad that Per with his very kind person and amazing talent is no longer here. In the Swedish math community we shared experiences of exclusivity as a special bond.
Per was a master of constructive mathematics, constructing an invariant subspace for any continuous linear operator T in Hilbert space H, by constructing a sequence of vectors converging to a vector for which repeated application of T does not span the whole space and thus forms an invariant subspace. Per was also a master of constructive music as combination of body/hand and soul/mind with ability to on the spot transpose any given sheet music to any key. We could meet in hands-on constructive math/piano but also lofty speculation, never any argument.
On Sept 8 and 26 another shock wave into the math community had been sent by OpenAI as constructive proofs of blow-up of solutions to the equations of Navier-Stokes (167 pages) and Euler (57 pages) performed by AI.
Per expressed that he had been lucky to not meet this seemingly formidable competition before, with AI able to construct proofs of any number of pages, beyond understanding by human mathematicians. Per could thus pass on to a heaven of math without shattered beliefs with his usual happy "that is also ok" and with his Musical Legacy completed in 15 CD on Spotify.
Pictures from Aug 20 2026:
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