Per Enflo 1944 - 2026 Mathematician and Pianist

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 H 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:

 

Banach and DFS and Clay Navier-Stokes Problem

This is an exercise in preparation for participation in a film about the Polish mathematician Stefan Banach who advanced functional analysis as mathematics describing relations between functions or analogies between analogies. My punch line is that the finite element method, as the subject of my work, is (nothing but) computational functional analysis following the spirit of Banach.

The crown of my work, together with Johan Hoffman and Johan Jansson, is Direct Finite Element Simulation DFS as solution of the Navier-Stokes-Euler equations without turbulence model or complicated wall model from a principle of best possible solution, in a situation where there is no exact solution. DFS brings revolutionary new capacity to Computational Fluid Dynamics CFD, which we (as a show case) claim resolves the Clay Navier-Stokes Problem by computation.

Functional analysis was formed by the mathematician Hilbert at the switch to modernity around 1900, with contributions from the Swedish mathematician Fredholm, and was further developed by Banach starting in 1920.  A prime objective was to justify mathematical models in the form of partial differential equations of solid and fluid mechanics and electromagnetics formulated during the 19th century by Laplace, Fourier, Navier, Stokes and Maxwell, by answering basic questions concerning existence and uniqueness of solutions, as well a construction of solutions by computation.

The basic element of functional analysis is a collection of functions named Hilbert space or Banach space equipped with a structure or geometry generalising that of ordinary three dimensional space. The solution of a given partial differential equation is then an element of a suitably chosen Hilbert or Banach space in basic cases determined by a principle of energy minimisation. The differential equation, which is impossible to solve directly by symbolic computation with pen and paper,  is thus reformulated into a minimisation problem over a function space, which allows construction of solutions as a limits of functions with decreasing energy computed according to the Banach Contraction Mapping Theorem.

Starting in the 1950s this form of computational functional analysis has been developed under the name of the finite element method into a universal method for computing solutions of the differential equations of science and engineering bringing revolutionary new capacities.  This success story was darkened only by Navier-Stokes-Euler equations of fluid mechanics, which were believed to demand computational power beyond anything which could be envisioned, the reason being the phenomena of turbulence and thin boundary layers involving small scales too costly to resolve computationally, the impossibilities presented in NASA CFD Vision 2030.

We show that with DFS the NASA CFD Vision 2030 is realised already today. By computational functional analysis in the spirit of Banach.

DFS and functional analysis gives a new perspective on differential equations representing ideal physics, however with uncomputable or non-existing exact solutions as in the case of Navier-Stokes-Euler,  and reformulations in terms of functional analysis with computable approximate solutions representing real physics.