Per Enflo solved one of the most long-awaited mathematical problems: the problem of an invariant subspace. This puzzle, which mathematicians had been working on for more than half a century, was solved by a Swedish mathematician who had already solved the discovered problem before.
Per Enflo is a well-known concert pianist and mathematician who was born in 1944 and is now professor emeritus at Kent State University in Ohio. He has had a remarkable career not only in mathematics but also in music.
The invariant subspace problem is one of the most difficult problems in linear algebra. If you’ve ever taken a college course in linear algebra in your freshman year, you’ve encountered concepts like vectors, matrices, and eigenvectors. If not, you can think of a vector as an arrow with length and direction living in a particular vector space.
A matrix is something that can transform a vector by changing the direction and/or length of a line. If a particular matrix transforms only the length of a particular vector (which means that the direction is either the same or reversed in the opposite direction), we call the vector an eigenvector of the matrix.
Another way to think of it is to say that a matrix converts eigenvectors (and any lines parallel to them) back to itself: these lines are invariant to that matrix. Collectively, we call these lines invariant subspaces of the matrix.
The invariant subspace problem is to find an invariant subspace of the matrix. This problem is one of the most difficult problems in linear algebra, and its solution is of great importance for many fields of science and technology, including quantum mechanics.
Per Enflo solved this problem in his new paper “On the problem of invariant subspaces in Hilbert spaces”. The article is only 13 pages long, and the list of references contains only one entry.
This is a significant achievement for mathematics, which is one of the most complex and abstract sciences. However, as the experts point out, the solution of this problem will not lead to direct practical applications.
Nevertheless, the solution of the invariant subspace problem is of great importance for theoretical mathematics and can lead to new discoveries in quantum mechanics and other sciences.
Per Enflo has already solved open problems in mathematics. He had solved the basis problem and the approximation problem, which had remained open for more than 40 years. By solving the approximation problem, Enflo solved a similar puzzle called the Mazur goose problem.
Polish mathematician Stanislaw Mazur in 1936 promised a live goose to anyone who solved his problem, and in 1972 he kept his word by giving the goose to Enflo.
Per Enflo is one of the best problem solvers in functional analysis. He has made a name for himself by solving open problems and has extensive experience with the invariant subspace problem.
Solving this problem is a significant achievement for mathematics and can lead to new discoveries in quantum mechanics and other sciences. However, as noted by experts, the solution of this problem will not lead to direct practical applications.