Seven decades on, mathematicians finally crack tricky problem | Science News

Researchers have proven a 70-year-old problem in mathematics. The research primarily has theoretical value, but has potential applications in computational mathematics and numerical computation. The researchers are now investigating avenues for practical applications.

The researchers were attempting to tackle the problem of figuring out the shape of a drum based on the noise it makes. (Image Credit: Sam te Kiefte/Unsplash).

New Delhi: An international team of mathematicians are looking for practical applications of a proof that primarily has theoretical value. The researchers were able to prove a 70-year-old problem in mathematics, formulated in 1954 by Hungarian-American mathematician George Pólya. The conjecture is about the estimation of frequencies of a round drum, or in terms of mathematics, the eigenvalues of a disk. The proof has been provided in an article published in Inventiones Mathematicae, a mathematical journal.

Pólya himself had confirmed the conjecture in 1961, for domains that can tile a space, such as triangles and rectangles. These shapes can be placed in an array to completely fill up any space. However, the same does not hold true for disks, which are difficult to tile with. Previously the conjecture had also been proven for some special domains in higher dimensions. Till the new proof was formulated the conjecture was known to hold true for only these cases. For the first time, the researchers have proved the conjecture for a non-tiling planar domain.

One of the authors of the paper, Iosif Polterovich says, “While mathematics is a fundamental science, it is similar to sports and the arts in some ways. Trying to prove a long-standing conjecture is a sport. Finding an elegant solution is an art. And in many cases beautiful mathematical discoveries do turn out to be useful—you just have to find the right application.” While the proof has mostly theoretical value, the researchers are investigating potential practical applications, including in the fields of computational mathematics and numerical computation. The proof was also developed through computer assistance.

What is Spectral Geometry

The mathematicians work in the field of spectral geometry, which is the branch of mathematics involved with understanding physical phenomena associated with wave propagation. Spectral geometry investigates how the eigenvalues or spectra are influenced by manifolds or geometric structures. The main concern in the field is understanding how the geometric properties of the manifold influences the eigenvalues. The work can be explained simply as figuring out the shape of the drum based on the sound it makes. The researchers came across links between spectral problems in the disk and some lattice counting problems.