New Discoveries on the Way to Infinity

Arno Kuijlaars is currently professor at the KU Leuven, Belgium. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will be a visiting professor at the Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden.

Arno Kuijlaars is one of the leading experts on asymptotic methods, which are methods that are used to analyze the long-term behavior of a system. The main goal of the project is the analysis of asymptotic behavior in certain types of random matrices.

Matrices are rectangular arrays of numbers, arranged in rows and columns. They were first used to solve systems of equations in ancient China and are widely used in many branches of mathematics, as well as being a key tool in the application of mathematics to many other fields of research, including physics and statistics.

For a given square matrix one can compute its eigenvalues, a set of numbers that are important in applications. For example, eigenvalues represent resonance frequencies in mechanical systems, as is the case for a vibrating string and its harmonics. When constructing a bridge, engineers need to know eigenvalues to assess the potential damage due to resonance. In quantum physics, the energy levels of atoms and molecules are determined by their eigenvalues.

Even though the significance of matrices lies mostly in their applications, they have become objects of mathematical research in themselves. There is often only incomplete information about the matrix elements, which can then be regarded as random numbers and leads to the theory of random matrices and the study of their eigenvalues, which are also random. The behavior of the eigenvalues is often the subject of interest, as the size of the matrix goes to infinity. This leads to often difficult questions in asymptotic analysis. The goal of the project is further investigation in this area.

Photo: KU Leuven