New results in operator theory

Alfonso Montes Rodríguez is currently professor at Universidad de Sevilla, Spain. 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.

Montes Rodríguez is an expert in mathematical analysis, a branch of mathematics in which finding methods to study differential equations is of great significance. Differential equations play a key role in formulating the laws of physics.  

The research proposed in the current project will center on operator theory and link to several other areas of mathematics, including dynamical systems, complex analysis, and harmonic analysis. Some of the questions studied are closely related to physics, for example Heisenberg’s uncertainty principle. This is the cornerstone of quantum physics, postulating that it is not possible to determine the exact position and momentum of a particle at the same time.

Montes Rodríguez and Håkan Hedenmalm at KTH have shown, in a previous joint publication, that the Perron-Frobenius operators arise naturally from the analysis of solutions to the one-dimensional Klein-Gordon’s equation. The equation describes a relativistic quantum particle with no spin. One of the goals of this project is to apply operator theory and so generalize such results by analyzing solutions of differential equations in several variables. In order to succeed, it will be necessary to develop new methods, as well as to find creative ways to extend and mesh the old ones in novel ways.

The second goal of the project is an attempt to prove one of the most famous problems in complex analysis, the Brennan conjecture, by applying methods of operator theory. Other researchers in the Department of Mathematics at KTH are also expected to participate in this effort.

Photo: CAS Oslo