Martin Vrabec

Project Title: Relativistic integrable systems and related structures

Integrable Hamiltonian systems describe such interacting physical systems for which precise information on their behaviour can be derived. For instance, in classical mechanics, an integrable system of interacting particles should admit a number of conserved quantities which can be used to find trajectories in a fairly explicit form. Integrable systems are rare, and they tend to point to important mathematical structures. Thus, they largely triggered the formulation and investigations of Double Affine Hecke Algebras (DAHAs) about 25 years ago, which is an important and rich area of mathematics nowadays.

This project aims to uncover and study new integrable systems and related algebraic structures. The project largely deals with relativistic systems, which have a strong motivation in physics since meaningful physical theories such as, for instance, electromagnetism, tend to be relativistically invariant. The project mainly deals with quantum mechanical systems but includes some analysis of classical mechanical systems, too. The integrable systems that we are interested in are related to the Calogero-Moser system, which describes pairwise interaction of particles on the line with inverse square distance potential. Its relativistic version, known as Macdonald-Ruijsenaars system, is closely related to DAHAs.