Quasi-invariants and integrable quantum mechanical systems

Symmetries, in the sense of being invariant under some form of transformation (e.g. rotations or translations), are of central importance for the natural sciences. This is particularly true in quantum mechanics, where symmetries already at an early stage were used to formulate and study key models and concepts. A striking example is Wolfgang Pauli’s exclusion principle, awarded the Nobel prise in 1945, which is closely connected with symmetry under permutations of particle positions.

Within quantum mechanics, models that can be solved exactly, that is to say without approximations and computer simulations, are exceedingly rare. Therefore, it came as a great surprise when Italian physicist Francesco Calogero in 1971 discovered a non-trivial model of particles moving in one dimension that can be solved exactly. It was later discovered that Calogero’s model has a number of “hidden” symmetries, responsible for its solvability, which can be described by so-called quasi-invariant polynomials. The concept of quasi-invariants was introduced relatively recent by mathematicians Oleg Chalykh and Alexander Veselov with the stated purpose of describing this type of symmetries of so called integrable (or solvable) quantum mechanical models. To date, only a somewhat indirect description of quasi-invariants is available and the corresponding symmetries are thus shrouded in mystery.

In this project, you will explore quasi-invariants and their connection with integrable quantum mechanical systems. The specific focus of the project can be tailored towards your (mathematical) interests. For example, you could focus on algebraic and geometric properties of quasi-invariants, with fascinating connections to commutative algebra, so-called Cherednik algebras and more or delve deeper into the theory of Calogero(-Moser) quantum many-body systems and their solutions.