Dr. Ian Marquette

University of Queensland, Australia

Construction of polynomial algebras related to superintegrable systems

Over the years, it has been discovered that symmetry algebras of superintegrable systems take the form of polynomial algebras. For examples, the integrals of 2D superintegrable models related to fourth and sixth PainlevĂ© transcendent lead to cubic algebras.

In recent years some progress have been made on particular cases of n-dimensional systems with quadratic integral of motion and their symmetry algebras, referred as higher rank quadratic algebras. Among them deformed Kepler-Coulomb systems, singular oscillator, generic model on n-sphere and on the pseudo sphere have been studied. It has been pointed out how these algebraic structures can allow to obtain the energy spectrum via deformed oscillator and Casimir invariants. However, these approaches rely on explicit realizations as differential operators. I will discuss how an algebraic construction of the symmetry algebra of the generic superintegrable systems on the 2-sphere can be generated from an underlying Lie algebra connected with intertwining operators. We obtain a cubic algebra which can be reduced to a quadratic algebra using Casimir invariants.

Tuesday 04 August 2021, 12AM Brisbane Time.

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Prof. Youjin Zhang

Tsingua University, China

Special Cubic Hodge Integrals and the Fractional Volterra Hierarchy

We show that the generating function of cubic Hodge integrals satisfying the local Calabi-Yau condition is the tau function of a particular solution of an integrable hierarchy called the fractional Volterra hierarchy. This integrable hierarchy is a certain generalization of the Volterra lattice hierarchy (also called the discrete KdV hierarchy) which is well known in the theory of nonlinear integrable systems.

Thursday 21 Jul 2020, 10AM Beijing Time.

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Prof. Emer. Masatoshi Noumi

Kobe University, Japan

Perturbative eigenfunctions for the elliptic Ruijsenaars difference operators

I will report recent progresses in the study of joint eigenfunctions for the commuting family of elliptic Ruijsenaars difference operators, on the basis of collaboration with Edwin Langmann (KTH, Sweden) and Junichi Shiraishi (Univerisity of Tokyo, Japan). We propose in particular two classes of perturbative eigenfunctions which are elliptic deformations of Macdonald polynomials and asymptotically free eigenfunctions in the trigonometric case, respectively.

Thursday 09 Jul 2020, 11AM Tokyo Time.

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