Prof. B. Xue

Zhengzhou University

Integrable Dynamic Systems with N-peakons

Since the discovery of Camassa-Holm equation, because of the special properties that peakon gets, it has received considerable attention in modern Mathematics and Physics. Many new integrable dynamic systems with N-peakon have been obtained, for instance, the DP equation, the Novikov equation, etc. In this talk, we will introduce the basic definition and some characters of peakon. Then some newly derived integrable dynamic systems with N-peakon will be presented. Meanwhile, new developments and hot points in the associated field are discussed.

October 01 2020, 10:00AM Beijing Time

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Prof. X. Chang

Chinese Academy of Sciences

On Frobenius-Stickelberger-Thiele polynomials and modified Camassa-Holm peakon lattice

In this talk, we will introduce some basic properties of the so-called Frobenius-Stickelberger-Thiele (FST) polynomials and highlight its roles in solving the multipeakons of the modified Camassa-Holm equation by use of inverse spectral method. The talk is based on joint works with Xingbiao Hu, Jacek Szmigielski and Alexei Zhedanov.

13 October 2020 10:00 AM Beijing Time

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Prof. Yuma Mizuno,

Tokyo Institute of Technology

Difference equations arising from cluster algebras

The theory of cluster algebras gives powerful tools for systematic studies of discrete dynamical systems. Given a sequence of quiver mutations that preserves the quiver, we obtain a finite set of algebraic relations, yielding a discrete dynamical system. Such a set of algebraic relations is called a T-system. In this talk, I will explain that T-systems are characterized by pairs of matrices that have a certain symplectic property. This generalize a characterization of period 1 quivers, which was given by Fordy and Marsh, to arbitrary mutation sequences. I will also explain the relation between T-systems and Nahm’s problem about modular functions, which is one of the main motivations of our study.

Tuesday 15 September 2020, 11:00AM Tokyo time.

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Prof. Atsuo Kuniba

The University of Tokyo

Generalized hydrodynamics for randomized box-ball system

Box-ball system (BBS) is a prominent example of soliton cellular automaton in one dimension.
By now its integrability has been well understood from the viewpoint of quantum groups, Bethe ansatz,
ultradiscretization and tropical geometry. In the last few years, randomized version of BBS has attracted
much attention in the light of generalized Gibbs ensemble and generalized hydrodynamics.
In this talk I will report on recent results in this direction. They include, as long as time permits,
limit shape of rigged configurations, exact solution to the speed equation, connection to the period matrix
of a tropical Riemann theta function, explicit description of density plateaux emerging from a domain wall
initial condition including their diffusive broadening.

The content is based on a joint work arXiv:2004.01569 with Grégoire Misguich and Vincent Pasquier.

Thursday 03 September 2020, 11:00AM Tokyo time.

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A/Prof. Milena Radnovic

University of Sydney, Australia

Integrable billiards and extremal polynomials.

In this talk, we will present a novel relationship between periodic trajectories of ellipsoidal billiards and the theory of generalised Chebyshev polynomials on systems of segments. Using that relationship, we prove fundamental properties of billiard dynamics, such as monotonicity of winding numbers and injectivity of the frequency map.

The results are obtained jointly with Vladimir Dragovic.

Tuesday 18 August 2020, 12:00PM Sydney time

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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 Painlevé 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, 12PM 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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