Dr. Cheng Zhang

Shanghai University, China

On the inverse scattering transform for integrable PDEs on the half-line

In this talk, we provide an inverse scattering transform for integrable PDEs on the half-line. The method is based on the notions of integrable boundary conditions as well as the Sklyanin-type double-row monodromy matrix. Taking the nonlinear Schr”odinger (NLS) equation as our primary example. Following the double-row monodromy matrix formalism, integrable boundary conditions for NLS are encoded into constraints between the so-called reflection matrices and the time-part of the Lax pair of NLS. This gives rise to a hierarchy of reflection matrices, which is accompanied by a hierarchy of integrable boundary conditions. Then, by establishing a scattering system for the double-row monodromy matrix for NLS on the half-line, we obtain possible analytic and spectral properties of the scattering functions. This allows us to set up the inverse description of the half-line problem with integrable boundary conditions in terms of a Riemann-Hilbert problem. Solutions to the Riemann-Hilbert problems lead to solutions of NLS on the half-line subject to arbitrary-order integrable boundary conditions. In particular, multi-soliton solutions can be derived for the model.

Thu 6 May 2021 12PM Sydney time

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Prof. Jan De Gier

The University of Melbourne, Australia

Transition probabilities and asymptotics for integrable two-species stochastic processes

I will discuss exact, multiple integral formulas for the transition
probability (Green’s function) of two different integrable two-species
stochastic particle models: the Arndt-Heinzel-Rittenberg (AHR) model
and the 2-TASEP whose generator is the $q\rightarrow 0$ limit of the
R-matrix related to $U_q(sl(3))$. We derive closed form formulas for
total crossing probabilities. In the case of the AHR I will sketch how an
asymptotic analysis of these expressions leads to a rigorous derivation
of universal hydrodynamic probability distribution functions. The latter
lie in the KPZ universality class and are related to distributions from
random matrix theory.

This is work in collaboration with Zeying Chen, Iori Hiki, William Mead,
Masato Usui, Michael Wheeler and Tomohiro Sasamoto.

Thu 22 April 2021 12PM Sydney time

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Dr. Shi-Hao Li

Sichuan University, China

Condensation algorithm of Pfaffians and discrete integrable system

Dodgson’s condensation algorithm of determinants is an efficient way to compute the value of determinant. In recent years, it was realised that the Dodgson’s condensation algorithm is actually a full discrete Toda lattice in 3-dimension with proper initial values. However, how to design a condensation algorithm of Pfaffians is still unknown. In this talk, I will present several different condensation algorithms of Pfaffians and some related discrete integrable systems will be introduced.

Tue 06 April 2021 1PM Sydney time

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Prof. Di Yang

USTC

On tau-functions for the KdV hierarchy

For an arbitrary solution to the KdV hierarchy, we give an expression of
the generating series of logarithmic derivatives of the tau-function of the
solution explicitly in terms of the so-called basic matrix resolvent. Based
on this we develop two new formulae for the generating series by
introducing a pair of wave functions of the solution. Applications to the
Witten–Kontsevich tau-function, to the generalized BGW tau-function,
as well as to a modular deformation of the generalized BGW
tau-function will be given.

*The talk is based on a series of joint works
with Marco Bertola, Boris Dubrovin and Don Zagier.*

Thursday 25 March 2021, 10AM Beijing Time

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Dr. D. Croydon

Kyoto University

Invariant measures for KdV and Toda-type discrete integrable systems

I will describe how to construct infinite versions of four
well-studied discrete integrable models (namely the ultra-discrete KdV equation,
the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation),
and discuss some arguments that are useful in identifying invariant measures for them.
In each case, the current configuration is represented as a certain path encoding,
and the dynamics of the system can be understood in terms of a version of Pitman’s
transformation (i.e. reflection in the past maximum), which is well-known amongst probabilists.
For identifying spatially homogeneous and independent random configurations that are invariant
under the dynamics in particular, I will present a ‘detailed balance’ criterion,
which is also relevant for various stochastic integrable systems.

*This talk is based on joint work with Makiko Sasada (University of Tokyo)
and Satoshi Tsujimoto (Kyoto University).*

Tuesday 9 March 2021, 1PM Sydney Time

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Dr. Gleb A. Kotousov

DESY, Theory Group

Density of states for an integrable spin chain

The focus of the talk is a certain critical 1D quantum spin chain that lies within the integrability class of a staggered six-vertex model. In the scaling limit the spin chain exhibits a continuous spectrum of conformal dimensions, which is characterised by a density of states. The latter was determined in the recent work arXiv:2010.10613. It turns out to involve the eigenvalues of a non-local integral of motion, the so-called reflection operator, from the quantum AKNS integrable structure. Apart from recounting the relevant results of that paper, I will briefly discuss their applications to the study of the non-compact CFT whose target space is the 2D Euclidean black hole (Hamilton’s cigar).

Thursday 25 February 2021, 1PM Sydney Time

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APIOS will be go on pause and resume in mid-February 2021. If you’re interested to give a talk or you would like to suggest a speaker, please contact one of our organisers. We will announce the first speakers in mid-January 2021.

Since in the first quarter of international travel will still be difficult we will continue to held the seminars fortnightly.

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Prof. Changzheng Qu

Ningbo University

Integrable Systems and Invariant Geometric Flows in affine-related Geometries

Invariant geometric flows in certain geometries have been studied extensively from different points of view. In this talk, we are mainly concerned with invariant geometric flows in centro-affine, centro-equiaffine and affine geometries etc. First, we show that the specific invariant geometric flows in those geometries are related respectively to the well-known integrable systems. Those integable geometric flows corresponding to some solutions of integrable systems will be studied. Second, the geometric formulations to integrability features of the resulting systems are investigated. Third, the geometric flows corresponding to the Camassa-Holm-type are also presented.

Thursday 26 November 2020 10AM Beijing time

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Prof. Artur Sergyeyev

Silesian University in Opava

Integrable (3+1)-dimensional systems from contact geometry

We present a large new class of (3+1)-dimensional integrable systems
using a novel kind of Lax pairs related to contact geometry, thus
showing inter alia that there is significantly more of such systems
than it appeared before. In particular, the class in question contains
two new explicit infinite families of (3+1)-dimensional integrable
systems and a first example of a (3+1)-dimensional system with a
nonisospectral Lax pair which is algebraic rather than rational in the
spectral parameter.

**References**

- A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry,
*Lett. Math. Phys.***108**(2018), no. 2, 359-376, arXiv:1401.2122 - A. Sergyeyev, Integrable (3+1)-dimensional system with an algebraic Lax pair,
*Appl. Math. Lett.***92**(2019), 196–200, arXiv:1812.02263

Thursday 10 November 2020, 1PM Sydney time

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Dr. Vera Roshchina

University of New South Wales

Faces of convex sets: dimensions and regularity

The facial structure of convex sets can be surprisingly complex, and unexpected irregularities of the arrangements of faces give rise to badly behaved sets. I will focus on specific properties of facial structure that capture irregularities (dimensions of faces, singularity degree, facial exposure and facial dual completeness). I will also talk about some classic results related to faces of convex sets, mention new results and counterexamples and will relate this to some open problems.

Thursday 29 October 1PM Sydney time

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