Workshop on Integrable Systems

The University of Sydney

2 December 2013

Programme
The Workshop will be held on 2 December 2013 at University of Sydney, Carslaw building, lecture room 360.
9 - 9:10 Nalini Joshi   Introduction
9:10 - 9:40 Kenji Kajiwara   Discrete mKdV and discrete sine-Gordon flows on discrete space curves

We report some recent results on the deformation of discrete space curves described by the discrete mKdV or the discrete sine-Gordon equations. We show that it is formulated as the torsion-preserving isoperimetric and equidistant deformation on the osculating plane. This is a collaborated work with J. Inoguchi, N. Matsuura and Y. Ohta.

9:45 - 10:15 Hidetaka Sakai   Isomonodromic deformation and 4-dimensional Painlevé type equations
10:20 - 10:50 Yasuhiko Yamada   Lax pairs for quantum Painlevé equations and their solutions

We will discuss quantized Lax pairs for Painlevé and Garnier equations. Using their relation to the conformal field theory, we formulate integral formulae for some special solutions of the Lax linear problems. General series solutions will be also considered.

10:50 - 11:20 coffee break
11:20 - 11:50 Peter Vassiliou   On Einstein metrics and the n-wave resonant interaction system
11:55 - 12:25 Geoff Prince   Solvable structures and the integration of PDEs

The integrability of some classes of PDEs in discussed in terms of the Frobenius integrability of the dual of their Vessiot distributions mediated by the presence of a solvable structure (a generalisation of group invariance). Various observations will be made about first integrals, conservation laws and travelling waves. This is work with Naghmana Tehseen at La Trobe.

12:25 - 14:15 lunch break
14:15 - 14:45 Omar Foda   Nekrasov's partition functions and minimal conformal field theories

I wish to introduce Nekrasov's partition functions and report on how they can be used to compute conformal blocks in minimal Virasoro conformal field theories. Based on work in progress with M Bershtein, Landau Institute, Moscow.

14:50 - 15:20 John Roberts   Non-QRT maps

I will discuss some characterisations and properties of birational maps that send a fibration of algebraic curves to itself (but not fibre-wise) or to another such fibration.

15:20 - 15:50 coffee break
15:50 - 16:20 Reinout Quispel   Some recent results on the Kahan discretization

The Kahan discretization for quadratic ordinary differential equations (ODEs) was introduced by Kahan, and shown to preserve the integrability of many integrable ODEs by Petrera, Pfadler and Suris. In this talk we explain and generalise some of these earlier results. (Joint work with Elena Celledoni, Robert McLachlan, David McLaren and Brynjulf Owren)

16:25 - 16:55 Holger Dullin   Is the pendulum equivalent to the reduced Euler top for some moments of inertia?

In a neighbourhood of regular Liouville tori any two Liouville integrable systems of the same dimension are equivalent in the sense that there is a symplectic diffeomorphism and a diffeomorphism that takes integrals into integrals which transforms the two systems into each other. Two Liouville integrable systems are equivalent if this equivalence holds globally. Using semi-global symplectic invariants defined near a separatrix we are going to answer the equivalence question posed in the title. (Joint work with George Papadopoulos)

Related lectures
Integrable Systems Seminar
5 December 2013 at 14h, AGR Carslaw 829

Masashi Yamaguchi   Rigidity index and q-middle convolution of q-difference equations

We consider 2 type transformation and those invariant of q-difference equations. First, We define the accessory parameters and rigidity index. Next, We define the q-convolution and q-middle convolution using gauge transformation. I will describe an analytic composition of q-convolution using Jackson integral. Moreover, I will report q-middle convolution preserves irreducibility, Fuchsian type and rigidity index. (Joint work with Hidetaka Sakai)