UK-Japan Winter School

**Integrable Systems and Symmetry**

**University of Manchester, 7-10 January 2010**

LECTURE COURSES

**Darryl Holm (Imperial College London):** *The shape of water, metamorphosis and infinite-dimensional geometric mechanics* (Lecture notes).
Whenever we say the words "fluid flows" or "shape changes" we enter the realm of infinite-dimensional geometric mechanics. Water, for example, flows. In fact, Euler's equations tell us that water flows a particular way. Namely, it flows to get out of its own way as adroitly as possible. The shape of water changes by smooth invertible maps called diffeos (short for diffeomorphisms). The flow responsible for this optimal change of shape follows the path of shortest length, the geodesic, defined by the metric of kinetic energy. Not just the flow of water, but the optimal morphing of any shape into another follows one of these optimal paths. These lectures will be about the commonalities between fluid dynamics and shape changes and will be discussed in the language most suited to fundamental understanding - the language of geometric mechanics. A common theme will be the use of momentum maps and geometric control for steering along the optimal paths using emergent singular solutions of the initial value problem for a nonlinear partial differential equation called EPDiff, that governs metamorphosis along the geodesic flow of the diffeos. The main application will be in the registration and comparison of Magnetic Resonance Images for clinical diagnosis and medical procedures.

**Alexandre Mikhailov (University of Leeds):** *Symmetries and classification of integrable nonlinear PDEs* (Lecture notes).
The classification stage is a very important and usually very difficult one in any area of research. The difficulty with integrable systems is that the very notion of integrability is not well defined even in the finite dimensional case, let alone for PDEs. One of the natural approaches is based on the idea of symmetry and turns out to be very fruitful. The course will be a review of various interesting ideas and tools used in the recent classification of integrable nonlinear PDEs, coming from differential algebra and classical invariant theory.

**Alexander Veselov (Loughborough University):** *Yang-Baxter maps and discrete integrability* (Lecture notes).
In contrast to the usual integrable dynamical systems and PDE's the discrete integrable systems until recently attracted little attention. Over the last decade the situation has changed completely. The discrete level is recognised to be more fundamental in many aspects than the continuous one. The links with classical geometry as well as modern algebraic structures made this area one of the most exciting of integrable systems. One of the new important notions here is the Yang-Baxter map, which is a map of the Cartesian square X ×X of a set X into itself, satisfying the Yang-Baxter relation.
The lecture course is an introduction
to the modern ideas of discrete integrability with its rich geometric links.

GUEST LECTURES

**Paulo Assis (University of Kent):** *Non-Hermitian Hamiltonians in field theories* (Lecture notes)

**Andy Hone (University of Kent):**
*Darboux transformations, Backlund transformations and symplectic correspondences*
(Lecture notes).
The KdV equation, with its infinite hierarchy of higher symmetries, is the archetypal example of an integrable partial differential equation. The key to the integrability of KdV is the existence of an associated linear system (Lax pair), part of which is a Schrodinger equation. The Schrodinger operator is covariant under the action of the Darboux transformation. Exploiting this covariance leads to the Backlund transformation for KdV, which can be used to generate sequences of solutions (in particular, multi-solitons) from a given seed.
The stationary or restricted flows of the KdV hierarchy provide examples of integrable finite-dimensional Hamiltonian systems, including Garnier systems, Neumann systems and Henon-Heiles systems. The goal of this talk is to describe the Darboux transformation for KdV, and explain how one can use it to obtain symplectic maps (or better to say, correspondences) that discretize these finite-dimensional systems. The integrable discretizations so obtained have special properties that characterize them as BTs (Backlund
transformations) for finite-dimensional systems, in the sense introduced by Kuznetsov and Sklyanin.

**Sara Lombardo (University of Manchester and VU University Amsterdam):**
*Classical invariant theory and automorphic Lie algebras*.
Symmetry has a very wide appeal: it is of much interest to mathematicians and physicists. In this talk we move from a geometrical idea of symmetry, namely symmetry groups of Platonic solids, to the algebraic concept of invariance. In particular, we first introduce the concept of automorphic Lie algebra and we then show that they can be obtained in a uniform way using the classical theory of invariants. It follows that sl_{2}-automorphic Lie algebras associated to the finite groups are isomorphic. The proof makes use of Clebsch-Gordan decomposition and transvectants, Molien functions and the trace form. The result is a crucial step towards the complete classification of sl_{2}-automorphic Lie algebras associated to finite groups. If time allows, some remarks will be made concerning the McKay correspondence.

**Reiko Miyaoka (Tohoku University):**
*Hypersurface geometry - with some applications* (Lecture notes).
We will introduce isoparametric hypersurface theory, and apply it to some special geometry.

**James Montaldi (University of Manchester):**
*Momentum maps and relative equilibria* (Lecture notes).
I will begin by going over the standard definition of momentum maps and describing some of their basic properties. Then I will discuss relative equilibria and how the geometry of the momentum map influences the families of such dynamical solutions.

**Ewan Morrison (University of Glasgow):** *Frobenius manifolds and evolution equations of hydrodynamic type* (Lecture notes)

**Jing Ping Wang (University of Kent):**
*Symbolic representation and symmetry integrability*
(Lecture notes).
In this talk I'll give an brief account of symbolic representation and how to apply it to the global classification of integrable evolutionary equations. In symbolic representation the existence of infinite hierarchy of symmetries is linked with factorisation properties of an infinite sequence of multi-variable polynomials. It provides a powerful tool for testing integrability of a given system and enables us to obtain the intrinsic structure of the symmetry hierarchy. Symbolic representation is suitable for studying integrability of wide classes of equations including noncommutative, non-evolutionary, non-local (integro-differential) , multi-component and multi-dimensional equations.

**Pavlos Xenitidis (University of Leeds):**: *Multidimensional consistency of discrete and continuous equations* (Lecture notes)

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Last update: 3 February 2010