The field of integrable equations is a broad area at the boundary of physics and mathematics. It was born with a quest for exact solutions to Newton’s equations of motion. Since the Kepler problem, which was solved by Newton himself, quite a few integrable models were found, including Euler, Lagrange and Kowalevsky tops. In the second half of the twentieth century the inverse scattering method was invented. This boosted the development into the realm of partial differential equations, leading to new concepts, e.g. the Lax pair, and exciting applications, e.g. solitons. The field is very much alive and has interactions with geometry, algebra, analysis and more recently also number theory. Its methods and ideas have been applied in condensed matter physics, statistical and quantum mechanics, quantum field theory and string theory. For almost half a century great effort has gone into recognising and classifying integrable equation as well as understanding the nature of integrability and proving integrability. Of course, these questions are intimately related. More recently the field has witnessed a shift of direction towards the discrete setting.
In the symmetry approach an equation is called integrable if it has infinitely many generalized symmetries. It was shown that the existence of a formal symmetry of sufficiently high order guarantees integrability. In this way the first complete list, of non-linear integrable Klein-Gordon models, was obtained. Among the many different approaches to recognition and classification of integrable equations, the symmetry approach has proven to be particularly successful.
The first global result, where the order of the equations is not fixed, was obtained by J.A. Sanders and J.P. Wang who classified homogeneous x- and t-independent scalar equations of positive weight [J. Diff. Equations, 1998]. Their method is based on the use of a symbolic calculus, introduced by I.M. Gel'fand and L.A. Dikiĭ [Russ. Math. Surveys, 1975], in which differential objects are represented by polynomials. For example, differentiation translates into multiplying with a sum of symbols and the first symmetry condition becomes a divisibility condition between certain polynomials called G-functions. This cleared the path to using methods and results from algebraic geometry, p-adic analysis, and number theory .
A conjecture of Fokas
Sanders and Wang formulated an implicit function theorem and used results from Diophantine approximation theory to perform the classification. Their exhaustive list contains ten integrable equations and it was proven that there are no other equations in the possession of symmetries. This result confirms the first part of Fokas's conjecture ‘If a scalar equation possesses at least one time-independent non-Lie point symmetry, then it possesses infinitely many. Similarly for n-component equations one needs n symmetries.’ [Stud. Appl. Mathematics, 1987].
Two component equations
An important problem, which is still open, is the classification of homogeneous 2-component equations. This is the main problem I aimed to solve in my PhD thesis. A major complication seemed to be the existence of multi-component equations with finitely many symmetries. Such equations are called almost integrable, in the spirit of Fokas's conjecture.
The first example of an almost integrable equation was presented by I.M. Bakirov. F. Beukers, J.A. Sanders and J.P. Wang used the symbolic method and p-adic analysis to show that his 4-th order 2-component equation does not have generalised symmetries at any order other than 6 [J. Diff. Equations, 1998]. The class of equations considered by Bakirov consists of triangular systems for functions u and v. The equation for v is a homogeneous linear evolution equation and the equation for u is an inhomogeneous version of the same equation with a term quadratic in v.
The G-function connected to such equations contains a parameter. Therefore one has to study sets of zeros. These zeros satisfy an equation that can be solved for the order: a Diophantine equation. The Lech-Mahler theorem implies that certain ratios are roots of unity. An algorithm of C.J. Smyth makes it possible to find these cyclotomic points [Numb. Th. Millennium, 2001]. This led to a finite list of integrable cases. Each equation in the list would still be integrable if its quadratic part contained derivatives of v. However, the list is not complete in this more general class of so called B-equations.
Most of the research I did during my PhD has been devoted to the classification and recognition of integrable and almost integrable B-equations [1,2,3,7]. Although this class of equations seems quite special, the Diophantine approach I employed and developed provides the key to classifying more general classes of evolution equations .
Connection between geometry and integrability
After my PhD Elizabeth Mansfield got me interested in the connection between finite dimensional geometry and integrability. Many integrable equations have been shown to describe the evolution of curvature invariants associated to a certain movement of curves in a particular geometric setting. In several examples the hierarchy of generalized symmetries of a curvature equation has been translated into a hierarchy of commuting geometric curves. Thus it seems that assigning to a curve its curvature functions leads to pairs of equivalent integrable equations.
Curvature functions are constructed, or defined, using a moving frame. This technique was introduced by Darboux, who studied curves and surfaces in Euclidean geometry, and was greatly developed by Cartan who used it in the context of generalizing Klein's Erlangen program. Cartan's intuitive constructions were made algorithmical by Fels and Olver. Given the action of a Lie-group on a flat manifold, their method yields a complete set of invariants, invariant differential operators, and the differential relations, or syzygies, they satisfy. This approach has lead to new applications that would not have been envisioned by Cartan, such as computer vision and numerical schemes that maintain symmetry. The Fels-Olver moving frame method seems to be appropriate to answer the question whether integrability lifts from curvature evolutions to curve evolutions.
Reductions of integrable partial difference equations
In 2006 I moved to Melbourne and started working on discrete equations. Two main classes may be distinguished: partial difference equations (PΔE), and ordinary difference equations (OΔE) or mappings. From a PΔE a mapping can be obtained by traveling wave reduction, one imposed periodic boundary conditions on a staircase. It is thought, and in certain cases proven, that if the PΔE is integrable, then the mapping derived from it is also integrable.
Almost integrable evolution equations
In my paper with J.A. Sanders  we have presented 7-th order B-equations with symmetries at order 11 and 29. The p-adic method of Skolem was used to prove that there are no other symmetries of this equation. Therefore it is a counterexample to the conjecture of Fokas. We also proved the existence of infinitely many equations that are almost integrable . Using resultants we provided a method to obtain all n-th order B-equations with a symmetry at order m . This was used to calculate all almost integrable equations of order 3<n<11 with a symmetry of order n<m<n+151. Refinements to the method of Skolem were made to prove that all these equations have exactly one higher symmetry, with the exception of the 7-th order equations which have two higher symmetries.
Classification of integrable B-equations
I solved the classification and recognition problems for B-equations . Using the algorithm of Smyth the problem was solved locally at low order n<24. Every zero of infinitely many G-functions could be described in terms of roots of unity pointing to the zero from 0 and -1, motivating the introduction of bi-unit coordinates. These were used to construct, at every order, a set of integrable B-equations that are not in a lower order hierarchy. Moreover, bi-unit coordinates provided the key to establishing that the hierarchies of symmetries are exhaustive, which was done in cooperation with Beukers. I also gave formulas for the number of integrable equations and proved that they are real, up to a complex scaling. The recognition problem was solved by giving a description of all B-equations that belong to a lower hierarchy. I gave a procedure to obtain the order of the hierarchy.
Global classification of 2-component approximately integrable evolution equations
The Lie algebra of pairs of differential polynomials is a graded algebra. The linear part has total grading 0, the quadratic terms have total grading 1, and so on. Gradings are used to divide the condition for the existence of a symmetry, [K,S]=0, into a number of simpler conditions: [K,S]=0 modulo quadratic terms, [K,S]=0 modulo cubic terms, and so on. This has been called the perturbative symmetry approach, and in the same spirit the idea of an approximate symmetry was defined. In  I have classified 2-component approximately integrable evolution equations globally, that is, the order of the equations can be arbitrarily high. This is achieved by applying the techniques developed in the special case of B-equations, where any approximate symmetry is a genuine symmetry.
A conjecture of Foursov
Foursov gave a classification of 3-rd order symmetrically coupled KDV-like equations [Inverse Problems, 2000]. One equation appeared to have an interesting symmetry structure. This equation is linearly equivalent to the KDV-equation coupled to a purely nonlinear equation with parameter q. Foursov conjectured that for all negative and rational q this equation has a hierarchy of even order polynomial symmetries. I showed that is the case . In fact, I proved a much stronger statement. There are several infinite sets of symmetries for any, possibly complex, value of q. These symmetries are not polynomial and do not necessarily commute.
My paper with E.L. Mansfield  is based on the Fels-Olver approach to moving frames. We give a method that determines, from minimal data, the curvature and evolution invariants that are associated to a curve moving in the geometry defined by the action of a Lie group. The syzygy satisfied by these invariants is obtained as a zero curvature relation in the relevant Lie algebra. An invariant motion of the curve is uniquely associated with a constraint specifying the evolution invariants as a function of the curvature invariants. The syzygy and this constraint together determine the evolution of curvature invariants. We prove that the condition for two curvature evolutions to commute appears as a differential consequence of the condition that the corresponding curve evolutions commute. This implies that integrability does not necessarily lift from the curvature evolution to the curve evolution. However, most commonly studied integrable curvature equations are homogeneous polynomials or rational functions of the differential invariants. Since in these classes the kernel of the differential operator is empty, pairs of integrable equations result.
Closed form expressions for integrals of high-dimensional mappings
The staircase method takes advantage of Lax matrices of a PΔE which depend on a variable k, usually called the spectral parameter, to construct a so called monodromy matrix whose trace yields integrals of motion for the travelling wave reduction. With O. Rojas and G.R.W. Quispel  we have provided closed form expression for the integrals of travelling wave reductions of the sine-Gordon and mKDV equations. Then, with Dinh Tran, we developed a systematic method to obtain closed form expressions, based on noncommutative Vieta-expansion. In  we applied this method to maps of the Adler-Bobenko-Suris classification and in  it was applied to the nth order Lyness-equation.
Functional independence and involutivity
We expressed the integrals in terms of multi-sums of products, novel combinatorial objects which have interesting properties. In  we have established the Poisson commutation relations between our multi-sums of products. This enabled us to prove involutivity of the integrals for certain lattice reductions. A paper on the functional independence is in the making.
In  we showed how to construct well-posed, or nearly well-posed, initial value problems for lattice equations defined on arbitrary stencils (as opposed to the ‘standard’ lattice equations defined on a square). At the same time we have expanded the notion of s-reduction to include all possible periodicities. In  we develop a method to construct well-posed initial value problems for systems of lattice equations (and apply it to a novel quotient-quotient-difference equation).
In  we prove directly, and in full generality, that the staircase method provides integrals for mappings, or correspondences, obtained as travelling wave reductions of integrable partial difference equations. We apply the method to a variety of equations, and systems of equations. Then we use symmetries of the lattice equations to dimensionally reduce the mappings obtained as travelling wave reductions. Taking dimensional reduction into account, our results support the idea that the staircase method provides sufficiently many integrals for the mappings to be completely integrable (in the sense of Liouville-Arnold). We also consider reduction on quad-graphs that differ from the lattice of integers points in the plane.
Growth of degrees
In  we study mappings obtained as periodic reductions of the lattice Korteweg-De Vries equation. For small periodicities we establish upper bounds on the growth of the degree of the numerator of their iterates. These upper bounds appear to be exact. We conjecture a similar growth for periodicities in all directions and have considered the growth in projective space as well.
Integrable lattice equations from elliptic orthogonal polynomials
The discrete time Toda equation and the Quotient-Difference equation arise in the standard theory of orthogonal polynomials, see for example [Papageorgiou, Grammaticos and Ramani, 1995]. By considering two-variable orthogonal polynomials on elliptic curve, we construct higher order analogues of these important equations in .