Geometric and Algebraic Aspects of Integrability

The inverse scattering transform (IST) is a method to solve the Cauchy problem associated with a class of nonlinear evolution equations. The lectures will give some history and background. The associated linear systems and methodology will be outlined for some physically significant equations such as the Korteweg-deVries (KdV), nonlinear Schrodinger (NLS), modified KdV (mKdV), and sine-Gordon equations. The recently discovered symmetry leading to the integrable nonlocal NLS equation and its IST will also be mentioned.

A presently emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of refinement of the discretization. Current interest in this field derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. The central role in this theory is played by discrete integrable systems. I these lectures we plan to reveal the integrable structure of discrete differential geometry. The corse is based on the book Alexander I. Bobenko, Yuri B. Suris, Discrete Differential Geometry: Integrable Structure, Graduate Studies in Mathematics, Vol. 98, AMS, 2008 http://page.math.tu-berlin.de/~bobenko/ddg-book.html

The existence of an infinite hierarchy of local symmetries can be deemed as a constructive definition of integrability for systems of partial differential equations. It enables us to formulate the necessary conditions which are strong enough to tackle the problem of classification of integrable systems. The derivation of these conditions will is the main focus of this crash lecture course. This approach is quite universal, although in the cases of differential-difference and partial difference equations we meet rather challenging problems.

The idea of a Lax pair associated with integrable systems is used to motivate twistor theory. This leads to a twistor description of integrable geometric systems such as the self-dual Einstein and self-dual Yang-Mills equations, and various reductions of them.

These lectures will describe some of the techniques that have been found useful for explicitly finding the integrals of exterior differential systems arising in geometry. These include Darboux’ method and its generalizations, compatible reductions, integrable extensions and Bäcklund transformations, conservation laws, and methods connected with analysis of the characteristic variety. Emphasis will be placed on illustrative examples and computations using the structure equations of Cartan. The first lecture will begin with a brief summary of the basics of exterior differential systems (EDSs) including involutivity, Cartan-Kähler theory, and the characteristic variety. This will be followed by an introduction to EDS formulations of some geometric problems via Cartan’s structure equations, illustrated by examples. The second lecture will focus mainly on examples, including minimal submanifolds, pseudo-holomorphic curves, Willmore geometry, and prescribed curvature problems in Riemannian and Finsler geometry.

Vortices are localised solitons in two dimensional space, and can be defined on any Riemann surface. The abelian vortex equations have an interpretation in terms of curvature and conical singularities, and are integrable in cases where the curvature has an appropriate constant value. Variants of the equations correspond to negative, zero and positive curvature. Many explicit solutions have been constructed using holomorphic maps between surfaces, but this method does not always work globally.

In this talk I shall show that the total energy of time-dependent uniton solutions of the Ward chiral model is proportional to the third homotopy class of the corresponding extended solutions. This explains the classical energy quantisation of moving solitons. I shall also discuss a compactified twistor fibration associated to the model, and use it to show that the second Chern number of the twistor bundle corresponds to the third homotopy class of the extended solution.

I will briefly review integrable background geometries and their associated gauge theories, which were introduced at the turn of the century as a framework to unify reductions of 4-dimensional selfduality equations. As time permits, I will also indicate some recent developments and generalizations.

We discuss a surprising relationship between Manakov operators, well-known in the theory of integrable systems on Lie algebras, and curvature tensors of projectively equivalent Riemannian and K\"ahler metrics of arbitrary signature. We demonstrate how using this relationship helps to solve a number of natural problems in differential geometry.

The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times. The Talbot effect, named after an optical experiment by one of the founders of photography, was first observed in optics and quantum mechanics, and leads to intriguing connections with exponential sums arising in number theory. Ramifications of these phenomena and recent progress on the analysis, numerics, and extensions to nonlinear wave models will be discussed.

Given a particular set of curves on a manifold, we can ask whether they are local geodesics of a pseudo-Riemannian metric. This is called the metrisability problem. I will present its solution for the integral curves of the Painleve equations and conclude that they are geodesics of a metric only for special choices of parameters, those for which the Painleve equations admit a first integral quadratic in first derivatives. In the second part of the talk I will present the solution to the following problem: given a torsion free affine connection on a surface, when does it admit 0, 1, 2 or 3 Killing 1-forms? The relation between existence of Killing forms and metrisability will be illustrated by the Painleve equations. Then, I will explain how one can interpret the existence of hamiltonian descriptions of hydrodynamic type systems as a particular case of this problem.

In this talk I will briefly review the classical literature on Darboux integrability; provide a generalization of the classical definition of Darboux integrability within the context of differential systems; and discuss recent developments regarding the group theoretical approach to Darboux integrability. Comparisons with the various notions of integrability for evolution equations will be made.

We prove that the following two conditions on scalar ODEs or systems of ODEs are equivalent: (a) linearization along each solution is trivializable; (b) any differential invariant of a given system under the group of contract transformations (in case of a scalar ODE) or point transformations (in case of systems) is its first integral. The latter condition was introduced by Elie Cartan, who called such differential equations as C-class equations. In particular, they can be either solved explicitly using their differential invariants or reduced to so-called systems of Lie type. The algebra of differential invariants, in its turn, can be computed constructively using the characteristic Cartan connection, which involves only differentiation and algebraic operations.

I will describe a novel construction of discrete confocal quadrics (discrete confocal coordinate systems). Our discretization respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (i.e. they allow a conformal parametrization along curvature lines). The construction is based on an integrable discretization of the Euler-Poisson-Darboux equation. This is a joint work with Suris, Schief and Techter.

The concept of Automorphic Lie Algebras (ALiAs) arises in the context of reduction groups introduced in the late 1970s by Mikhailov. ALiAs are obtained by imposing a finite symmetry group G on a Lie algebra over a field of rational functions [1], [2]. Since their introduction the goal has been to classify them and the case in which the symmetry group is finite and acts on sl_n by inner automorphisms has been recently classified [3]. Past work shows remarkable uniformity between Lie algebras associated to different reduction groups. For example, if the base Lie algebra consists of traceless 2 × 2 matrices and the representation of G is irreducible then the ALiA is independent of the reduction group [2]. In this talk I will recall the concept of ALiAs, discuss a number of open questions from the algebraic point of view and introduce a cohomology of root systems [4]. The talk is based on joint work with Vincent Knibbeler and Jan Sanders (Vrije Universiteit Amsterdam). [1] S. Lombardo, A.V. Mikhailov, Reduction groups and Automorphic Lie Algebras, Comm. Math. Phys. 258(1): 179–2002, 2005 [2] S. Lombardo, J. A. Sanders, On the Classification of Automorphic Lie Algebras, Comm. Math. Phys. 299, 793-824, 2010 [3] V. Knibbeler, S. Lombardo, J. Sanders, Higher dimensional Automorphic Lie Algebras, Journal of Foundation of Computational Mathematics, Springer, 2016 [4] V. Knibbeler, S. Lombardo, J. Sanders, Automorphic Lie Algebras and Cohomology of Root Systems, arXiv:1512.07020

Applying r-matrix approach to particular Jacobi algebra instead of Poisson algebra, it is possible to construct integrable hierarchies of (3+1)-dimensional dispersionless systems. The price one has to pay is the loss of Hamiltonian structure of considered systems.

The simplest model of a bicycle is a segment of fixed length that can move, in n-dimensional Euclidean space, so that the velocity of the rear end is always aligned with the segment (the rear wheel is fixed on the frame). The rear wheel track and a choice of direction uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. The two track are related by the bicycle (Darboux, Backlund) transformation which defines a discrete time dynamical system on the space of curves. I shall discuss the symplectic, and in dimension 3, bi-symplectic, nature of this transformation and, in dimension 3, its relation with the filament equation. An interesting problem is to describe the curves that are in the bicycle correspondence with themselves (in this case, given the front and rear tracks, one cannot tell which way the bicycle went). In dimension two, such curves yield solutions to Ulam's problem: is the round ball the only body that floats in equilibrium in all positions? I shall discuss F. Wegner's results on this problem and relate them with the planar filament equation. Open problems and conjectures will be emphasized.

The algebraic entropy, first defined as a coarse characterisation of the complexity of birational maps, has become a valuable test of integrability of (bi)rational dynamics, and an object of interest per se. In particular it has been conjectured to always be the logarithm of an algebraic integer. We will describe three ways of calculating it exactly: firstly by a heuristic and quick way, then by the singularity analysis developed for two dimensional systems, an thirdly by a novel approach, also deeply linked to the singularity structure of the system, but applicable in all dimensions, including infinite dimensional systems like the so called 'discrete delay Painlev\'e equations.

We study F-manifolds equipped with a pair of flat connections (and a pair of F-products), that are required to be compatible in a suitable sense. In the first part of the talk we show that bi-flat F-manifolds in dimension three are locally parameterized by solutions of the full Painlevé IV,V and VI equations, according to the Jordan normal form of the operator of multiplication by the Euler vector field. In the second part of the talk we discuss some examples related to complex reflection groups. Based on joint works with Alessandro Arsie.

This talk concerns integrable structures in N=4 super Yang-Mills theory. I will discuss a twistor correspondence that applies on the one hand to solutions to the self-dual Yang-Mills equations that are stationary and axisymmetric that is based on a parametrised family of Joukowski (or Zhukovski) transformations. This can be used to obtain tau functions that give rise to certain near BPS anomalous dimensions. In particular we see that such anomalous dimensions satisfy integrable differential equations. This is joint work with Andrea Ferrari.

For an arbitrary semisimple Frobenius manifold we construct an integrable hierarchy of Hamiltonian partial differential equations. In the particular case of quantum cohomology the tau-function of a solution to the Hodge hierarchy generates the intersection numbers of the Gromov--Witten classes and their descendents along with the characteristic classes of Hodge bundles on the moduli spaces of stable maps. For the one-dimensional Frobenius manifold the Hodge hierarchy is a deformation of the Korteweg-de Vries hierarchy depending on an infinite number of parameters. We conjecture that this hierarchy is a universal object in the class of scalar Hamiltonian integrable hierarchies possessing tau-functions. This is a joint work with Boris and Di Yang

In my joint work with B. Bakalov we have proposed a certain vertex algebra approach to the higher genus reconstruction associated with a semi-simple Frobenius manifold. Our construction gives a differential operator constraint for each state in the so called W-algebra. The problem is that in general the W-algebra is difficult to compute, so our construction has only limited applications. In this talk, I would like to discuss a case in which the Eynard--Ornatin recursion can be used to determine states in the W-algebra.

In 1986 Its and Novokshenov studied the asymptotics and the movable poles of real solutions on the real positive line of the Painleve III equation of type (0,0,4,-4). They had results on the behaviour near 0 and near infinity. I will combine this with the global geometry of the moduli spaces for initial data and monodromy data. This will lead to facts on the movable poles (and movable zeros) on the whole positive real line. Behind this is an interpretation of the corresponding isomonodromic connections as TERP-structures (or noncommutative Hodge structures) and results on them by T. Mochizuki, Sabbah and myself. The results are joint work with M.A. Guest.

In previous work with Alexander Its and Chang-Shou Lin we have solved a special case of the 2D Toda equations, which we call the tt*-Toda equations. This special case has various interpretations: harmonic maps of a surface, the topological-antitopological fusion equations of Cecotti-Vafa and Dubrovin, the Hitchin equations (Higgs bundles). We shall review this example, and describe/explain a convexity property of the solutions, which arises from the third point of view, i.e. moduli spaces of flat connections. This is joint work with Nan-Kuo Ho.

In 1987 Hitchin discovered a new family of algebraic integrable systems, solvable by spectral curve methods. One novelty was that the base curve was of arbitrary genus. Later on it was understood how to extend Hitchin's viewpoint, allowing poles in the Higgs fields, and thus incorporating many of the known classical integrable systems, which occur as meromorphic Hitchin systems when the base curve has genus zero. However, in a different 1987 paper, Hitchin also proved that the total space of his integrable system admits a hyperkahler metric and (combined with work of Donaldson, Corlette and Simpson) this shows that the differentiable manifold underlying the total space of the integrable system has a simple description as a character variety Hom(\pi_1(\Sigma), G)/G of representations of the fundamental group of the base curve \Sigma into the structure group G (typically G=GL_n(C)). This misses the main cases of interest classically, but it turns out there is an extension. In work with Biquard from 2004 Hitchin's hyperkahler story was extended to the meromorphic case, upgrading the speakers holomorphic symplectic approach from 1999. Using the irregular Riemann-- Hilbert correspondence the total space of such integrable systems then has a simple explicit description in terms of monodromy and Stokes data, generalising the character varieties. The construction of such ``wild character varieties'', as algebraic symplectic varieties, was recently completed in work with D. Yamakawa. In this talk I will describe some simple examples of wild character varieties including some cases of complex dimension 2, familiar in the theory of Painleve equations, although their structure as new examples of complete hyperkahler manifolds (gravitational instantons) is perhaps less well-known. This abstract viewpoint on integrable system also leads to a theory of Dynkin diagrams for them, which we hope can be used in their classification.

Dispersive Poisson brackets and Poisson pencils on formal loop spaces play a important role in the description of integrable hierarchies, especially in the setting of hierarchies of topological type. We will first review the general framework and motivation for the study of such objects, including main results like Getzler’s theorem and the notion of central invariants by Liu, Dubrovin and Zhang. Then we will discuss our recent work, which include the proof, using spectral sequences techniques, of the triviality of the bi-Hamiltonian cohomology of semisimple Poisson brackets of DN type. That in turn implies the existence of arbitrary order dispersive deformations, starting from any choice of central invariants, thus solving a conjecture of Dubrovin et al. Finally we will briefly describe our recent results on the generalisation to the multivariable setting and outline some open problems. Based on joint works with H. Posthuma, S. Shadrin and with M. Casati, S. Shadrin.

We study monopoles in three-space and instantons on a multi-Taub-NUT space with some gauge group G. The index bundle of the associated family of Dirac operators defines a bow representation as well as an integrable system associated with it. Solutions of this integrable system are in one-to-one correspondence with the gauge equivalence classes of these monopoles or instantons. This view leads to a natural Kaehler potential on their moduli space. The first part of this talk is based on the recent work with Mark Stern and Andres Larrain-Hubach

Electronic Itinerary included in delegate information pack

We study the Kadomtsev-Petviashvili equation in the limit of small dispersion and we show that the critical behaviour of the solutions is universal and it is described by a Painleve’ transcendent.

We present several numerical studies of solutions to PDEs from the family of nonlinear Schrödinger and Korteweg-de Vries equations. We study the formation of dispersive shocks and of potential blow-ups in the solutions. Conjectures for the asymptotic description of break-up and blow-up of the solutions is presented.

Combining an old idea of Olver and Rosenau with the classification of second and third order homogeneous Hamiltonian operators (Ferapontov, Pavlov, V. 2014-2016) we classify compatible trios of two-component homogeneous Hamiltonian operators. The trios yield pairs of compatible bi-Hamiltonian operators whose structure is a direct generalization of the bi-Hamiltonian pair of the KdV equation. The bi-Hamiltonian pairs give rise to multi-parametric families of bi-Hamiltonian systems. We recover known examples and we find apparently new integrable systems. (Joint work with P. Lorenzoni, A. Savoldi.)

Hydrodynamic type systems of the Jordan blocks matrix form associated with the confluent Lauricella functions are considered. It is shown that such nondiagonalizable systems can be obtained from the generic diagonal hydrodynamic type systems via the confluence process similar to that for the Gauss hypergeometric function. These systems are integrable and govern dynamics of critical points for confluent Lauricella type functions. Such systems appear in the change of type transitions for the quasi-linear systems of PDEs of mixed type. Gradient catastrophe and its regularization for pure parabolic systems are considered. Differential reductions of such systems are discussed too.

We present the main features of the inverse spectral transform (IST) of integrable dispersionless PDEs in multidimensions, stressing the main differences from the classical IST of soliton PDEs. We use such a novel IST to solve the Cauchy problem, construct the longtime behavior of solutions, and investigate analytically wave breaking phenomena, when they take place. This theory, applicable to several distinguished examples, like the dispersionless Kadomtsev - Petviashvili, the 2D dispersionless Toda, and the heavenly equations, has been developed in collaboration with S. V. Manakov, and this talk is dedicated to his memory. If time permits, we also discuss some rigorous aspects of this theory, recently understood in collaboration with P. G. Grinevich and D. Wu, and applications to two Cauchy problems for nonlinear wave phenomena in Nature.

We introduce the technique combining the features of integration schemes for SDYM equations and multidimensional dispersionless integrable equations to get SDYM equations on the conformally self-dual background. Generating differential form is defined, the dressing scheme is developed. Some special cases and reductions are considered.

I will consider GL(2,R)-structures in even dimensions and present a general construction of almost-complex manifolds associated to the structures. The integrability of the structures is expressed in terms of the torsion and curvature of the GL(2,R)-structures. As a application I will present an integrable system whose solutions are in a one-to-one correspondence with torsion-free GL(2,R)-structures in dimension 4.

We will discuss the question whether it is possible to read off dispersionless integrability via geometry of the equation or its solution space. We will demonstrate the affirmative answer in several particular important cases, related to projective, Lagrangian and Grassmann geometries. This will be done via relations to Einstein-Weyl geometry in 3D, self-duality in 4D, and GL2-structures. Based on joint work with E.Ferapontov, B.Doubrov, and V.Novikov.

This question is difficult because the 6-sphere has an almost complex structure which would not be kahler. I will explain, using old ideas, that a complex structure would be "even" while the almost complex structure is "odd", yielding a contradiction.

Starting from the non-commutative discrete KP system we construct, by applying periodic constraints, a family of solutions of the functional Yang-Baxter equation. In the commutative case these maps were studied in relatation to geometric crystals. Their counterpart on the level of integrable discrete systems is provided by recently introduced non-commutative Gelfand-Dikii systems.

Somos sequences are integer sequences that are generated by recurrence relations that are quadratic (or bilinear) in the dependent variables. They naturally arise in the theory of integrable systems, as finite-dimensional reductions of the discrete KP or BKP equation. In the case of three-term Somos (or Gale-Robinson) recurrences, they also come equipped with the structure of a cluster algebra. Here we review current progress in understanding these sequences from algebraic, geometric and arithmetic viewpoints.

We give a construction of completely integrable (2m)-dimensional Hamiltonian systems with m cubic integrals in involution. Applying to the corresponding quadratic Hamiltonian vector fields the so called Kahan-Hirota-Kimura discretization scheme, we arrive at birational (2m)-dimensional maps. We show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure он R^{2m}, and possess m independent integrals of motion, which are perturbations of the original integrals. Thus, these maps are completely integrable in the Liouville-Arnold sense. Moreover, under a suitable normalization of the original m-tuples of commuting vector fields, the m-tuples of maps commute and share the invariant symplectic structure and m integrals of motion.

We propose a novel approach to the classification of integrable differential/difference as well as fully discrete equations in 3D based on the requirement that hydrodynamic reductions of the corresponding dispersionless limit are `inherited’ by the full dispersive equation. The only constraint of the method is that the initial ansatz possesses a non-degenerate dispersion less limit (this is the case for all known Hirota-type equations). Based on the method of deformations of hydrodynamic reductions, we classify 3D integrable Hirota type equations within various particularly interesting subclasses.

A quantisation of the KP equation on a cylinder is proposed that is equivalent to an infinite system of one-dimensional bosons carrying masses m=1,2,... The Hamiltonian is Galilei-invariant and includes the cubic split/merge (m1,m2)<->(m1+m2) terms for all combinations of particles with masses m1, m2 and m1+m2, with a special choice of coupling constants. The Bethe eigenfunctions for the model are constructed. The consistency of the coordinate Bethe ansatz, and therefore, the quantum integrability of the model is verified for the sectors up to the total mass M=8.

We will describe differential-difference bispectral families that can be constructed starting from the hypergeometric equation. This can be viewed as a generalisation of Wilson's adelic Grassmannian and its trigonometric analogue studied by Haine, Horozov and Iliev. The constructed bispectral operators generalise various previously known examples related to the exceptional Jacobi polynomials. This is joint work with A Khalid (Leeds).

In this talk I shall discuss the relationship between orthogonal polynomials with respect to semi-classical weights, which are generalisations of the classical weights and arise in applications such as random matrices, and integrable systems, in particular the Painlev\'e equations and discrete Painlev\'e equations. It is well-known that orthogonal polynomials satisfy a three-term recurrence relation. I will show that for some semi-classical weights the coefficients in the recurrence relation can be expressed in terms of Hankel determinants, which are Wronskians, that also arise in the description of special function solutions of Painlev\'e equations. The determinants arise as partition functions in random matrix models and the recurrence coefficients satisfy a discrete Painlev\'e equation. The semi-classical orthogonal polynomials discussed will include a generalization of the Freud weight and an Airy weight.

We introduce a class of Z_N graded discrete Lax pairs, with N x N matrices, linear in the spectral parameter and classify the associated discrete integrable systems. Several well known examples belong to our scheme for N=2 (discrete MKdV equation, Hirota's discrete sine-Gordon equation, discrete potential KdV, Schwarzian KdV equation), so many of our systems may be regarded as generalisations of these. Even at N=3, several new integrable systems arise. We also present continuous isospectral deformations these Lax pairs, giving compatible differential-difference systems, which play the role of continuous symmetries of the discrete systems. Master symmetries and their respective hierarchies can be constructed. We present two nonlocal symmetries of our discrete systems, which give rise to the two-dimensional Toda lattice, with our nonlocal symmetries being the B\"acklund transformations and our discrete system being the nonlinear superposition formula (for the generic case). This talk is joint work with Pavlos Xenitidis and based on the paper: arXiv:1411.6059 [nlin.SI].

Let G be a complex reductive algebraic group. In this talk, we will discuss a general conjecture that there is a natural action of the double affine Hecke algebra of type G on a (quantized) character variety of the complement of a knot in S^3. We will explain a motivation behind this conjecture and give an explicit construction, which is inspired by the theory of quantum integrable systems. The main implication is the existence of 3-variable polynomial knot invariants that specialize to the famous Jones polynomial and its colored versions introduced by Witten, Reshetikhin and Turaev. Time permitting, we show that, in the classical limit (q=1), our conjecture follows from (and essentially reduces to) a known conjecture in knot theory due to G. Brumfiel and H. Hilden (1990). (The talk is based on joint work with P. Samuelson.)

Joint work with Joseph Grant. We give a presentation of an Artin braid group of type ADE for any quiver which is mutation-equivalent to a Dynkin quiver of the same type. We show how these presentations can be understood topologically in types A and D using a disk and a disk with a cone point of order two respectively.

We study symplectic properties of monodromy map of second order linear equation with meromorphic potential with simple poles and zeros on a Riemann surface. The periods of the square root of the quadratic differential on the canonical cover of the Riemann surface provide a natural system of Darboux coordinates on the cotangent bundle $T^*\Mcal_{g,n}$. Furthermore, the canonical symplectic structure on $T^*\Mcal_{g,n}$ induces the Goldman Poisson structure on the corresponding character variety under the monodromy map of second order differential equation on a Riemann surface. Being combined with results of S.Kawai this result shows the symplectic equivalence of embeddings of the cotangent bundle into the space of projective structures given by the Bers (quasi-fuchsian) and the Bergman projective connections. The case of holomorphic quadratic differentials was considered in a recent joint work with M.Bertola and C.Norton.

I will describe how to solve geometric Cauchy problems for surfaces associated to Riemannian and Lorentzian harmonic maps using special potentials in the DPW method and its generalizations.

Recently differential Galois integrability conditions have appeared to be very effective for various Hamiltonian systems. The necessary conditions for the integrability of a Hamiltonian system in the Liouville sense are given in terms of properties of the differential Galois group of variational equations obtained by linearisation of equations of motion in a neighbourhood of a particular solution. The fundamental Morales-Ramis theorem of this approach says that if a Hamiltonian system is meromorphically integrable in the Liouville sense in a neighbourhood of a phase curve corresponding to a particular solution, then the identity component of thedifferential Galois group variational equations along this phase curve is Abelian. In the case of natural Hamiltonian systems on $2n$ dimensional Euclidean phase space with the standard kinetic energy and with homogeneous potentials of integer degrees generically exist certain particular solutions. Moreover differential Galois groups of variational equations along these solutions can be calculated effectively. Successful integrability analysis of Hamiltonian systems with homogeneous potentials in flat Euclidean spaces motivated us to look for systems in curved spaces with similar properties. We propose two classes of Hamiltonians that can be considered as such analogues of homogeneous Hamiltonian systems in flat spaces. Moreover we present differential Galois integrability conditions for more general case when potentials are homogeneous of rational degrees functions of variables i.e. they are algebraic functions. Obtained integrability conditions are applied to certain classes of potentials and integrable cases are identified.

We begin this seminar by reviewing one-dimensional potential scattering, and discuss the repulsive and attractive Poeschl-Teller potentials g(g-1)/sinh^2(x) and -g(g-1)/cosh^2(x) as explicit examples. After relating this to the hyperbolic integrable N-particle Calogero-Moser systems, we sketch the relativistic version of the latter, and generalizations of the Poeschl-Teller results to this relativistic setting. We present an overview rather than details, linking in particular to factorized scattering and the sine-Gordon quantum field theory.

V-systems are special configurations of vectors defined in terms of linear algebra. They were introduced by Veselov in the end of 90th in relation with Frobenius manifolds. V-systems generalize Coxeter root systems. The class of V-systems is closed under natural operations of taking subsystems and restrictions. A number of examples of V-systems is known but classification remains open. After explaining these properties of V-systems I am going to discuss more recent study of the special subclass of harmonic V-systems. The corresponding arrangements of hyperplanes are shown to be free in the sense of Saito, and the basic logarithmic fields have explicit potentials for classical families of V-systems. The talk is based on joint works with A.P. Veselov.