Programme:

Speakers

Sophie Morier-Genoud (Paris) - q-analogues of real numbers

In recent joint work with Valentin Ovsienko we defined q-analogues of real numbers. Our construction is based on a q-deformation of the Farey graph. I will explain the construction and give the main properties. In particular I will mention links with the combinatorics of posets, cluster algebras, Jones polynomials...

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Giovanni Felder (Zurich) - The integrable Boltzmann system

Ludwig Boltzmann, in his search for an example of a chaotic dynamical system, studied the planar motion of a particle subject to a central force bouncing elastically at a line. As recently noticed by Gallavotti and Jauslin, the system is actually integrable if the force has an inverse-square law. I will review the construction of the second integral of motion and present the results: the orbits of the Poincaré map are periodic or quasi-periodic and anisochronous, so that KAM perturbation theory (Moser's theorem) applies, implying that for small perturbations of the inverse-square law the system is still not chaotic. The proof relies on mapping the Poincaré map to a translation by an element of an elliptic curve. A corollary is the Poncelet property: if an orbit is periodic for given generic values of the integrals of motions then all orbits for these values are periodic.

Rod Halburd (UCL) - Variants of the Painlevé property and integrable subsystems

We use global results about functions that are meromorphic in regions of the plane to find individual solutions of differential, difference and delay-differential equations whose only movable singularities are poles. We also allow for simple global branching. In this way we can find or describe subsets of solutions of equations that are in general non-integrable.

Vsevolod Adler (Moscow) - Stationary solutions of non-autonomous symmetries of integrable equations

The talk is about some recent results on Painleve-type reductions for integrable equations. My first example is a reduction obtained as a stationary equation for master-symmetry of KdV equation. It is equivalent to some fourth order ODE and numerical experiments show that some of its special solutions may be related to the Gurevich-Pitaevskii problem on decay of initial discontinuity. The second example is about non-Abelian Volterra lattices. Here we study several low-order reductions and demonstrate their relation with non-Abelian analogues of discrete and continuous Painleve equations.

Anna Felikson (Durham) - Mutations of non-integer quivers: finite mutation type

Given a skew-symmetric non-integer (real) matrix, one can construct a quiver with non-integer weights of arrows. Such a quiver can be mutated according to usual rules of quiver mutation introduced within the theory of cluster algebras by Fomin and Zelevinsky. We classify non-integer quivers of finite mutation type and prove that all of them admit some geometric interpretation (either related to orbifolds or to reflection groups). In particular, the reflection group construction gives rise to the notion of non-integer quivers of finite and affine types. We also study exchange graphs of quivers of finite and affine types in rank 3. The talk is based on joint works with Pavel Tumarkin and Philipp Lampe.

Oleg Chalykh (Leeds) - Twisted Ruijsenaars model

The quantum Ruijsenaars model is a q-analogue of the Calogero—Moser model, described by n commuting partial difference operators (quantum hamiltonians) h_1, …, h_n. As it turns out, for each natural number l>1 there exists another integrable system whose quantum hamiltonians have the same leading terms as the l-th powers of h_1, …, h_n. I will discuss several ways of arriving at this generalisation.