# Workshop Programme

## for period 11 - 15 May 2009

### Algebraic Theory of Difference Equations

11 - 15 May 2009

Timetable

 Monday 11 May 09:00-09:30 Registration 09:30-10:30 Singer, M (North Carolina State) Galois theories for difference equations I. (Tutorial) The aim of these two talks is to show how group theory can be used to derive properties of solutions of difference equations. The tools to do this are provided by Galois theories, which allow us to link groups to difference equations. I will give an elementary introduction to two such theories and show how they can be used to describe algebraic properties of sequences satisfying recurrence relations as well as differential properties of functions satisfying functional equations. For example, I will show how one can reprove and generalize Hoelder's result that the Gamma function satisfies no algebraic differential equation. 10:30-11:00 Tea/Coffee 11:00-12:00 Scanlon, T (UC, Berkeley) Model theory and difference equations I. (Tutorial) 12:00-14:00 Lunch at University House 14:00-15:00 Ramis, J-P (Paul Sabatier Toulouse III) Invariants for linear difference and q-difference equations I. (Tutorial) 15:00-16:00 Umemura, H (Nagoya) Principle of galois theory. We proposed a general Galois Theory of differential equations in char. 0 in 1996. The principle in this theory works also in different contexts. Namely (1) Difference equations in char. 0, (2) Differential equation in char. p > 0, (3) Difference equation in char. p > 0. General Galois Theory allows us to analyze group theoretically integrability of dynamical systems on algebraic varieties defined over a field of char. p > 0 as well as char. 0. 16:00-16:30 Tea/Coffee 16:30-17:30 Casale, G (Rennes 1) Galois theory for nonlinear difference equations I will present Malgrange's definition of Galois pseudogroup for a nonlinear difference equation and explaine how it can be used to obtain integrability condition on linearisation of the equation. In case of a symplectique map one gets the discrete version of Morales-Ramis theorem : If such a map is Liouville integrable then the difference Galois group of its linearisation along a (good) particular solution is almost commutative. 17:30-18:30 Wine Reception, School of Mathematics
 Tuesday 12 May 09:30-10:30 Singer, M (North Carolina State) Galois theories for difference equations II. (Tutorial) The aim of these two talks is to show how group theory can be used to derive properties of solutions of difference equations. The tools to do this are provided by Galois theories, which allow us to link groups to difference equations. I will give an elementary introduction to two such theories and show how they can be used to describe algebraic properties of sequences satisfying recurrence relations as well as differential properties of functions satisfying functional equations. For example, I will show how one can reprove and generalize Hoelder's result that the Gamma function satisfies no algebraic differential equation. 10:30-11:00 Tea/Coffee 11:00-12:00 Scanlon, T (UC, Berkeley) Model theory and difference equations II. (Tutorial) 12:00-14:00 Lunch at University House 14:00-15:00 Ramis, J-P (Paul Sabatier Toulouse III) Invariants for linear difference and q-difference equations II. (Tutorial) 15:00-16:00 Sauloy, J (Paul Sabatier Toulouse III) Using q-difference equations study nonabelian H 1over elliptic curves The local analytic classification of $q$-difference equations involves a $q$-analogue of Birkhoff-Malgrange-Sibuya theorem to the effect that the space of isoformal analytic classes is isomorphic to the $H^{1}$ of the so-called "Stokes sheaf", a sheaf of non abelian groups over the elliptic curve $E_{q} := \mathbf{C}^{*}/q^{\mathbf{Z}}$. On the other hand, the local analytic Galois theory of $q$-difference equations involves invariants of a new type, linked to "$q$-alien derivations". I hope to be able to use these new invariants to study the $H^{1}$ of some non abelian sheaves over $E_{q}$. 16:00-16:30 Tea/Coffee 16:30-17:30 Chatzidakis, Z (Paris 7 - Denis-Diderot) Difference fields and descent of difference varieties In this talk I will state and explain a few results of descent of difference varieties which can be obtained using model theoretic tools. The typical statement is the following: Let K_1 and K_2 be difference fields, and V_i, i=1,2, difference varieties defined over K_i. Assume that there is a dominant rational difference morphism from V_1 onto V_2. Then V_2 in turns dominates some difference variety defined over K=K_1\cap K_2. This result is of course not true as stated, and we explain which hypotheses make it valid. In particular it gives an alternate proof of a result of M. Baker on algebraic dynamics and generalises it to higher dimensions. This is joint work with Ehud Hrushovski. Extended abstract, at http://www.logique.jussieu.fr/~zoe/papiers/Leeds09.pdf 17:30-18:00 Tomasic, I (QMUL) Fields with automorphism and measure Scanlon conjectured that a field equipped with a measure (or a suitable Euler characteristic) must be pseudofinite. I did not know how to prove the above, so I generalised it to the following. A difference field with measure must be a model of ACFA. I will present my efforts toward proving this conjecture.
 Wednesday 13 May 09:30-10:33 Halburd, R (UCL) Integrable systems and difference equations (Tutorial) This will be a general overview aimed at non-specialists. 10:30-11:00 Tea/Coffee 11:00-12:00 van der Put, M (Groningen) Families of singular connections and the Painlevé equations The theme of this talk is a systematic construction of the ten isomonodromic families of connections of rank two on the projective line inducing Painlevé equations and some families of connections living above elliptic curves. These are obtained by considering the complex analytic Riemann--Hilbert morphism from a moduli space of connections to a categorical moduli space of analytic data (i.e., ordinary monodromy, Stokes matrices and links), here called the monodromy space. Our method extends the work of Jimbo, Miwa and Ueno. 12:00-16:30 Free time 16:30-17:00 Sokolov, V (Landau Institute for Theoretical Physics) Termination of hydrodynamic type chains 17:00-17:30 Nishioka, S (Tokyo) Irreducibility of q-Painlevé equation of type $A_6^{(1)}$ in the sense of order I introduce a result on the irreducibility of q-Painlevé equation of type $A_6^{(1)}$ in the sense of order using the notion of decomposable extensions. The equation is one of the special non-linear q-difference equations of order 2 with symmetry $(A_1+A_1)^{(1)}$ and is also called q-Painlevé equation of type II. The decomposable difference field extension is a difference analogue of K. Nishioka's which was defined to prove the irreducibility of the first Painlevé equation in the sense of Nishioka-Umemura. The strongly normal extension of difference fields defined by Bialynicki-Birula is decomposable. I proved that transcendental solutions of the equation in a decomposable extension may exist only for special parameters, and that all of them satisfies the identical well-known Riccati equation if we apply the Bäcklund transformations to it appropriate times. 17:30-18:00 Ovchinnikov, A (Illinois, Chicago) Differential Tannakian categories and fiber functors We define a differential Tannakian category (without using fiber fubctors) and show that under a natural assumption it has a fiber functor. If in addition this category is neutral, that is, the target category for the fiber functor are finite dimensional vector spaces over the base field, then it is equivalent to the category of representations of a (pro-)linear differential algebraic group. Our approach generalises Deligne’s fiber functor construction for the usual Tannakian categories. 18:00-18:30 Juan, L (Texas Tech) Picard-Vessiot extensions with specified Galois group In joint work with Ted Chinburg and Andy Magid we address the problem of recognizing a Picard-Vessiot extension E of a differential field F from weaker information than the structure of E as a differential field. Our work includes a differential counterpart of the normal basis theorem in polynomial Galois theory and the construction of an invariant that depends on the differential Galois group of the extension. I will present the main results and some examples.
 Thursday 14 May 09:30-10:30 Matzat, B-H (Heidelberg) Frobenius modules for groups of Lie type Frobenius modules encode many of the typical phenomena in positive characteristic. They enjoy a difference Galois theory with linear algebraic group schemes as partners similar to differential modules. Here we construct very general Frobenius modules with groups of Lie type as Galois groups. This applies among others to ordinary Galois extensions, p-adic differential equations and t-motives. 10:30-11:00 Tea/Coffee 11:00-12:00 di Vizio, L (CNRS, IMJ, Paris) Arithmetic theory of q-difference equations We propose a global theory of q-difference equations over a finite extension of the field of rational functions k(q). In the first part of the talk we will give a definition of a q-difference analogue of the theory of G-functions and establish a regularity result for the associated operators, obtained as a combination of a q-analogue of the Andre'-Chudnovsky Theorem and Katz Theorem. In the second part of the paper, we combine the results pf the forst part with some formal q-analog Fourier transformations, obtaining a statement on the irrationality of special values of the formal q-Laplace transformation of a G_q-function. 12:00-14:00 Lunch in University House 14:00-15:00 Kowalski, P (Wroclaw) Ax's theorem for additive power series Ax's theorem is a power series version of Schanuel's conjecture. It is a statement about the transcendence degree of the values of the exponential map on a linearly independent sequence of power series. I will discuss an analogous statement where the role of the exponential map is played by additive power series (in positive characteristic). 15:00-16:00 Gekhtman, M (Notre Dame) Poisson geometry of directed networks and integrable lattices Recently, Postnikov used weighted directed planar graphs in a disk to parametrize cells in Grassmannians. We investigate Poisson properties of Postnikov's map from the space of edge weights of a planar directed network into the Grassmannian. We show that this map is Poisson if the space of edge weights is equipped with a representative of a 6-parameter family of universal quadratic Poisson brackets and the Grasmannian is viewed as a Poisson homogeneous space GL(n) equipped with an appropriately chosen R-matrix Poisson-Lie structure. Next, we generalize Postnikov's construction by de ning a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. This family includes, for a particular kind of networks, the Poisson bracket associated with the trigonometric R-matrix. We use a special kind of directed networks in an annulus to study a cluster algebra structure on a certain space of rational functions and show that sequences of cluster transformations connecting two distinguished clusters are closely associated with Backlund-Darboux transformations between Coexeter-Toda flows in GL(n). This is a joint work with M. Shapiro and A. Vainshtein. 16:00-16:30 Tea/Coffee 16:30-17:30 Halburd, R (UCL) Admissibility of solutions of discrete dynamical systems For discrete equations on a number field, the rate of growth of the heights of iterates is a good detector of integrability. In the case of a rational number, the height is just the maximum of the absolute value of the denominator and numerator. A solution is called admissible if its height grows much faster than the heights of the coefficients in the equation. For certain classes of equations it is shown that the existence of a single slow-growing admissible solution is enough to guarantee that the equation is a discrete Painleve equation. Inadmissible solutions are also explored. These solutions correspond to pre-periodic orbits for classical (autonomous) dynamical systems. The classical theory is extended to better understand these solutions in the non-autonomous setting. 19:00-22:00 Drinks and Banquet at University House
 Friday 15 May 09:30-10:30 Hardouin, C (Heidelberg) Galois theory of q-difference in the roots of unity For $q \in \mathbb{C}*$ non equal to $1$, we denote by $\sigma_q$ the automorphism of $\mathbb{C}(z)$ given by $\sigma_q(f)(z)=f(qz)$. As q goes to $1$, a q-difference equation w.r.t. $\sigma_q$ goes to a differential equation. The theory related to this fact is also called \textit{confluence} and one part of its study is the behaviour of the related Galois groups during this process. Therefore it seems interresting to have a good Galois theory of q-difference equation for q equal to a root of unity. Because of the increasing size of the constant field at these points, such construction has been avoided for a long time. Recently P.Hendricks has proposed a solution to this problems but his Galois groups were defined over very transcendant fields. We propose here a new approach based, in a certain sense, on a q-deformation of the work of B.H. Matzat and Marius van der Put for Differential Galois theory in positive characteristic. We consider also a family of \textit{iterative difference operator} instead of considering, just one difference operator, and by this way we stop the increasing of the constant field and succeed to set up a Picard-Vessiot Theory for q-difference equations where q is a root of unity and relate it to a Tannakian approach. 10:30-11:00 Tea/Coffee 11:00-12:00 Taimanov, I (Novosibirsk State) The Moutard transformation and its applications We discuss the Moutard transformation which is a two-dimensional generalization of the Darboux transformation of the Schroedinger operator and expose some applications of this transformations to spectral theory and soliton equations. In particular, we expose examples of Schroedinger operators with smooth fast-decaying potentials and nontrivial kernels in $L_2$ and of blowing up solutions of the Novikov--Veselov equation, a two-dimensional generalization of the Korteweg--de Vries equation. These results we obtained jointly with S.P. Tsarev. 12:00-14:00 Lunch at University House 14:00-15:00 Laine, I (Joensuu) Tropical Nevalinna theory and ultra-discrete equations 15:00-16:00 Novikov, S (Maryland) New discretization of complex analysis