09:30 to 10:30 M Singer ([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 to 11:00 Tea/Coffee 11:00 to 12:00 T Scanlon ([UC, Berkeley])Model theory and difference equations II. (Tutorial) 12:00 to 14:00 Lunch at University House 14:00 to 15:00 J-P Ramis ([Paul Sabatier Toulouse III])Invariants for linear difference and q-difference equations II. (Tutorial) 15:00 to 16:00 J Sauloy ([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 to 16:30 Tea/Coffee 16:30 to 17:30 Z Chatzidakis ([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 to 18:00 I Tomasic ([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.
 09:30 to 10:33 R Halburd ([UCL])Integrable systems and difference equations (Tutorial) This will be a general overview aimed at non-specialists. 10:30 to 11:00 Tea/Coffee 11:00 to 12:00 M van der Put ([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 to 16:30 Free time 16:30 to 17:00 V Sokolov (Landau Institute for Theoretical Physics)Termination of hydrodynamic type chains 17:00 to 17:30 S Nishioka ([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 to 18:00 A Ovchinnikov ([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 to 18:30 L Juan ([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.
 09:30 to 10:30 C Hardouin ([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 to 11:00 Tea/Coffee 11:00 to 12:00 I Taimanov ([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 to 14:00 Lunch at University House 14:00 to 15:00 I Laine ([Joensuu])Tropical Nevalinna theory and ultra-discrete equations 15:00 to 16:00 S Novikov ([Maryland])New discretization of complex analysis