Koji HASEGAWA (Tohoku University)

Quantization of discrete Painlevé/Garnier system via affine quantum group

A discretization of Painlevé VI equation was obtained by Jimbo and Sakai in 1996. There are two ways to quantize it:
1) use the affine Weyl group symmetry of $D_5^{(1)}$ i.e. based on the work by Kajiwara-Noumi-Yamada,
2) Lax formalism i.e. monodromy preserving point of view.
We will review the both approaches in this talk. The second one uses a periodic reduction of the non-autonomous generalization of quantum discrete sine-Gordon (=sl$_2$-Toda field) system of Kashaev-Reshetikhin, and it turns out that the resulting equation is equivalent to the one from the first approach. Furthermore, one can generalize this to the Garnier case (i.e. higher rank, many singular points case) with the help of the universal R matrix.

Arata KOMYO (Osaka University)

The moduli spaces of parabolic connections with a quadratic differential and isomonodromic deformations

The moduli spaces of parabolic connections are referred to as phase spaces of the differential equations determined by the isomonodromic deformations. In this talk, we extend these moduli spaces by considering the moduli spaces of parabolic connections with a quadratic differential. We show that the extended moduli spaces are equipped with the structure of twisted cotangent bundles. This extension of the moduli spaces is related to the classical trick of turning a time dependent Hamiltonian flow into an autonomous one by adding variables.

Gen KUROKI (Tohoku University)

Quantized $\tau$-functions generated by the Bäcklund transformations

We construct quantum (non-commutative and q-difference version of) Painlevé tau-functions generated by the Bäcklund transformations and prove that they satisfy the quantum Hirota-Miwa equation.

Frank LORAY (Université de Rennes1)

Classification of algebraic solutions of irregular Garnier systems

We prove that algebraic solutions of Garnier systems in the irregular case are of two types. The classical ones come from isomonodromic deformations of linear equations with diagonal or dihedral differential Galois group; we give a complete list in the rank N = 2 case (two indeterminates).The pull-back ones come from deformations of coverings over a fixed degenerate hypergeometric equation; we provide a complete list when the differential Galois group is $\mathrm{SL}_2(\mathbb{C})$. By the way, we have a complete list of algebraic solutions for the rank N = 2 irregular Garnier systems.

Yuki MATSUBARA (Kobe University)

On the Cohomology of The Moduli Space of Parabolic Connections

We consider the moduli space of logarithmic connections of rank 2 on the projective line minus 5 points with fixed spectral data. We compute the cohomology of such moduli space, and this computation will be used to extend the results of Geometric Langlands correspondence due to D. Arinkin to the case where the this type of connections have five simple poles on P1. In this talk, I will review the Geometric Langlands Correspondence in the tames ramified cases, and after that, I will explain how the cohomology of above moduli space will be used.

Takao SUZUKI (Kindai University)

Cluster algebra and generalized $q$-Painlevé VI systems of type $A$ (joint work with N. Okubo)

The cluster algebra was introduced by Fomin and Zelevinsky. It is a variety of commutative ring described in terms of cluster variables and coefficients. Its generating set is defined by an operator called a mutation which transforms a seed consisting of cluster variables, coefficients and a quiver. Then new cluster variables (resp. coefficients) are rational in original cluster variables and coefficients (resp. coefficients). Hence we can obtain various discrete integrable systems from mutation-periodic quivers as relations satisfied by cluster variables and coefficients. Recently Okubo derived the $q$-Painlevé VI equation from the mutation-periodic quiver with 8 vertices. In this talk we consider an extension of his $q$-$P_{\rm VI}$ quiver. Then we can find some compositions of iterative mutations and permutations such that the quiver is invariant. They provide four types of generalized $q$-Painlevé VI systems which contain the known three; the $q$-Garnier system, a similarity reduction of the lattice $q$-UC hierarchy and a similarity reduction of the $q$-Drinfeld-Sokolov hierarchy.

Szilard SZABO (Budapest University of Technology and Economics)

Perversity equals weight for Painlevé systems

An important conjecture in non-Abelian Hodge theory by de Cataldo, Hausel and Migliorini asserts that the weight filtration on the cohomology spaces of a character variety agrees with the perverse Leray filtration on the cohomology spaces of the corresponding Dolbeault moduli space. We prove an analogous result for wild character varieties and the corresponding irregular Hitchin systems associated to the Painlevé cases, and speculate on an extension of our result to a more general setup.