会议摘要（Abstract）

Discrete and continuous Painlevé equations have attracted a lot of attention in recent decades, since they define (new) transcendental functions, and exhibit rich mathematical structures, in algebraic geometry, representation theory and asymptotic analysis. The higher analogues of the Painlevé equations, including the isomonodromic Garnier systems have an even richer structure, involving connections with multivariate special functions, including higher-genus Abelian functions, and multiple orthogonal polynomials and an expected higher-dimensional variant of the underlying algebraic geometry that was established for the usual Painlevé equations. The connections with integrable systems, via higher-order similarity reduction, may form a key to a further understanding of these more complicated systems, but so far the study of Garnier and higher Painlevé systems has been lagging behind. One of the aims of the workshop is to repair this inbalance. By bringing together experts as well as interested researchers, we aim at creating a platform where many of the open questions can be discussed and begin to be tackled. Thus, we hope the workshop can act as a launching pad for opening a systematic research program into these systems.

举办意义(Description of the aim)

Background:

In recent decades the Painlevé equations, and their discrete analogues have been studied extensively, both from the point of view of integrable systems as well as in physics (random matrix models and statistical mechanics) and in algebraic geometry (rational surfaces of initial conditions) and representation theory (affine Weyl groups). In contrast the higher order Painlevé equations have not attracted (yet) a similar level of attention. These higher order ordinary differential and difference equations emerged as multi-phase similarity reductions from integrable hierarchies (in the continuous case), as well as from constructions from integrable partial difference equations (in the discrete case). They are of interest, as they are expected to yield novel transcendental functions which asymptotically go to higher-genus Abelian functions, whereas the usual Painlevé equations tend to elliptic functions (genus one) in the long-time range.

Garnier in 1912 constructed a higher analogue of Fuchs’ isomonodromic deformation problem for Painlevé VI with multiple moving singularities and multiple dependent variables. This results in coupled systems of 2nd order ODE's, compatible through a system of linear PDEs, which can be viewed as the PVI hierarchy. Apart from the Painlevé the property of this higher order system, the limiting behaviour of the solutions lead to hyper-elliptic integrals. The Garnier system remained relatively unexplored until the work by Okamoto in the 1970s, who focused on its Hamiltonian aspects. It has special solutions in terms of multivariate hypergeometric functions (Lauricella, etc.) (see e.g. the monograph "From Gauss to Painlevé" by Iwasaki et al.). More recently a q-Garnier system (Sakai, 2005), and an elliptic variant (Ormerod & Rains, 2015) were established, while reductions from KdV and Boussinesq systems to discrete and continuous Garnier systems were given (Nijhoff & Walker, 2001; Tongas & Nijhoff, 2005), and the isomonodromy problem was studied (Dubrovin & Mazzocco , 2000). Some classification results on algebraic solutions of Garnier systems were obtained (Diarra & Loray, 2015) and in recent years some quantum Painleve and Garnier systems have been investigated (Nagoya et al, 2004 and 2008, Novikov & Sukeimanov, 2016). Apart from these isolated results, the study of the Garnier systems and higher rank Schlesinger systems (isomonodromic matrix systems) has remained relatively sparse.

Objectives:

The workshop aims at stimulating the research into the higher Painlevé equations and Garnier systems, by bringing together experts as well as interested researchers and bring to the fore open problems, challenges and possible new directions. This is meant to be a mostly explorative venture in the hope that some synergies can bring about progress in this largely not yet developed area. The following directions will be highlighted:

● Reduction from integrable PDEs and PΔEs: While continuous and discrete Painlevé equations often arise as reductions from integrable partial difference and differential equations, for Garnier systems this remains mostly to be established. Isolated precedents comprise Garnier systems derived from higher-order similarity reduction of continuous & discrete KdV and Boussinesq hierarchies.

● Lagrangian multiform aspects: While the second order discrete and continuous Painlevé equations possess a conventional Lagrangian description, the newly established Lagrangian multiform theory (Lobb & Nijhoff, 2009), which provides a natural variational formalism for multi-time integrable systems, is directly applicable to the case of Garnier systems.

● Connection with higher-genus abelian functions: `Garnier transcendents’ tend to higher-genus Abelian functions in asymptotic limits. The study of the singularity structure of those solutions may help to link the isomonodromy theory to algebra-geometric techniques on Riemann surfaces.

● Special solutions in terms of multivariate hypergeometric functions: For special parameter values of the Garnier systems solutions exist in terms of multivariate hypergeometric functions. This remains to be done for discrete Garnier systems leading possibly to multivariate elliptic hypergeometric functions;

● Algebraic geometry of spaces of initial conditions; Lifting the celebrated work by Sakai on the classification of discrete and continuous Painlevé equations within the context of the algebraic geometry of rational surfaces and affine Weyl groups, to the case of Garnier systems. Some work has begun (e.g. Takenawa, 2024), but requires further developement.

● Applications in random matrix theory and physics: Continuous and most distinctly discrete Painlevé equations have played an important role in random matrix ensembles and in the theory of semi-classical orthogonal polynomials. These relations were often motivated from physics, e.g. in 2D quantum gravity and string theory. So far, there have been little appearance of Garnier systems in this context, but the structures are ready to be explored for such connections.