会议摘要（Abstract）

Elliptic discrete integrable systems are among the richest of the whole class of integrable equations, both continuous as well as discrete. Their solutions in terms of special functions involve novel features, such as bi-elliptic addition formulae and elliptic generalisations of classical functions and orthogonal polynomials. A lot of progress have been made on various aspects of the theory, but some important elliptic models have received not as much attention as they deserve. One of the aims of the workshop is to repair this inbalance. Thus, the workshop will bring together experts who have worked on various aspects of integrability and elliptic function theory, and thus aim at bringing progress in dealing with these challenging but rewarding model systems.

在整个连续和离散可积方程中，椭圆离散可积系统被认为蕴含着最为丰富的信息。关于这类系统的求解，涉及到特殊函数的一些新的性质，如双-椭圆加法公式、经典的函数与正交多项式的椭圆推广等。对于椭圆离散可积系统理论的研究在许多方面都取得了不小的进展，但一些重要的椭圆可积模型尚未获得应有的关注。这也是此次研讨会的举办背景之一。研讨会将汇集可积系统和椭圆函数理论各个方面的专家学者，交流在这些具有挑战性的可积模型方面的研究进展，发展相关理论。

举办意义(Description of the aim)

Elliptic integrable systems are, in principle, exactly solvable model equations defined through ordinary or partial difference equations and that contain parameters associated with an elliptic curve. In the continuous case the outstanding examples are the Krichever-Novikov equation and the fully anisotropic Landau-Lifschitz equation, where in the former case the equation contains an arbitrary quartic polynomial in the dependent variable, to which one can associate an elliptic curve, while in the latter case the anisotropy parameters in general position are associated with the moduli of an elliptic curve. The discovery of discrete analogues of these equations and their significance within the framework of integrable systems forms the main motivation for this proposal. In addition, classes of ordinary difference equations of discrete-time many-body systems exhibit features which are radically different and richer than those of the corresponding continuous-time many-body systems. (Recently a novel example of such an elliptic many-body system arising as reduction of the BKP equation was found by Zabrodin). Furthermore, there is an entire classification of difference equations of Painlevé type (i.e. equations whose general solutions are meromorphic w.r.t. the movable singularities) at the top of which stands the elliptic Painlevé equation discovered in 1999 by H. Sakai. About a year earlier V. Adler discovered a discrete analogue of the Krichever-Novikov equation, which stands at the top of the well-known Adler-Bobenko-Suris (ABS) classification of quadrilateral lattice equations, which are four-point partial difference equations integrable in the sense of being multidimensionally consistent (i.e., consistently embeddable in a higher-dimensional lattice). Regarding the Landau-Lifschitz equations, there are three independent versions known: one proposed by Nijhoff and Papageorgiou in 1989, another constructed by Adler and Yamilov in 1996, and yet a third proposed by Adler himself in 2000. Almost nothing is known to date about these three models, not even if/how they are interrelated. Nonetheless, it would be of great interest to know their solution structures and their connection with other integrable systems. Finally, there is also a three-dimensional system of equations, namely an elliptic version of the Kadomtsev-Petviashvili equation, which was first proposed by Date, Jimbo and Miwa in 1983 and further studied recently by Fu and Nijhoff.

Apart from elliptic models, there are also classes of elliptic solutions of integrable lattice equations that exhibit novel features, such as a new concept of elliptic Nth root of unity, appearing in elliptic solutions of higher rank lattice equations of Gel’fand-Dickey type. 

There are many outstanding problems in the theory of elliptic discrete systems, and resolving them has become urgent, as they stand at the top of the tree of integrable systems: knowing their resolution would imply the resolution of similar questions for the degenerate cases, which comprises almost all integrable equations.

In recent years, various novel mathematical techniques and methods have been developed to study these discrete systems and their solutions, bringing together ideas stemming from several branches of mathematics and physics, that are usually distinct, now come together: asymptotic analysis, algebraic geometry, representation theory, spectral/isomonodromy analysis, random matrix theory, exactly solvable models, theory of special functions and combinatorial geometry. These will form the unifying themes of the workshop, which will comprise the following topics:

--Elliptic solutions of higher-rank and multidimensional lattice equations;

--The construction of higher-rank elliptic lattice systems;

--Bi-elliptic addition formulae (as appearing in the soliton solutions of the ABS Q4 equation);

--Elliptic discrete time many-body systems;

--The solution structure of the variants of the lattice Landau-Lifschitz equations;

--Elliptic Lax pairs and isomonodromic deformation problems;

--Elliptic orthogonal polynomials and corresponding elliptic discrete integrable systems;

--Reductions to elliptic autonomous and non-autonomous ordinary difference equations, such as the QRT map and elliptic Painlevé equation;

--Algebro-geometric solutions of integrable lattice equations, and higher-genus lattice systems;

--Elliptic solutions of recurrence relations for integer (Somos) sequences;

--Connections with models of quantum theory and statistical mechanics.

椭圆离散可积系统指包含与椭圆曲线相关的参数的精确可解的常差分方程或偏差分方程。在连续情况下，典型的例子是Krichever-Novikov方程和各向异性Landau-Lifschitz方程。前者方程中包含有关于因变量的任意四次多项式，可以将椭圆曲线与之关联；而后者涉及的各向异性参数与椭圆曲线的模参数相关。这些方程的离散形式的发现及它们在可积系统框架内具有的重要意义构成申请此次研讨会的主要背景。在描述离散时间的多体系统的常差分方程中表现出与相应的连续时间多体系统截然不同且更丰富的性质。最近，Zabrodin给出了一个从BKP方程约化获得椭圆多体系统的新例子。目前关于Painlevé型 (即其通解相对于可移动奇点是亚纯函数的方程) 差分方程已有完整的分类，分类中处于顶端的方程是1999年由H. Sakai发现的椭圆离散Painlevé方程。大约在Sakai的工作的一年之前，V. Adler发现了Krichever-Novikov方程的离散形式，该方程处于后来给出的Adler-Bobenko-Suris (ABS)著名的关于多维相容的四边形方程分类的顶端 (多维相容性定义了一种离散可积性，指低维方程可以相容地嵌入到更高维空间)。关于Landau-Lifschitz方程，目前有三个已知的独立的离散版本：一个由Nijhoff和Papageorgiou于1989年提出，一个由Adler和Yamilov于1996年构建，还有一个由Adler于2000年给出。迄今为止，人们对这三个离散的Landau-Lifschitz模型知之甚少，甚至还不清楚它们之间是否存在联系或如何关联。十分期待理解它们的解结构以及它们与其他可积系统的联系。最后，还有一个与Kadomtsev-Petviashvili方程相关的椭圆三维方程组，由Date、Jimbo和Miwa于1983年提出，最近由Fu和Nijhoff对其做了进一步研究。

除了椭圆模型以外，离散可积方程还存在多种类型的椭圆函数解，并涉及到一些新的特征。例如在Gelfand-Dickey型高阶离散方程的椭圆孤子解的研究中引入了新的“椭圆单位根”的概念。

    椭圆离散系统理论中存在许多尚未解决且亟待研究的问题。发展椭圆可积系统的研究方法具有重要意义。事实上，椭圆可积系统处于可积系统分类的顶端，这类系统通过退化几乎包括了所有的可积方程。

近年来，不断有新的数学技巧和方法出现，用于研究这类系统及其精确解。这些研究逐步将来自于数学和物理的若干看似不同的分支结合起来，包括：渐近分析、代数几何、表示理论、谱/等单值性分析、随机矩阵理论、精确可解模型、特殊函数理论和组合几何等。这些贯穿于此次研讨会的主题内容，具体地，我们将关注：

--高阶和高维离散方程的椭圆函数解；

--高阶椭圆离散系统的构造；

--双-椭圆加法公式(出现于ABS的Q4方程的孤子解中)；

--椭圆离散多体系统；

--各种离散Landau-Lifschitz 方程的解的结构；

--椭圆Lax 对和等单值形变问题；

--椭圆正交多项式与相应的椭圆离散可积系统；

--椭圆自治与非自治常差分方程的约化，如QRT映射与椭圆Painlevé方程;

--离散可积系统的代数几何解与高亏格离散系统；

--整数(Somos)序列的递推结构的椭圆解；

--量子理论模型与统计力学方面的联系。