Geometric aspects of integrability

在现代可积系统理论中，有几个重要的分类问题，这些问题可以用几何术语重新表述。最著名的例子是弗罗贝尼乌斯流形理论，它是由杜布罗文在与二维拓扑场论相联系时提出的。对数Frobenius结构的分类可以归结为对某些装备超平面排列的描述，从而推广了Coxeter排列。这个问题在三维空间中仍然是一个开放的问题，也将是未来研究的方向之一。另一个方向与杨-巴克斯特映射的概念有关，在离散可积动力学中起着非常重要的作用。在最简单的代数情况下，当集合是射影直线时，相应的映射可以被证明与del Pezzo曲面的对称性有关，这可以用于分类。

In the modern theory of integrable systems there are several important classification problems, which can be reformulated in geometric terms. The most celebrated example is the theory of Frobenius manifolds, which was introduced by Dubrovin in relation with 2D topological field theory. The classification of the logarithmic Frobenius structures can be reduced to the description of certain equipped hyperplane arrangements, generalising the Coxeter arrangements. This problem is still open already in dimension 3 and will be one of the directions of the proposed research.Another direction is related to the notion of the Yang-Baxter maps, playing a very important role in discrete integrable dynamics. In the simplest algebraic case, when the set is projective line, the corresponding maps can be shown to be related to the symmetries of the del Pezzo surfaces, which can be used for their classification.