Geometric aspects of integrability

在现代可积系统理论中，有几个重要的分类问题可以用几何术语重新表述。最著名的例子是Frobenius流形理论，它是由Dubrovin在二维拓扑场论中提出的。对数Frobenius结构的分类可以简化为某些装备超平面排列的描述，推广了Coxeter排列。这个问题在三维空间中仍然是开放的，并且将是提出的研究方向之一。另一个方向与Yang-Baxter映射的概念有关，它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，当集合是射影线时，对应的映射可以显示为与del Pezzo曲面的对称性相关，这可以用于它们的分类。

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.