Geometric aspects of integrability

可积性的几何方面

• 批准号: 2613790
• 金额: --
• 依托单位: Loughborough University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2613790
• 关键词: Geometric; aspects; integrability

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.

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。最著名的例子是弗罗贝纽斯流形的理论，它是由Dubrovin介绍的与二维拓扑场论的关系。对数弗罗贝纽斯结构的分类可以归结为某些装备超平面安排的描述，推广了考克斯特安排。这一问题在三维空间中仍然是一个未解决的问题，也是今后研究的方向之一，另一个方向是与Yang-Baxter映射有关的问题，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，相应的映射可以被证明与del Pezzo曲面的对称性有关，这可以用于它们的分类。