Geometric aspects of integrability

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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.