We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F(t1, ..., tn) of WDVV equations of associativity polynomial in t1, ..., tn-1, exp tn.

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Extended affine Weyl groups and Frobenius manifolds

For the root system of type $B_l$ and $C_l$, we generalize the result of \cite{DZ1998} by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the Dynkin diagram instead of a particular choice of \cite{DZ1998}.

Extended affine Weyl groups and Frobenius manifolds -- II

https://people.sissa.it/~dubrovin/Papers/2008/dubrovin_liu_zhang_2008.pdf

Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures

In 1891 A. Hurwitz considered the problem of enumerating the ramified coverings of the two-dimensional sphere by two-dimensional surfaces with fixed types of branching over the branch points. In the original setting the problem was reformulated in terms of characters of the symmetric group. Recently it turned out that the problem is also very closely connected with diverse physical theories, with singularity theory, and with the geometry of the moduli spaces of complex curves. The discovery of these relationships has led to an enlargement of the class of cases in which the enumeration yields explicit formulae, and a clarification of the nature of the classical results. This survey is devoted to a description of the contemporary state of this thriving topic and is intended for experts in topology, the theory of Riemann surfaces, combinatorics, singularity theory, and mathematical physics. It can also serve as a guide to the modern literature on coverings of the sphere.

Ramified coverings of the two-dimensional sphere and the intersection theory in spaces of meromorphic functions on algebraic curves

Branched covers of s2 and braid groups

The geometry of a bifurcation diagram in the base of a versal deformation of a singularity is studied for single singularities on a manifold with boundary. In particular, vector fields and groups of diffeomorphisms are studied which are defined in a neighborhood of a bifurcation diagram as are stratification of a bifurcation diagram and decomposition of singularities.

O. V. Lyashko

Papers

Geometry of bifurcation diagrams