Frobenius manifolds lie at the crossroads of many areas of mathematics and physics, such as singularity theory, quantum field theory, mirror symmetry, and enumerative geometry as well as integrable systems. They were introduced by Boris Dubrovin as a geometric way to understand the nonlinear associativity equations (the Witten-Dijkgraaf-Verlinde-Verlinde, or WDVV, equations) that arise in certain topological quantum field theories in which the “observables” of the theory give topological information about certain manifolds.

The link with the theory of integrable systems has developed over the last fifteen years and gives a new perspective on the theory. Soliton equations, such as the KdV and Toda equations are seen as “deformations” of more basic “dispersionless” equations, or more accurately, equations of hydrodynamic type. This approaches merges geometry, deformation theory and the theory of bi-Hamiltonian structures to provide an extremely elegant mathematical theory.

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