Past lectures

Abstract

Frobenius manifolds were discovered in the beginning of '90s as an axiomatization of certain structures appearing in 2D topological quantum field theory, in singularity theory, in the Hamiltonian theory of integrable systems, and in the theory of Gromov - Witten invariants of symplectic manifolds. The theory of Frobenius manifolds establishes remarkable relationships between these, sometimes rather distant, mathematical theories. The best understood one is the class of semisimple Frobenius manifolds. An insight into their geometrical structure allows to construct, using the theory of linear differential equations with rational coefficients, a complete set of local invariants of semisimple Frobenius manifolds. Applying the theory of Riemann - Hilbert problems one obtains the global classification of semisimple Frobenius manifolds. These ideas will be presented in the lectures. All the necessary constructions and results from geometry of Frobenius manifolds and from the theory of Riemann - Hilbert problem will be explained. A number of open problems will be formulated regarding the analytic theory of Frobenius manifolds as well as their applications to the bihamiltonian geometry of loop spaces.

References

B.Dubrovin, Geometry of 2D topological field theories. In: Integrable Systems and Quantum Groups. Eds. M.Francaviglia, S.Greco, Springer Lecture Notes in Math., vol. 1620 (1996), pp. 120 - 348.

B.Dubrovin, Painleve' transcendents and topological field theory. In: "The Painleve' property: one century later", R.Conte (Ed.), Springer Verlag, 1999, p.287-412.