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An invitation to Frobenius manifolds

ARGOMENTI: Seminari

SEMINARIO DOTTORATO
Wednesday 25 February 2009 h. 15:00, room 2AB/40
Luca Philippe MERTENS (Ph.D. in Pure Math., SISSA Trieste)
"An invitation to Frobenius manifolds"

-Abstract
Frobenius manifolds are geometric structures encoding the dispersionless limit of a bihamiltonian integrable hierarchy. They were introduced by B. Dubrovin to study the remarkable connection between 1+1 integrable systems, 2D topological field theories and Gromov-Witten invariants of symplectic manifolds discovered by E. Witten and M. Kontsevich for the case of the KdV hierarchy. The first part of the talk will be an introduction to key ideas and definitions. We will define an appropriate class of integrable hierarchies and we will show how one can associate a Frobenius manifold to them, and vice versa. We will present in detail the example of the Toda Hierarchy, encoding the Gromov-Witten Invariants of the complex projective line. The second part of the talk will focus on recent developments of the theory related to 2+1 integrable systems. This class of hierarchies naturally lead to the notion of infinite dimensional Frobenius manifold. We will present the case of the 2D Toda hierarchy, highlighting main differences with respect to the finite dimensional case and pointing out future applications of the theory.

Rif. int. C.Marastoni, T. Vargiolu, M. Dalla Riva

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