Quantum cohomology, the theory of Frobenius manifolds and the relations to integrable systems have been flourishing areas since the early 1990s. A conference was organized at the Max-Planck-Institute for Mathematics to bring together leading experts in these areas. This volume originated from that meeting and presents the state of the art in the subject.

Frobenius manifolds are complex manifolds with a multiplication and a metric on the holomorphic tangent bundle, which satisfy several natural conditions. This notion was defined in 1991 by Dubrovin, motivated by physics results. Another source of Frobenius manifolds is singularity theory. Duality between string theories lies behind the phenomenon of mirror symmetry. One mathematical formulation can be given in terms of the isomorphism of certain Frobenius manifolds. A third source of Frobenius manifolds is given by integrable systems, more precisely, bihamiltonian hierarchies of evolutionary PDE's. As in the case of quantum cohomology, here Frobenius manifolds are part of an a priori much richer structure, which, because of strong constraints, can be determined implicitly by the underlying Frobenius manifolds.

This volume is suitable for graduate students and research mathematicians interested in geometry and topology.

A publication of Vieweg+Teubner. The AMS is exclusive distributor in North America. Vieweg+Teubner Publications are available worldwide from the AMS outside of Germany, Switzerland, Austria, and Japan.

Readership

Graduate students and research mathematicians interested in geometry and topology.

Table of Contents

• A. Douai and C. Sabbah -- Gauss-Manin systems, Brieskorn lattices and Frobenius structures (II)
• J. Fernandez and G. Pearlstein -- Opposite filtrations, variations of Hodge structure, and Frobenius modules
• E. Getzler -- The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants
• A. B. Givental -- Symplectic geometry of Frobenius structures
• C. Hertling and Yu. I. Manin -- Unfoldings of meromorphic connections and a construction of Frobenius manifolds
• R. Kaufmann -- Discrete torsion, symmetric products and the Hilbert scheme
• X. Liu -- Relations among universal equations for Gromov-Witten invariants
• A. Losev and Yu. I. Manin -- Extended modular operad
• S. Merkulov -- Operads, deformation theory and F-manifolds
• A. Polishchuk -- Witten's top Chern class on the moduli space of higher spin curves
• K. Saito -- Uniformization of the orbifold of a finite reflection group
• I. Satake -- The Laplacian for a Frobenius manifold
• B. Siebert -- Virtual fundamental classes, global normal cones and Fulton's canonical classes
• A. Takahashi -- A note on BPS invariants on Calabi-Yau 3-folds
• List of Participants