Activity on Frobenius manifolds, quantum cohomology, and singularities

dedicated to Yuri I. Manin on the occasion of his 65th birthday

May - August 2002

Organizers: B. Dubrovin, C. Hertling, Yu. Manin, M. Marcolli, K. Saito

Frobenius manifolds are manifolds with a rich additional structure, a multiplication and a metric on the tangent bundle. Frobenius manifolds turn up in quantum cohomology, where they encode information of Gromov-Witten invariants of genus 0, and in singularity theory, as base spaces of unfoldings of singularities. Isomorphism of certain of the Frobenius manifolds on both sides is one version of mirror symmetry.

The activity will bring together researchers on these and related subjects. The main event of it will be a

Workshop on July 8-19, 2002.

The expected participants include: Barannikov, Batyrev, Behrend, Bertola, Caporaso, Dubrovin, Fantechi, Fernandez, Gathmann, Gerasimov, Getzler, Givental, Hertling, Ionel, Jarvis, Joe, Kaufmann, Kim, Kimura, Kock, Lee, Liu, Mann, Manin, Marcolli, Matsuo, Merkulov, Noumi, Ohta, Ohyama, Okounkov, Ono, Pandharipande, Polishchuk, Pribik, Sabbah, K. Saito, M.-H. Saito, Satake, Siebert, Soibelman, Strachan, Takahashi, Terao, Tsuchiya, Zhang.

Program

Monday July 8:

• 09:50-10:00 Greeting
• 10:00-10:50 A.B. Givental: Quantum Riemann-Roch and Lefschetz
• 10:50-11:20 tea break
• 11:20-12:10 S. Merkulov: Operads, deformation theory and homotopy Frobenius manifolds
• 14:10-15:00 A. Okounkov: Gromov-Witten theory of target curves I
• 15:10-16:00 R. Pandharipande: Gromov-Witten theory of target curves II
• 16:00-16:30 afternoon tea

• 10:00-10:50 B. Siebert: Affine manifolds and mirror symmetry
• 10:50-11:20 tea break
• 11:20-12:10 E. Ionel: Counting embedded curves in Calabi-Yau 3-folds
• 14:10-15:00 Yu. Manin: Moduli spaces of painted stable curves and deformations of Frobenius manifolds
• 15:10-16:00 Y.P. Lee: Quantum K-theory and Frobenius manifold
• 16:00-16:30 afternoon tea

• 10:00-10:50 X. Liu: Quantum product on the big phase space
• 10:50-11:20 tea break
• 11:20-12:10 T. Kimura: Comparing cohomological field theories
• 14:10-15:00 T. Jarvis: New cohomological field theories arising from quasi-homogeneous polynomials
• 15:10-16:00 A. Polishchuk: Witten's top Chern class on {\overline{M}^{1/r}_{g,n}}
• 16:00-16:30 afternoon tea

• 10:00-10:50 B. Dubrovin: Moduli of Frobenius manifolds and normal forms of integrable PDEs
• 10:50-11:20 tea break
  
• 11:20-12:10 Y. Zhang: Quantum cohomology of CP^1 and the Toda hierarchy
• 14:10-15:00 E. Getzler: The equivariant Toda lattice and the equivariant Gromov-Witten invariants of the projective line
• 15:10-16:00 A. Takahashi: BPS invariants on Calabi-Yau 3-folds
• 16:00-16:30 afternoon tea

• 10:00-10:50 M.-H. Saito: Geometric characterizations of equations of Painlevé type
• 10:50-11:20 tea break
• 11:20-12:10 M. Noumi: Some remarks on the discrete symmetry of Painlevé equations
• 14:10-15:00 A. Matsuo: On the symmetry of vertex operator algebras
• 15:00-15:30 tea break
• 15:30-16:20 A. Tsuchiya: Conformal Field Theory associated with regular vertex operator algebra

• 10:00-10:50 K. Saito: Elliptic flat structure (heighest weight representations of elliptic algebras)
• 10:50-11:20 tea break
• 11:20-12:10 M. Bertola: Invariant algebra of Jacobi groups, relation with current algebras on elliptic curves and (multiple) flat structures
• 14:10-15:00 A. Gerasimov: Quantization of Special geometry and Holomorphic Anomaly equation
• 15:10-16:00 I. Satake: The Laplacian on the deformation space of singularity
• 16:00-16:30 afternoon tea

• 10:00-10:50 C. Hertling: Unfoldings of meromorphic connections and a construction of Frobenius manifolds
• 10:50-11:20 tea break
• 11:20-12:10 C. Sabbah: Gauss-Manin systems, Brieskorn lattices and Frobenius structures
• 14:10-15:00 H. Terao: The Hodge filtration and the contact-order filtration of derivations of Coxeter arrangements
• 15:10-16:00 I. Strachan: Bi-Hamiltonian structures on caustics and discriminants
• 16:00-16:30 afternoon tea

• 10:00-10:50 K. Ono: Floer homology of periodic Hamiltonian systems with integer coefficient
• 10:50-11:20 tea break
• 11:20-12:10 A. Gathmann: Relative Gromov-Witten invariants and invariants of hypersurfaces
• 14:10-15:00 R. Kaufmann: Orbifolding and second quantization
• 15:10-16:00 K. Behrend: Equivariant vector fields and the Cohomology of stable map spaces
• 16:00-16:30 afternoon tea

• 10:00-10:50 D. Joe: Equivariant mirrors and Virasoro conjecture for the flag manifolds
• 10:50-11:20 tea break
• 11:20-12:10 B. Kim: Counting Gromov-Witten invariants for Grassmannians using C^*-actions
• 14:10-15:00 J. Kock: Tangency quantum cohomology
• 15:10-16:00 A.B. Givental: A_{n-1} singularities and nKdV hierarchies
• 16:00-16:30 afternoon tea

• 10:00-10:50 J. Fernandez: Frobenius modules and Hodge asymptotics
• 10:50-11:20 tea break
• 11:20-12:10 B. Fantechi: Orbifold cohomology for global quotients
• 14:10-15:00 S. Barannikov: t.b.a.
• 15:00-15:30 afternoon tea

Enquiries may be addressed to:
C. Hertling hertling@mpim-bonn.mpg.de

The activity is financed by the Sofja Kovalevskaja Award of the Award winner Prof. Dr. Matilde Marcolli. This Award is under the sponsorship of the Alexander von Humboldt Foundation, the Federal Ministry of Education and Research and the Programme for Investment in the Future (ZIP) of the German Government.