MODULI OF CURVES, QUANTUM COHOMOLOGY AND TOPOLOGICAL FIELD THEORY

Seminar outline:

Part I: Moduli of curves and stable maps.

Teichmuller space of genus g. Deformation of Riemann surfaces via deformation of Fuchsian groups, deformation of conformal structures, and deformation of complex structures. Complex analytic theory of Teichmuller spaces. Compactification of the moduli space. Stable maps and their moduli spaces. Gromov-Witten invariants. A construction of moduli spaces and their compactification. Compactness and transversality theorems. Composition low and the quantum cohomology ring. Mirror symmetry conjecture. Applications to enumerative geometry.

Part II: Topological field theory.

Frobenius pairs. Dubrovin structures. Frobenius structures. Darboux-Egoroff equations. Euler field. Potential of Dubrovin structure. WDVV equations and their simplest solutions. Flat metrics and co-metrics generated by Dubrovin structure. Example: Dubrovin structure on the orbit spaces of Coxeter groups. Relation to Gromov-Witten invariants and quantum cohomology.

Prerequisites:

Basic courses in linear algebra, differential geometry and topology.

Literature:

M. Audin. Cohomologie quantique, Seminaire Bourbaki, 1995-96, n. 806; M. Audin. An introduction to Frobenius manifolds, moduli spaces of stable maps and quantum cohomology, Preprint, IRMA, Universite Louis Pasteur, Strasbourg, 1998;B. A. Dubrovin. Geometry of 2D topological field theories, In: Lect. Notes Math. 1620, Springer, 1996, pp. 120-348;W. Fulton and R. Pandharipande. Notes on stable maps and quantum cohomology, In: Algebraic Geometry, Santa Cruz 1995 (Proc. Symp. Pure Math. vol. 62, part 2), AMS, 1997, pp. 45-96;Y. Imayoshi and M. Taniguchi. An introduction to Teichmuller spaces, Springer, 1992;S. M. Natanzon. Geometry of two-dimensional topological field theories, Independent Moscow University, 1998;Quantum fields and strings: A course for mathematicians. P. Deligne et al., eds., AMS, 1999;Y. Ruan and G. Tian. A mathematical theory of quantum cohomology; J. Diff. Geom. 42 (1995), no. 2, pp. 259-367.