Advances in Algebraic Geometry Motivated by Physics

✍ Scribed by Previato E. (ed.)

• Publisher: AMS
• Year: 2001
• Tongue: English
• Leaves: 310
• Series: Contemporary Mathematics 276
• Category: Library

✦ Synopsis

Our knowledge of objects of algebraic geometry such as moduli of curves, (real) Schubert classes, fundamental groups of complements of hyperplane arrangements, toric varieties, and variation of Hodge structures, has been enhanced recently by ideas and constructions of quantum field theory, such as mirror symmetry, Gromov-Witten invariants, quantum cohomology, and gravitational descendants.

These are some of the themes of this refereed collection of papers, which grew out of the special session, "Enumerative Geometry in Physics," held at the AMS meeting in Lowell, MA, April 2000. This session brought together mathematicians and physicists who reported on the latest results and open questions; all the abstracts are included as an Appendix, and also included are papers by some who could not attend.

The collection provides an overview of state-of-the-art tools, links that connect classical and modern problems, and the latest knowledge available.

Readership: Graduate students and research mathematicians interested in algebraic geometry and related disciplines.

✦ Table of Contents

A. I. Suciu -- Fundamental groups of line arrangements: Enumerative aspects
Enumerative or reality problems
S. J. Kovács -- Number of automorphisms of principally polarized abelian varieties
F. Sottile -- Rational curves on Grassmannians: Systems theory, reality, and transversality
Variational and moduli problems
D. Abramovich and A. Bertram -- The formula 12=10+2×1 and its generalizations: Counting rational curves on F2
D. Abramovich and F. Oort -- Stable maps and Hurwitz schemes in mixed characteristics
L. Caporaso -- On modular properties of odd theta-characteristics
E. Cattani and J. Fernandez -- Asymptotic Hodge theory and quantum products
H. Clemens -- On rational curves in n-space with given normal bundle
R. Vakil -- A tool for stable reduction of curves on surfaces
Mirror symmetry and Gromov-Witten invariants
D. A. Cox, S. Katz, and Y.-P. Lee -- Virtual fundamental classes of zero loci
T. J. Jarvis, T. Kimura, and A. Vaintrob -- Gravitational descendants and the moduli space of higher spin curves
B. Kreußler -- Homological mirror symmetry in dimension one
A. R. Mavlyutov -- The Hodge structure of semiample hypersurfaces and a generalization of the monomial-divisor mirror map
A. Polishchuk and A. Vaintrob -- Algebraic construction of Witten's top Chern class
A. Postnikov -- Symmetries of Gromov-Witten invariants
S. Rosenberg and M. Vajiac -- Gauge theory techniques in quantum cohomology
C. Woodward -- Gromov-Witten invariants of flag manifolds and products of conjugacy classes
Appendix
E. Previato -- The Lowell meeting