The Cohomology Rings of Moduli Spaces

点击翻译按钮获取中文摘要

DMS-0432701Linda ChenThe investigator will study cohomology rings and Chow rings of various moduli spaces and parameter spaces in algebraic geometry. In particular, she will study Gromov-Witten invariants of toric Deligne-Mumford stacks, (equivariant) cohomology of quot schemes, Chow groups of Fulton-Macpherson spaces, and the birational geometry of moduli spaces of curves. She will also study extensions of Schubert calculus which arise in the study of quantum cohomology. These are distinct projects, but are similar in the fact that each of the spaces involved has a rich combinatorial structure to be analyzed.It is a fundamental problem in algebraic geometry to understand objects satisfying certain geometric criteria, often naturally parametrized by an algebraic scheme or stack. Such moduli and parameter spaces have recently been at the forefront of mathematical and scientific research, much of which has grown out of Kontsevich's introduction of the moduli space of stable maps in the 1990's. They are a source of particular excitement as part of a collision of the fields of algebraic geometry, algebraic combinatorics, symplectic geometry, and theoretical physics.