The Cohomology Rings of Moduli Spaces

研究人员将研究代数几何中各种模空间和参数空间的上同调环和Chow环。特别是，她将研究Toric Deligne-Mumford堆栈的Gromov-Witten不变量，Quot格式的(等变)上同调，Fulton-Macpherson空间的Chow群，以及曲线的模空间的双曲几何。她还将研究舒伯特微积分的扩展，这是在量子上同调研究中出现的。这些都是不同的项目，但相似的事实是，每个涉及的空间都有丰富的组合结构需要分析。理解满足某些几何标准的对象是代数几何中的一个基本问题，通常由代数方案或堆栈自然地参数化。这种模空间和参数空间近来一直处于数学和科学研究的前沿，其中很大一部分起源于康采维奇在1990年代的《S》中引入了稳定映射的模空间。它们是代数几何、代数组合学、辛几何和理论物理领域碰撞的一部分，特别令人兴奋。

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.