Enumerative Geometry and Topology of Moduli Spaces

DMS-0401128 Michael Thaddeus 该项目包括量子上同调主题的几个变体，旨在构建有趣的模空间并研究其枚举几何。 该项目的大部分内容涉及从曲线到循环群的全纯映射空间。 这是有限维的，可以像稳定映射空间一样紧致化，允许定义和计算循环群的Gromov-Witten不变量。 这应该允许一个富有成效的推广非常活跃的领域的量子上同调的齐次空间的无限维设置仿射李群。 量子上同调的另一个变体是陈阮的弦上同调或轨道上同调：我们将寻求这个理论的一个版本的韦尔定理，并研究B场或grebe的作用。 另一个是科茨最近发展的量子黎曼-罗克理论，它将应用于曲线的对称乘积，这是代数几何的研究，数学中最经典的部分之一，涉及多项式方程的求解。 但它是由全纯曲线理论中的一些最新思想所告知的，特别是Gromov-Witten理论，该理论试图计算给定曲面（或高维空间）上给定形状或位置的曲线的数量。 这在很大程度上是由量子场论推动的，希望该项目最终能与物理学的思想联系起来。

DMS-0401128Michael Thaddeus This project encompasses several variations on the theme of quantum cohomology, aiming to construct interesting moduli spaces and study their enumerative geometry. The bulk of the project concerns the space of holomorphic maps from a curve to a loop group. This is finite-dimensional and can be compactified like a space of stable maps, allowing Gromov-Witten invariants of loop groups to be defined and calculated. This should allow a fruitful generalization of the very active field of quantum cohomology of homogeneous spaces to the infinite-dimensional setting of affine Lie groups. Another variation of quantum cohomology is the stringy or orbifold cohomology of Chen-Ruan: a version of the Weil conjectures for this theory will be sought, and the role of a B-field or grebe investigated. Yet another is the quantum Riemann-Roch theory recently developed by Coates, which will be applied to symmetric products of curves.This is research in algebraic geometry, one of the most classical parts of mathematics, concerned with finding solutions of polynomial equations. But it is informed by some of the latest ideas in the theory of holomorphic curves, and especially the Gromov-Witten theory, which seeks to count the numbers of curves on a given surface (or a space of higher dimension) of a given shape or position. This in turn is largely motivated by quantum field theory, and it is hoped that the project will ultimately make contact with ideas from physics.