Moduli of Pointed Curves and Relative Stable Maps

研究人员建议使用Kontsevich的“稳定映射”空间（最初是由数学物理学在过去十年中激发的）和相关的boundaries来解决各种领域的问题。传统上，稳定映射的空间一直是用Deligne和Mumford定义的基本“曲线的模空间”来研究的。 本文提出了从曲线到簇的映射研究，反过来研究曲线的模空间。 一些拟议中的工作可能会依赖于Jun Li最近对Kontsevich工作的扩展，即在代数范畴中定义“相对稳定映射”空间。 它们可以被认为是有孔的表面（想象一个球，一个甜甜圈，或者一个法式煎饼），有成对的点“粘在一起”。 本世纪初，物理学中弦理论的思想导致了“稳定映射”的引入，将某些类型的节点曲线映射参数化到另一个空间。 这一发展已被证明是令人难以置信的富有成效的，引发了各种领域的进步。调查员的研究领域是使用这些想法在领域ofalgebraic几何，特别是与应用到许多其他领域（如枚举几何，算术几何，组合数学，和物理）。

The investigator proposes to use Kontsevich's space of "stable maps"(originally motivated by mathematical physics last decade) and relatedobjects to tackle problems in a variety of fields. Traditionally, thespace of stable maps has been studied using facts about the fundamental"moduli space of curves" defined by Deligne and Mumford. The investigatorproposes to conversely study the moduli space of curves by studying mapsfrom curves to varieties. Some of the proposed work will likely rely onJun Li's recent extension of Kontsevich's work, the definition of a spaceof "relative stable maps" in the algebraic category.It has long been known that nodal algebraic curves are a powerful tool inalgebraic geometry. They can be thought of as surfaces with holes(picture a ball, a donut, or a french cruller) with pairs of points "gluedtogether". Earlier this decade, ideas from string theory in physics ledto the introduction of "stable maps", parametrizing certain kinds of mapsof nodal curves into another space. This development has proved to beincredibly fruitful, sparking advances in a variety of fields. Theinvestigator's area of research is the use of these ideas in the field ofalgebraic geometry, in particular with applications to many other fields(such as enumerative geometry, arithmetic geometry, combinatorics, andphysics).