Moduli of Pointed Curves and Relative Stable Maps

研究者建议使用Kontsevich的“稳定地图”空间（最初是由近十年的数学物理学激发的）和相关对象来解决各种领域的问题。传统上，稳定映射的空间是利用Deligne和Mumford定义的基本“曲线模空间”的事实来研究的。本文提出通过研究曲线到变量的映射，反过来研究曲线的模空间。一些被提议的工作可能会依赖于李军最近对Kontsevich工作的扩展，即代数范畴中“相对稳定映射”空间的定义。结点代数曲线是代数几何中的一个强大工具，这一点早已为人所知。它们可以被认为是有孔的表面（想象一个球、一个甜甜圈或一个法式油条），上面有一对对点“粘在一起”。本世纪初，物理学中弦理论的思想引入了“稳定映射”，将某些类型的节点曲线映射参数化到另一个空间。事实证明，这一发展成果丰硕，激发了各个领域的进步。研究者的研究领域是这些思想在代数几何领域的应用，特别是在许多其他领域的应用（如枚举几何、算术几何、组合几何和物理）。

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).