Moduli of Pointed Curves and Relative Stable Maps

这位研究人员建议使用康采维奇的“稳定地图”空间(最初是由过去十年的数学物理推动的)和与物体相关的空间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定义的基本的“曲线的模空间”来研究的。研究人员提出通过研究曲线到簇的映射来研究曲线的模空间。一些拟议的工作可能会依赖于李军最近对Kontsevich工作的扩展，即在代数范畴中定义了“相对稳定映射”的空间。很早就知道，节点代数曲线是代数几何中的一个强大工具。它们可以被认为是带有孔洞的表面(想象一个球、一个甜甜圈或一个法式油炸圈饼)，上面有两个“粘在一起”的点。本世纪初，从物理学中的弦理论到“稳定映射”的引入，将节点曲线的某些类型的映射参数化到另一个空间。事实证明，这一发展取得了令人难以置信的成果，在各个领域引发了进展。研究人员的研究领域是在代数几何领域中使用这些思想，特别是在许多其他领域(如计数几何、算术几何、组合学和物理学)中的应用。

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).