Higher-Dimensional Analogs of Stable Curves

该项目建立在PI的工作基础上，将稳定的n点曲线推广到更高的维度：2维的稳定对和具有群作用的稳定对：环面、阿贝尔、半阿贝尔和约化。他将研究稳定对及其模的几个新应用，包括环托雷利映射；不规则周期多面体瓷砖的表征约化变种的环面退化及其在镜像对称上的应用。代数几何的主要主题-研究者工作的主要领域-是描述多项式方程的解，这些方程对数学及其在科学和工程中的应用具有根本的重要性。稳定曲线，其高维的推广研究者将继续研究，已经看到令人兴奋的应用在数论和弦理论的物理。

DMS-0401795Valery AlexeevThe project builds on PI's work towards generalizations of stable n-pointed curves to higher dimensions: stable pairs of dimension two and stable pairs with group action: toric, abelian, semiabelian and reductive. He will investigate several new applications of stable pairs and their moduli, including toric Torelli map; characterization of non-regular periodic polyhedral tilings; toric degenerations of reductive varieties with applications to mirror symmetry.The main subject of algebraic geometry - the primary field of the investigator's work - is to describe solutions of polynomial equations that are of fundamental importance to mathematics and its applications to science and engineering. Stable curves, whose higher-dimensional generalizations the investigator will continue to study, have seen exciting applications in number theory and in string theory in physics.