Higher-Dimensional Analogs of Stable Curves

DMS-0401795 Valery Alexeev该项目建立在PI的工作上，致力于将稳定的n点曲线推广到更高的维度：二维的稳定对和具有群作用的稳定对：复曲面，阿贝尔，半阿贝尔和还原。他将研究稳定对及其模的几个新应用，包括复曲面Torelli映射、非正则周期多面体镶嵌的表征;还原簇的环面退化及其在镜像对称中的应用。代数几何的主要课题--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。稳定曲线在数论和物理学中的弦论中有着令人兴奋的应用，我们将继续研究它的高维推广。

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.