Calabi-Yau Threefolds and Birational Geometry

格罗斯9700761首席研究员将继续他对Calabi-Yau三重奏的研究，这是一个复杂的三重折，第一个陈级消失。他们几何学的各个方面都将被讨论。他将继续发展形变理论的工具，以了解Calabi-Yau三重模空间的各种相互联系。这些工具还将被用于研究有理Gorenstein三重奇点的形变理论，这可以给出对四维双曲面几何的见解。Calabi-Yau三重折叠也将考虑到最近提出的建造Strominger，Yau和Zaslow以实现镜像对称。在其他问题中，这将涉及到研究特殊的拉格朗日环面的性质，以及特殊的拉格朗日子流形在Calabi-Yau流形的几何中所起的作用。这是在代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的25年里，它已经取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。

Gross 9700761 The principal investigator will continue his study of Calabi-Yau threefolds, which are complex threefolds with vanishing first Chern class. Various aspects of their geometry will be addressed. He will continue developing tools of deformation theory in order to understand how various moduli spaces of Calabi-Yau threefolds interconnect. These tools will also be applied to study the deformation theory of rational Gorenstein threefold singularities, which can give insights into four-dimensional birational geometry. Calabi-Yau threefolds will also be considered in light of the recently proposed construction of Strominger, Yau, and Zaslow for mirror symmetry. Among other issues this will involve studying the nature of special Lagrangian torus fibrations and the role that special Lagrangian submanifolds play in the geometry of Calabi-Yau manifolds. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.