Lagrangian Torus Fibration of Calabi-Yau Manifolds and Application to Mirror Symmetry

摘要：DMS - 0104150PI：阮卫东。阮建议研究Calabi-Yau流形和更一般的Kaehler流形的拉格朗日振动结构，以及它们在镜像对称和其他Kahler几何问题中的应用。该研究的目的是进一步理解更一般的Calabi-Yau流形的拉格朗日环面振动结构，探索Kaehler流形中simticgeometry和complex geometry的相互作用，如Kontsevich的分类镜像对称程序所示。镜像对称起源于弦理论，它说我们有两种看起来完全不同的理论，描述着同样的物理现象。如果我们对同一个宇宙有两个不同的模型，我们可以期望其中一个模型可以填补理解另一个模型的空白。这个项目是为了理解这种对称原理。这一建议的结果应该引起各个学科的数学家以及弦理论物理学家的兴趣。

Abstract for DMS - 0104150PI: Wei-Dong RuanDr. Ruan proposes to study the Lagrangian fibration structure of Calabi-Yau manifolds and of more general Kaehler manifolds, and their applications in mirror symmetry and other Kahler geometry problems as well. The objective of the proposed research is to further inderstand the Lagrangian torus fibration structures for moregeneral Calabi-Yau manifolds, exploring the interplay of sympleticgeometry and complex geometry in Kaehler manifolds, as indicated by Kontsevich's categorical mirror symmetry program. Mirror symmetry has its origin in string theory andsays that we have two theories, which appear to be completely distinct, describing the same physics. If we have two different models of the same universe, one can expect that one of the models can fill in the gaps in understanding the other model. This project is towards understanding this symmetry principle.Results from this proposal should be of interest to mathematicians fromvarious disciplines as well as physicists in string theory.