Geometry of Mirror Symmetry

摘要奖：DMS-0405859首席研究员：埃里克Zaslow这个项目解决镜像对称的几何方面。 该PI将研究Strominger-Yau-Zaslow猜想的几何形状，通过观察在大的复杂结构极限附近的Calabi-Yauthreefold的结构图，在那里特殊的拉格朗日环面纤维化崩溃。 第一个方案是在Y型奇异轨迹外的开球上生成一个真实的Monge-Ampere度量，然后研究它的单值性。 这一区域的知识是必不可少的，以获得更好的理解卡-丘，包括盘瞬子修正的作用。 相关的工作是研究一般Ricci平坦流形中平坦或Ricci平坦环面纤维化的几何，以及研究特殊Lagrange子流形和其他校准子流形通过改变度量远离Ricci平坦子流形的方法的计数问题。 最后一个项目是研究拓扑顶点下的可积层次。该项目研究镜像对称物理现象背后的数学。 镜像对称是指两种不同的物理理论对自然界做出相同的预测。 在数学上，等价性背后有着深刻的联系，这个项目着眼于等价性的几何学。 理解镜像对称的几何学将导致对数学结构的更广泛的理解，并且可以想象物理学。 例如，镜像对称有助于拓扑场论的计算，拓扑场论已经被证明与粒子物理学基础的传统规范理论有计算相关性。 在一个更基本的数学层面上，理解镜像对称的几何形状可以导致对卡拉比在1957年提出并在1979年被Yauin证明存在的特殊结构的具体描述。 尽管如此，我们对几何学的这一重要方面知之甚少。 数学和物理之间丰富的相互作用可以教给我们很多东西。

AbstractAward: DMS-0405859Principal Investigator: Eric ZaslowThis project addresses geometric aspects of mirror symmetry. ThePI will study the geometry of the Strominger-Yau-Zaslowconjecture by looking at the conjectural picture of a Calabi-Yauthreefold near the large complex structure limit, where thespecial Lagrangian torus fibration collapses. The resultingmanifold is a real integer affine manifold with singularities.The first project will produce a real Monge-Ampere metric on anopen ball outside a Y-shaped singular locus, then study itsmonodromy. A knowledge of this region is essential to gaining aglobal understanding of the Calabi-Yau, including the role ofdisc instanton corrections. Related projects are to study thegeometry of flat or Ricci-flat torus fibrations in Ricci-flatmanifolds in general, and to study the counting problem ofspecial Lagrangian and other calibrated submanifolds by varyingthe metric away from the Ricci flat one. The final project isinvestigate integrable hierarchies underlying the topologicalvertex.The project studies the mathematics behind the physicalphenomenon of mirror symmetry. Mirror symmetry is when twodifferent physical theories give rise to the same predictionsabout nature. Mathematically, there are deep connections behindthe equivalence, and this project looks at the geometry of theequivalence. Understanding the geometry of mirror symmetry willlead to a broader comprehension of the structure of mathematics,and conceivably physics. For instance, mirror symmetry aids inthe calculation of topological field theories, which have beenshown to have computational relevance to the conventional gaugetheories that underly particle physics. At a more basic,mathematical level, understanding the geometry of mirror symmetrycan lead to a concrete description of the special structuresconjectured by Calabi to exist in 1957 and proved to exist by Yauin 1979. Still, we know very little about this important aspectof geometry. The rich interplay of mathematics and physics hasmuch to teach us.