Calculation of Gross-Siebert mirror rings for log Calabi-Yau manifolds.

镜像对称性是由弦理论家在周围首次提出的，它假定两个不同的几何对象之间存在二元性。镜像对称现在对代数、微分和辛几何产生了广泛的影响，在数学中是一个非常广泛和活跃的领域。格罗斯和西伯特最近发展了一种非常普遍的构造镜面的理论方法，但到目前为止，使用这种方法的显式计算很少。王的项目旨在开发一套计算格罗斯-西伯特镜面的工具来记录Calabi-Yau流形。Wang将首先考虑以光滑反正则因子为边界的一对Fano簇的函数环。这将导致包含有理曲线的新公式在两点满足反正则除数。这也将与Fan，Wu和You最近的工作联系在一起，他们用orbilold不变量构造了一个类似的理论。当边界光滑时，Wang发展了一个令人满意的理解后，Wang将把这些技巧推广到具有正常交叉点边界的新例子。这将导致新的镜对结构，并极大地提高我们对Gross-Siebert镜结构的理解。

Mirror symmetry, first introduced by string theorists around 1989, posits a duality between two different geometric objects. Having now had a broad impact on algebraic, differential and symplecticgeometry, mirror symmetry is a very broad and active field in mathematics. Gross and Siebert have recently developed a theoretical approach to construction of mirrors in great generality, but to datethere have been very few explicit calculations using this approach.Wang's project aims to develop a suite of tools for calculation of the Gross-Siebert mirrors to log Calabi-Yau manifolds. Wang will initially consider the ring of functions associated to a pair ofa Fano variety with a smooth anti-canonical divisor as boundary. This will lead to new formulas involving rational curves meeting the anti-canonical divisor at two points. It will also makeconnection to recent work of Fan, Wu and You, who have constructed a similar theory using orbifold invariants.After developing a satisfactory understanding when the boundary is smooth, Wang will extend these techniques to new examples with a normal crossings boundary. This will lead to new constructions of mirror pairs and greatly enhance our understanding of the Gross-Siebert mirror construction.