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

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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.