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

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 symplectic
geometry, 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 date
there 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 of
a 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 make
connection 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.