Mirror symmetry in Algebraic Geometry

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Mirror symmetry is a phenomenon first discovered by string theorists in 1989. This phenomenon, when described in mathematical language, posits that certain kinds of geometric objects, known as Calabi-Yau manifolds, come in pairs X, Y. Further, mirror symmetry posits an intricate relationship between the geometry of the members of the pair. This relationship can be summarized by the rough statement that the symplectic geometry of X is isomorphic to the complex geometry of Y. Mathematical explorations of mirror symmetry have led to deep and profound insights in algebraic, logarithmic, symplectic and differential geometry, as well as algebraic combinatorics and representation theory. Working with Siebert, I have developed a program (colloquially referred to as the Gross-Siebert program) for exploring the underlying geometry of mirror symmetry using methods from algebraic and tropical geometry. I propose to use the techniques developed with Siebert to make significant breakthroughs in our understanding of mirror symmetry. Recently, we gave a general mirror construction for log Calabi-Yau pairs and maximally unipotent degenerations of Calabi-Yau manifolds. This allows the possibility of dramatic progress in the subject. The construction of a mirror goes by way of the construction of its coordinate ring. These coordinate rings can be viewed as the degree zero part of a `relative quantum cohomology ring.' I plan to generalize the construction of this ring to all degrees. A proof of mirror symmetry at genus zero could then be realised by constructing an isomorphism between this ring and the ring of polyvector fields of the mirror. I will also use the new constructions of mirrors to develop powerful new methods for constructing algebraic varieties, and I will develop a range of practical techniques for understanding the mirrors constructed. I also propose to explore a range of other enumerative invariants in the context of the Gross-Siebert program.