Theta Functions and Log Calabi Yau Varieties

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One of the central open questions in geometry, growing out of theoretical physics, is the mirror symmetry conjecture which states that a class of geometric objects, of fundamental importance in many branches of mathematics, as well as theoretical physics, so called Calabi-Yau manifolds, come in natural pairs, which are "mirror " in that one geometric aspect of one is equivalent to a quite different geometric aspect of its pair. The conjecture is important both because Calabi-Yau manifolds appear in many diverse parts of mathematics, but also because the mirror aspect unites two branches of mathematics (symplectic and complex geometry, which measure the two different aspects) in ways not previously anticipated. Perhaps the most fundamental question is how, given one such object, one obtains its mirror partner. The main focus of the proposed research is a detailed program for constructing the mirror to a given Calabi-Yau. Under this award, the PI will continue mentoring postdocs and training of graduate students. In addition, he will also mentor high school students in mathematics.The award is based on the PI's conjecture (joint with Gross, Hacking, and Siebert) that certain manifolds (log Calabi Yaus with maximal boundary, or compact Calabi Yaus close to a the large complex structure limit point of the moduli space) come with canonical functions, vastly generalizing the classical theta functions for polarized Abelian varieties. The proposal is to construct these theta functions and use them to prove a surprisingly simple conjecture on moduli of polarized Calabi Yau pairs, and to extend to higher dimensions the PI's construction (joint with Gross, Hacking, and Siebert) of a geometric compactification of the moduli space of polarised K3 surfaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.