This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The main objective of the proposed research is to find a geometrically meaningful compactification of the moduli space of polarized K3 surfaces, obtained as an instance of Mumford's theory of toroidal compactifications. The theory requires a fan structure on a certain cone (equivariant with respect to a certain discrete group), and the motivating observation is that elementary ideas from Mori theory and mirror symmetry produce a canonical such fan: Namely the cone in question turns out to be the cone of effective divisors in the total space of a natural moduli space, Dolgachev's mirror to moduli of polarized K3s, and the Mori fan is a canonical fan structure on the effective cone of any variety. The hope is that (near its boundary) the associated toric variety carries a canonical universal family of pairs of K3 surface with divisor. The existence of such a family would naturally generalize the mirror symmetry programs of Kontsevich-Soibelman and Gross-Siebert, and at the same time unifying this work with Thurston's work on triangulations of the 2-sphere, and Looijenga's conjecture on smoothings of cusp singularities.
According to quantum field theory the physical universe is controlled by a so called Calabi-Yau manifold -- a certain kind of geometric object. As a result it is important to understand the set of such objects -- the so called Moduli space of Calabi-Yau manifolds, and in particular to understand the behavior of this space at infinity, i.e. near its boundary, or equivalently, to understand how Calabi-Yau manifolds can degenerate. One case is well understood: A complex 1-dimensional Calabi-Yau manifold is the surface of a doughnut, and as one moves to the boundary of the moduli space, one loop on the doughnut shrinks, the limiting point on the boundary of the moduli space corresponds to crushing this loop to a point -- which yields the space you obtain by taking a real 2 dimensional sphere and bending it until two points touch each other. The goal of the project is to understand what happens in the next higher dimension, complex dimension two, and thus to describe the boundary behavior of the moduli space of so called K3 surfaces.