M. Gross plans to study the geometry of mirror symmetry for Calabi-Yau manifolds. This will be done from the perspective of an algebro-geometric version of the Strominger-Yau-Zaslow conjecture introduced by M. Gross and B. Siebert. Associated to certain sorts of large complex structure limit degenerations of Calabi-Yau manifolds one can define a dual intersection complex, which is an affine manifold with singularities. Conversely, given an affine manifold with singularities, it is possible to build the degenerate fibre of such a degeneration, along with the structure of a log scheme. Gross first plans to complete the correspondence between affine manifolds with singularities and large complex structure limit degenerations of Calabi-Yau varieties by showing that these log schemes can be smoothed. Furthermore, Gross plans to compute invariants of these smoothings in terms of structures on the affine manifolds. One expects Hodge numbers, and more importantly, variations of Hodge structure, can be calculated directly from computations on the affine manifold. The ultimate goal will be to compare these results with calculations of Gromov-Witten invariants for the mirror, thereby eventually providing an explanation for mirror symmetry.
The work proposed by M. Gross lies at the intersection of string theory and geometry. String theory replaces the traditional notion of the point particle with a small loop of string, moving through space-time. To make string theory compatible with quantum mechanics, space-time must be ten-dimensional. Since space-time appears four-dimensional, one expects six of these dimensions to be a very small `curled up' geometric object. These geometric objects are called Calabi-Yau manifolds. In the early 1990s, string theorists proposed a remarkable association between completely different Calabi-Yau manifolds: certain calculations extremely difficult to perform on one Calabi-Yau manifold could be completed by performing completely different, and much easier, calculations on a different Calabi-Yau manifold. This discovery was known as mirror symmetry. Since this time, many geometers have been trying to understand the mathematics behind this miraculous observation. The work of M. Gross hopes to give mathematical insight and explanation for the phenomenon of mirror symmetry.
