Affine Manifolds, Log Geometry, and Mirror Symmetry

University Of California-San Diego, La Jolla CA

Investigators

Abstract

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 an intersection complex, which is an affine manifold with singularities. Conversely, given an affine manifold with singularities, Gross and Siebert showed it is possible to build explicitly such a degeneration in terms of "tropical" data on the affine manifold with singularities. M. Gross plans to study applications of this construction, and in particular intends to work towards the goal of showing that this construction provides a geometric explanation for mirror symmetry. Ultimately, it will be shown that the same tropical structures determine both numbers of rational curves and period calculations on the two different sides of 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.