Calabi-Yau manifolds and mirror symmetry

University Of California-San Diego, La Jolla CA

Investigators

Abstract

ABSTRACT DMS - 0204326. M. Gross plans to study the geometry of mirror symmetry for Calabi-Yau manifolds. This will be done from the perspective of the Strominger-Yau-Zaslow conjecture and algebro-geometric analogues of this conjecture currently being developed by M. Gross and B. Siebert. These ideas will be pursued with a goal of finding a general mirror symmetry construction, associating to any polarized maximally unipotent degeneration of Calabi-Yau manifolds a mirror family of polarized Calabi-Yau manifolds. Implications of these ideas to understanding the behaviour of Ricci-flat metrics near maximally unipotent degenerations and the counting of rational curves using the notion of log Gromov-Witten invariants will also be pursued. 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.