Abstract : |
Mirror symmetry is a phenomenon in geometry first noticed by string theorists
around 1990. This was first of all a duality between Calabi-Yau manifolds,
compact complex manifolds with nowhere vanishing top dimensional holomorphic
differential forms. Such manifolds were used by string theorists to
compactify 10 dimensional space-time, and this duality interchanged
fundamentally different topological invariants of the manifolds.
Furthermore, work of physicists Candelas, de la
Ossa, Green and Parkes used this duality to provide a spectacular,
though then conjectural, calculation of the number of rational curves
on a Calabi-Yau manifold. Since then these conjectural calculations
have been verified, however there is yet to be a fundamental mathematical
understanding of this phenomenon.
I will explore the geometric relationship between mirror pairs of
Calabi-Yau manifolds, by way of the Strominger-Yau-Zaslow conjecture,
which proposed that mirror symmetry could be explained by the existence
of dual torus fibrations on mirror pairs of Calabi-Yau manifolds, with
fibres having special properties, including volume minimizing properties.
I will discuss progress on this conjecture in the past few years, and
in particular illustrate how the conjecture works on a purely topological
level by showing it explains mirror symmetry in the most famous example,
the quintic threefold in projective four-space and its mirror.
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