Calabi-Yau Manifolds: Families, Fibrations, and Degenerations

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The aim of this project is to investigate the properties of a certain type of manifold, called a Calabi-Yau manifold. The special properties of Calabi-Yau manifolds mean that they appear in many areas of pure mathematics, from a central position in classification problems in algebraic geometry, to the use of 1-dimensional Calabi-Yau manifolds (a.k.a. elliptic curves) in number theory, and the special role played by 3-dimensional Calabi-Yau manifolds in mathematical physics and string theory.The project itself will involve the construction and study of Calabi-Yau manifolds. The first part of the project will involve a study of 2-dimensional Calabi-Yau manifolds, which are more commonly known as K3 surfaces. In particular, the project will focus on the behaviour of K3 surfaces as they vary in families, with a focus on how taking limits causes these families to degenerate.This theory will then be applied to construct 3-dimensional Calabi-Yau manifolds which admit fibration structures by the K3 surfaces studied in the first part. Finally, the properties of these K3-fibred Calabi-Yau threefolds will be explicitly studied.