Tyurin Degenerations of K3 Surfaces

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The aim of this project is to investigate the properties of a certain type of surface, called a K3 surface. K3 surfaces are the two-dimensional case of a class of manifold called Calabi-Yau manifolds, whose special properties 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 K3 surfaces and 3-dimensional Calabi-Yau manifolds in mathematical physics and string theory.The proposed project involves the construction and study of degenerations of K3 surfaces. These degenerations involve mathematically deforming K3 surfaces until they break up, or degenerate, into a collection of simpler pieces. The project will study a certain class of degenerations, called Tyurin degenerations, which play a special role in the mathematics of mirror symmetry and yet are currently poorly understood. The aim of the project is to construct new examples of Tyurin degenerations, study their properties, and work towards a broader classification theory for them.