Tyurin Degenerations of K3 Surfaces

这个项目的目的是研究一种特定类型的曲面的性质，称为K3曲面。K3曲面是一类称为Calabi-Yau流形的二维流形，它的特殊性质意味着它们出现在纯数学的许多领域：从代数几何中分类问题的中心位置，到一维Calabi-Yau流形(又名。椭圆曲线)，以及K3曲面和三维Calabi-Yau流形在数学物理和弦理论中所起的特殊作用。这些退化涉及对K3曲面进行数学变形，直到它们分解或退化为一组更简单的碎片。该项目将研究一类被称为Tyurin简并的简并，它们在镜像对称的数学中扮演着特殊的角色，但目前人们对它知之甚少。该项目的目的是构建新的Tyurin退化实例，研究它们的性质，并致力于为它们建立更广泛的分类理论。

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.