Noncommutative surfaces and Calabi-Yau algebras

非交换代数几何将交换环与方案之间对应的某些方面推广到非交换环的集合。特别地，非交换射影几何使用在非交换梯度环上的梯度模的范畴的局部化来代替相干束的范畴。本课题的主要目标是推进非交换射影曲面的分类。我们继续研究非交换投影平面的两族类曲面，特别是Sklyanin型的曲面。Calabi-Yau代数是一类重要的非交换代数，具有良好的同调性质，与可交换的Calabi-Yau变种有关。例如，在某些情况下，它们可以被视为提供奇异变量的非交换解析。在本项目中，我们还提出研究与非交换射影几何密切相关的若干类分级Calabi-Yau代数。从几何空间到实数的所有函数的集合具有自然的加法和乘法。一个有加法和乘法的集合称为环，上面的例子称为空间的坐标环。它的代数性质通常反映并给出了空间的基本几何信息。非交换几何的目标，也就是这个项目的主题，是推广到不直接作为空间坐标环出现的环，因为这些环中两个元素的乘积取决于你乘它们的顺序。虽然没有潜在的空间，但这些环的代数可以有很深的结构，使人们仍然可以用几何的方式来研究它们。例如，某些非交换环可以被认为是奇异交换空间的非交换分辨率（空间不是光滑的，但有尖锐的点，分辨率使其平滑）。我们的主要项目涉及非交换曲面（二维空间）的分类和Calabi-Yau代数的结构，Calabi-Yau代数是最初由理论物理中的弦理论驱动的非交换环。

Noncommutative algebraic geometry generalizes certain aspects of the correspondence between commutative rings and schemes to the setting of noncommutative rings. In particular, noncommutative projective geometry uses a localization of the category of graded modules over a noncommutative graded ring as a substitute for a category of coherent sheaves. A major goal of this project is to advance the classification of noncommutative projective surfaces. We continue to study surfaces in the birational classes of noncommutative projective planes, especially those of Sklyanin type. Calabi-Yau algebras are an important class of noncommutative algebras with good homological properties, which are related to commutative Calabi-Yau varieties. For example, in certain cases they can be seen as providing noncommutative resolutions of singular varieties. In this project, we also propose to study certain classes of graded Calabi-Yau algebras which are closely connected to noncommutative projective geometry.The set of all functions from a geometric space to the real numbers has a natural addition and multiplication. A set with an addition and multiplication is called a ring, and the example above is called the coordinate ring of the space. Its algebraic properties often reflect and give information about the underlying geometry of the space. The goal of noncommutative geometry, which is the subject of this project, is to generalize to rings which do not directly arise as coordinate rings of spaces, because the product of two elements in these rings depends on the order you multiply them. Though there is no underlying space, the algebra of these rings can have deep structure that allows one to still study them in a geometric way. As an example, certain noncommutative rings can be thought of as giving a noncommutative resolution of a singular commutative space (the space is not smooth but has sharp points, and the resolution smooths it out). Our main project concerns the classification of noncommutative surfaces (2-dimensional spaces) and the structure of Calabi-Yau algebras, which are noncommutative rings motivated originally by string theory in theoretical physics.