Noncommutative surfaces and threefolds

在代数几何中，交换阶代数与射影格式之间有着重要的对应关系。非交换射影几何将这种对应的各个方面推广到非交换环境，例如通过使用环上的梯度模的范畴来代替实际的几何空间。由于人们对非交换的射影曲线已经有了很好的认识，所以非交换曲面和特殊类型的三折曲面是当前研究的主要课题。本课题的主要目标是研究非交换射影平面的非交换类曲面，特别是Sklyanin型的非交换射影平面。非交换性是两个操作以不同顺序执行时产生不同结果的现象。物理学中一个重要的例子是位置和动量算符不能交换；这是海森堡测不准原理的数学公式的一部分。非交换几何的主题试图给出一些几何直觉背后的非交换操作的集合是封闭的加法和乘法（这些被称为环）。然后，对点、线和其他几何概念的研究可以帮助阐明我们对这种环的理解。

In algebraic geometry, there is an important correspondence between commutative graded algebras and projective schemes. Noncommutative projective geometry generalizes aspects of this correspondence to the noncommutative setting, for example by using the category of graded modules over the ring as a substitute for an actual geometric space. Noncommutative projective curves are well-understood, so noncommutative surfaces and special kinds of threefolds are the major subjects of current interest. A major goal of this project is to study surfaces in the birational classes of noncommutative projective planes, especially those of Sklyanin type.Noncommutativity is the phenomenon of two operations giving different results when performed in a different order. An important example in physics is the fact that the position and momentum operators do not commute; this is the part of the mathematical formulation of the Heisenberg Uncertainty principle. The subject of noncommutative geometry attempts to give some geometric intuition behind collections of noncommuting operations which are closed under addition and multiplication (these are called rings). Then the study of points, lines, and other such geometric concepts can help to illuminate our understanding of such rings.