Noncommutative surfaces and threefolds

在代数几何中，交换分次代数与射影概型之间有着重要的对应关系。 非对易射影几何将这种对应推广到非对易环境，例如使用环上的分次模范畴作为实际几何空间的替代。非对易射影曲线是人们熟知的，因此非对易曲面和特殊类型的三重曲面是当前研究的主要课题。 本项目的一个主要目标是研究非对易射影平面的双有理类中的曲面，特别是Sklyanin型的曲面。非对易性是两个运算在不同的顺序下给出不同结果的现象。 物理学中的一个重要例子是位置和动量算符不对易;这是海森堡测不准原理的数学表述的一部分。 非交换几何的主题试图给出一些几何直观背后的非交换运算的集合，这些运算在加法和乘法下是封闭的（这些被称为环）。 然后研究点、线和其他类似的几何概念可以帮助我们理解这样的环。

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.