Noncommutative Algebra and Algebraic Geometry

我们建议研究代数几何与非对易代数之间的相互作用。代数几何是一门古老的学科，它研究由多项式描述的几何形状以及与之相关的多项式代数。我们称这些形状为变种。多项式的图形就是一种简单的例子。我们研究了由多个变量的多项式描述的高维形式。 在非对易代数中，我们假设有一些变量，其中xy不等于yx。非对易代数的最简单例子是由矩阵给出的。在非对易代数几何中，我们采用正则代数几何的方法和结果，并试图将它们推广到变量不交换的情况。 建议的研究是研究品种上的顺序。这些是非交换代数，其中我们结合了由多项式和矩阵描述的簇。一个阶会给我们一个矩阵代数，它的项是多项式。因为我们的矩阵是固定的有限大小，所以与通常的代数几何有很大的交互作用。这允许我们将深入的非平凡结果推广到非对易环境。 代数几何中的一个中心问题是分类。我们建议将已知的一维和二维分类扩展到更高的维度。我们还建议研究正则代数和代数的奇点。如果一个簇没有凸起或称为奇点的奇点，它就是正则的。有纯粹的代数方法来确定一簇是规则的，还是具有奇点的。有许多方法来测量奇点。我们建议研究这些不存在奇点的性质，并研究非对易代数几何中的奇点。

We propose to study the interaction between algebraic geometry and noncommutative algebra. Algebraic geometry is an old subject where one studies geometric shapes described by polynomials and the related algebra of polynomials. We call these shapes varieties. A graph of a polynomial is a simple example of a variety. We study higher dimensional versions of these described by several polynomials in several variables. In noncommutative algebra we assume that we have variables where xy is not equal to yx. The simplest example of a noncommutative algebra is given by matrices. In noncommutative algebraic geometry we take the methods and results of regular algebraic geometry and try to extend them to the setting where our variables do not commute. The proposed research is to study orders over varieties. These are noncommutative algebras where we combine the varieties described by polynomials and matrices. An order will give us an algebra of matrices whose entries are polynomials. Since our matrices are of some fixed finite size, there is much interaction with usual algebraic geometry. This allows us to extend deep non-trivial results to the noncommutative setting. One central problem in algebraic geometry is classification. We propose to extend what is known of classification in one and two dimensions to higher dimensions. We also are proposing to study regular algebras and singularities of algebras. A variety is regular if it has no bumps or kinks called singularities. There are purely algebraic ways to determine if a variety is regular, or it has singularities there are many ways to measure the singularity. We propose to study these properties of not having any singularities, and to study the singularities in noncommutative algebraic geometry.