Noncommutative algebraic geometry

代数几何是研究多元多项式的解空间的几何。它包括由多项式定义的平面上常见形状的几何，如抛物线、球面和曲线。人们试图通过映射或参数化将解与其他空间联系起来来研究解。这门学科受到多项式代数的高度控制。关于解空间的几何描述直接对应于关于正在求解的多项式的代数描述。在非交换代数中，乘积的顺序得到不同的答案。然而，XY不等于YX。人们可以做同样类型的代数，用多项式来做几何目的。这种代数是由几何问题和直觉驱动的。它被称为非交换代数几何。它使用了应用于非交换代数的交换代数几何的技巧和思想。例如，我们可以尝试用矩阵来解非对易方程，并试图理解解的空间。更具体地说，方程YX-XY=1没有矩阵解，但方程YX+XY=0有很多解。我们希望了解解的空间。除了在非交换代数中有直接的应用之外，代数几何有一些应用。代数几何中出现的一些重要空间可以描述为具有矩阵的非对易方程的解。建议的研究涉及这种相互作用发生时的空间。一个重要的问题是知道我们什么时候可以将解空间参数化。例如，抛物线y等于x的平方上的点被参数化为x=t，y等于t的平方，但不可能将y的平方等于x的三次方的解参数化为一，我们有一个几何条件来刻画何时可以将某些类型的非对易方程的矩阵解参数化。我们将试图证明这些条件确实成立。

Algebraic geometry is the study of the geometry of solutions spaces of polynomials in several variables.  This includes the geometry of familiar shapes like parabolas, spheres, and curves in the plane defined by a polynomial.  One tries to study the solutions by relating them to other spaces via mappings or parametrizations.  This subject is highly controlled by the algebra of polynomials.  Geometric statements about the space of solutions correspond directly to algebraic statements about the polynomials one is solving.  In noncommutative algebra the order of the product yields different answers.  So xy is not equal to yx.  However, one can do the same type of algebra that is done with polynomials for geometric purposes.  This algebra is motivated by geometric problems and intuition.  It is called noncommutative algebraic geometry.  It uses the techniques and ideas of commutative algebraic geometry applied to noncommutative algebra.  For example we may try to solve noncommutative equations with matrices and try to understand that space of solutions.  More concretely, the equation yx-xy=1 has no matrix solutions, but the equation yx+xy=0 has many.  We wish to understand the space of solutions.  In addition to having immediate applications in noncommutative algebra, there are applications to algebraic geometry.  Important spaces that arise in algebraic geometry can be described as solutions of noncommutative equations with matrices.  The proposed research deals with the spaces were this interaction occurs.  One important problem is to know when we can parametrize the solutions spaces.  For example, the points on the parabola y equal x squared are parametrized by x=t, y equals t squared, but it is impossible to parametrize the solutions of y squared equals x cubed minus one, with polynomials.  We have a geometric conditions that should characterize when one can parametrize the matrix solutions of certain types of noncommutative equations.  We will try to prove that these conditions do indeed hold true.