Noncommutative Algebraic Geometry

代数几何是研究多元多项式的解空间的学科。这包括由多项式定义的平面中的抛物线、球体和曲线等熟悉形状的几何图形。人们试图通过映射或参数化将它们与其他空间联系起来来研究这些解。这门学科受到多项式代数的高度控制。关于解空间的几何表述直接对应于关于所解的多项式方程的代数表述。在非对易代数中，以不同的顺序进行运算会产生不同的结果。例如，穿上鞋子，然后穿上袜子，就不同于按常规顺序做这件事。更正式地说，我们知道x乘以y不一定等于y乘以x。非对易代数，就像在研究多项式的代数一样，经常受到几何问题和直觉的启发。在这个背景下，这个领域被称为非对易代数几何，它使用了(对易)代数几何应用于非对易代数的技术和思想。例如，我们可以试着用矩阵解非对易方程，并试着理解解的空间。更具体地说，描述位置和动量之间的量子力学关系的方程YX-XY=1没有矩阵解，但方程YX-XY=0有许多矩阵解。一般而言，非对易代数几何的一个重要目标是了解变量不需要交换的多项式方程的解空间。除了在非对易代数中有直接的应用外，在代数几何和物理中也有应用。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。这需要本科生和研究生以及博士后研究员的工作。他们开发代数和计算方面的技术专长，并将学习如何协作、传播结果和解决多层次的技术问题。他们高度发达的技能组合将使他们成为各个领域学术研究的宝贵补充，或者允许他们在密码学或编码等不同领域的工业工作。

Algebraic geometry is the study of solution 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 polynomial equations one is solving.In noncommutative algebra, applying operations in different orders can yield different results. For example, putting on one's shoes and then putting on one's socks is not the same as doing it in the conventional order. More formally, we have that x times y is not necessarily equal to y times x. Noncommutative algebra, just as in the study of the algebra of polynomials, is often motivated by geometric problems and intuition. In this setting, the field is called noncommutative algebraic geometry and 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, which describes the quantum mechanical relation between position and momentum, has no matrix solutions, but the equation yx+xy=0 has many. In general, an important goal of noncommutative algebraic geometry is to understand the space of solutions to polynomial equations where the variables need not commute. In addition to having immediate applications in noncommutative algebra, there are applications to algebraic geometry and physics.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. This requires the work of undergraduate and graduate students, and postdoctoral fellows. They develop technical expertise in algebra and computation and will learn how to collaborate, disseminate results, and solve multi-layered technical problems. Their highly developed skill sets will make them valuable additions to academic research in various fields, or allow them to work in industry in diverse areas such as cryptography, or coding.