Regular Algebras

该奖项支持Michaela Vancliff从事非交换代数的研究，特别强调由正则代数理论和非交换代数几何产生的问题。她对被视为几何空间的分级模范畴感兴趣，某些分级模扮演几何对象的角色。通过所谓的线性方案将线性几何模块(点模块、线模块等)参数化。Vancliff计划研究高维线性格式的结构和作用如何概括点格式的结构和作用。在以前的NSF支持下，Vancliff与T.Cassidy合作，提出了代数几何技术，允许容易地构造任意有限整体维的某些正则代数(推广了分次Clifford代数)；将这种代数命名为分次斜Clifford代数。Vancliff打算研究这样的具有有限多个点的全局四维正则代数和一个单参数线模族，以此作为对全局四维“一般”二次正则代数所产生的线型进行分类的一步。她与B.Shelton(在之前的NSF支持下)的初步研究表明，这样的代数应该具有恰好由六条椭圆曲线组成的直线方案，因此如果发现这在一般情况下成立，那么它将模仿全局维的通用二次正则代数的点方案(其中它是一条椭圆曲线)。多项式形式的方程组及其解在几乎每一个科学领域中都扮演着关键的角色，例如统计力学、基本粒子物理、量子力学、机器人、结晶学、网络等。通常，解不能通过实验找到，并且通常它们不是数字而是函数(例如，微分运算符或矩阵)，因此，一般而言，它们不交换。寻找在非对易变量中找到任何多项式型方程组的所有解的方法的科学称为非对易代数。为了找到解决方案，主要思路如下。人们把这样一个方程式系统与一个叫做“代数”的实体联系在一起，它编码了原始方程式的所有性质。与这个代数相关的是“模”，它们编码了方程解的所有性质。因此，为了找到所有的解，我们应该找到相关代数的所有模。在许多应用中，以这种方式产生的代数往往共享多项式环所满足的某些性质；这种代数被称为正则代数，是Vancliff项目的主要焦点。非对易代数几何的目标之一是利用几何技巧找出正则代数的某些模(点模、线模等)，然后用这些模找出给出原始方程组的解的模。Vancliff的基本目标是改进这些几何技术，并更好地了解它们与模块类别的结构之间的关系。

This award supports the research of Michaela Vancliff to work in non-commutative algebra, with special emphasis on problems arising from the theory of regular algebras and non-commutative algebraic geometry. She is interested in the graded-module category viewed as a geometric space, with certain graded modules playing the role of geometric objects. The linear geometric modules (point modules, line modules, etc) are parametrized by so-called linear schemes. Vancliff plans to study how the structure and role of higher-dimensional linear schemes generalize the structure and role of point schemes. Under prior NSF support, in work with T. Cassidy, Vancliff produced algebro-geometric techniques that allow the easy construction of certain regular algebras (that generalize graded Clifford algebras) of any finite global dimension; naming such algebras graded skew Clifford algebras. Vancliff intends to study such regular algebras of global dimension four that have finitely many points and a one-parameter family of line modules as a step towards classifying the line schemes that arise for ``generic'' quadratic regular algebras of global dimension four. Her initial research with B. Shelton (under prior NSF support) suggests that such an algebra should have a line scheme that consists of exactly six elliptic curves, so if this is found to hold in general, then it would mimic the point scheme of generic quadratic regular algebras of global dimension three (where it is one elliptic curve).Systems of polynomial-style equations and their solutions play a critical role in almost every scientific field, such as statistical mechanics, elementary particle physics, quantum mechanics, robotics, crystallography, networking, etc. Often, the solutions cannot be found by experimentation, and often they are not numbers but are functions (e.g., differential operators or matrices), and so, in general, they do not commute. The science of seeking methods that find all solutions to any system of polynomial-style equations in non-commuting variables is called non-commutative algebra. To find the solutions, the main idea is as follows. One associates to such a system of equations an entity, called an ``algebra'', that encodes all the properties of the original equations. Associated to this algebra are ``modules'', and these encode all the properties of the solutions to the equations. So, in order to find all the solutions, one should find all the modules for the associated algebra. In many of the applications, the algebras that arise in this way tend to share certain properties satisfied by the polynomial ring; such algebras are called regular algebras and are the main focus of Vancliff's projects. One of the goals of non-commutative algebraic geometry, the subfield in which Vancliff works, is to use geometric techniques to find certain modules (point modules, line modules, etc) of the regular algebra, and then to use those modules to find the modules giving the solutions to the original system of equations. Vancliff's underlying goal is to improve on these geometric techniques and to understand better how they relate to the structure of the category of modules.