Regular Algebras

该奖项支持研究的Michaela Vancliff工作在非交换代数，特别强调的问题所产生的理论，作为正规代数和代数几何。她对被视为几何空间的分级模块类别感兴趣，其中某些分级模块扮演几何对象的角色。线性几何模（点模、线模等）都是由所谓的线性方案参数化的。Vancliff计划研究高维线性方案的结构和作用如何推广点方案的结构和作用。在先前NSF的支持下，与T. Cassidy，Vancliff生产代数几何技术，使容易建设某些AS-正规代数（推广分次Clifford代数）的任何有限的全球层面，命名这样的代数分次斜Clifford代数。Vancliff打算研究这样的AS-正则代数的全球第四个维度，有许多点和一个单参数家庭的线模块作为一个步骤，对分类的线计划出现的“通用”二次AS-正则代数的全球第四个维度。她和B最初的研究。谢尔顿（在先前的NSF支持下）建议这样的代数应该有一个由正好六条椭圆曲线组成的线方案，所以如果发现这在一般情况下成立，那么它将模仿全局维数为3的一般二次AS-正则代数的点方案（其中它是一条椭圆曲线）。数学一般是对模式的研究，并且经常通过方程组来描述这样的模式。例如，多项式式方程及其解的系统在几乎每个科学领域中发挥着关键作用，例如统计力学、基本粒子物理学、量子力学、机器人技术、晶体学、网络等。微分算子或矩阵），因此，一般来说，它们不交换。寻找非交换变量的任何多项式方程组的所有解的方法的科学称为非交换代数。为了找到解决方案，主要思想如下。人们将这样一个方程组与一个实体联系起来，称为“代数”，它编码了原始方程的所有属性。与这个代数相关联的是“模”，它们编码了方程解的所有性质。 所以，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具有交换多项式所满足的某些性质;这样的代数被称为AS-正则代数，并且是Vancliff项目的主要焦点。研究这类代数及其模的目标之一是利用几何技巧找到AS-正则代数的某些模（点模、线模等），然后利用这些模找到给出原方程组解的模。Vancliff的根本目标是改进这些几何技术，并更好地理解它们如何与模块类别的结构相关。

This award supports the research of Michaela Vancliff to work in non-commutative algebra, with special emphasis on problems arising from the theory of AS-regular algebras and 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 AS-regular algebras (that generalize graded Clifford algebras) of any finite global dimension, naming such algebras graded skew Clifford algebras. Vancliff intends to study such AS-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 AS-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 AS-regular algebras of global dimension three (where it is one elliptic curve).Mathematics in general is the study of patterns and frequently such patterns are described via systems of equations. For instance, 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 commuting polynomials; such algebras are called AS-regular algebras and are the main focus of Vancliff's projects. One of the goals of the study of such algebras and their modules is to use geometric techniques to find certain modules (point modules, line modules, etc) of the AS-regular algebra, and then to use those modules to find the modules that give 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.