Regular Algebras

正则代数

• 批准号: 0457022
• 负责人: Michaela Vancliff
• 金额: --
• 依托单位: University of Texas at Arlington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0457022&HistoricalAwards=false
• 关键词: Regular; Algebras

This award supports the research of Michaela Vancliff to work innon-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. She intends to produce algebro-geometric techniques that allow the easy construction of regular algebras of global dimension four that have finitely many points and a one-parameter family of line modules; such techniques would allow researchers in the field to easily create examples on which to test their conjectures. An underlying theme of her research is to classify the line schemes that arise for "generic" quadratic regular algebras of global dimension four. Vancliff is also interested in connections between this type of geometry and that of various Poisson-geometric structures. 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 a certain algebra; one that encodes all the properties of the original equations. Associatedto this algebra are modules, and these encode all the properties of the original solutions. Hence, 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; they 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.

该奖项支持Michaela Vancliff在非交换代数方面的研究，特别强调正则代数和非交换代数几何理论中出现的问题。她感兴趣的是将分级模块类别视为一个几何空间，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）通过所谓的线性格式参数化。Vancliff计划研究高维线性格式的结构和作用如何推广点格式的结构和作用。她打算产生代数几何技术，使全局四维的正则代数易于构造，这些正则代数具有有限多个点和单参数的线模族；这样的技术将使该领域的研究人员能够很容易地创建例子来检验他们的猜想。她研究的一个潜在主题是对全局四维的“一般”二次正则代数的线方案进行分类。Vancliff还对这种几何结构与各种泊松几何结构之间的联系感兴趣。多项式式方程组及其解在几乎所有科学领域都起着至关重要的作用，如统计力学、基本粒子物理、量子力学、机器人、晶体学、网络等。通常，解决方案不能通过实验找到，而且它们通常不是数字而是函数（例如，微分算子或矩阵），因此，一般来说，它们不能交换。寻找在非交换变量中找到任何多项式型方程组的所有解的方法的科学被称为非交换代数。为了找到解决方案，主要思路如下。人们把这样的方程组与某种代数联系起来；它编码了原始方程的所有属性。与这个代数相关的是模块，这些模块编码了原始解的所有属性。因此，为了找到所有的解，我们应该找到相关代数的所有模块。在许多应用中，以这种方式产生的代数倾向于共享某些性质；它们被称为正则代数，是凡格里夫项目的主要焦点。非交换代数几何是凡利夫工作的子领域，其目标之一是使用几何技术找到正则代数的某些模块（点模块，线模块等），然后使用这些模块找到给出原始方程组解的模块。Vancliff的潜在目标是改进这些几何技术，并更好地理解它们与模块类别结构的关系。