Regular Algebras

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。Vancliff的主要目标是增加正则代数的知识体系，并进一步发展现有的几何技术。她特别感兴趣的是将此类代数的分级模块范畴视为一个几何空间，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）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。她还对这类几何分析与各种泊松几何结构之间的联系感兴趣。其中一些活动需要开发基本的计算算法，并使用计算机代数包实现它们，以便能够对线性方案进行显式计算。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。解决方案往往是无法通过实验或其他方法找到的结果，它们往往不是数字而是函数（例如，微分算子或矩阵），因此，一般来说，它们不会交换。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工作被称为非交换代数。寻找解决方案的主要思路如下。人们把这样一个方程系统与一个抽象的对象联系起来，称为代数，它编码了系统的所有属性。与代数相关的抽象对象称为模块，这些模块编码了解的所有属性。因此，为了找到所有的解，我们应该找到并理解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Vancliff项目的主要焦点。非交换代数几何是凡利夫工作的子领域，它的目标之一是利用几何技术找到正则代数的一些模（点模、线模等），然后用这些模找到给出原方程组解的模。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。

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. Vancliff's main objective is to add to the body of knowledgeon regular algebras and to further the existing geometric techniques.In particular, she is interested in the graded-module category of suchan algebra 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 ofhigher-dimensional linear schemes generalize the structure and role ofpoint schemes. She is also interested in connections between this typeof geometric analysis and that of various Poisson-geometric structures. Some of these activities entail the development of fundamental computational algorithms, and their implementation using a computer-algebra package, in order to enable explicit computation oflinear schemes.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. The solutions are often results thatcannot be found by experimentation nor other methods, and often they are not numbers but are functions (e.g., differential operators or matrices), and so, in general, they do not commute. The business of seeking methods that find all solutions to any system of polynomial-style equations in non-commuting variables is called non-commutative algebra. The main idea to find the solutions is as follows. One associates to such a system of equations an abstract object called an algebra, which encodes all the properties of the system. To the algebra is associated abstract objects called modules,and these encode all the properties of the solutions. Hence, in orderto find all the solutions, one should find and understand all the modules for the associated algebra. In many of the applications, thealgebras that arise in this way tend to share certain properties; theyare called regular algebras and are the main focus of Vancliff'sprojects. One of the goals of non-commutative algebraic geometry, thesubfield in which Vancliff works, is to use geometric techniques to find some of the 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 ofmodules.