Powers in Commutative Algebra: Approaches, Properties, and Applications

交换代数的幂：方法、性质和应用

• 批准号: RGPIN-2018-05004
• 负责人: Cooper, Susan
• 金额: $ 2.33万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758500
• 关键词: Powers; Commutative; Algebra; Approaches; Properties

My research explores the intersections of Commutative Algebra, Combinatorics and Geometry. The overlap of these areas allows us to solve problems using different viewpoints. Putting the areas together empowers us to develop new techniques and questions. A simple example of the intersection of Algebra and Geometry is the use of equations to describe lines in the plane. Similarly, there is an overlap between Algebra and Graph Theory. A graph is a collection of nodes with edges connecting some of the nodes. We assign to each node a variable and to each edge a product of variables called a monomial. Doing so leads to the study of monomial ideals, a collection of combinations of monomials. When we look at points where polynomials evaluate to zero, we need to capture the complexity of the vanishing. For example, 0 is a vanishing point of the polynomial functions f(x) = x and g(x) = x^2. However, the vanishing at g(x) = x^2 is considered more "complicated". To study the complexity, we look at collections of polynomials called ideals and their regular and symbolic powers. At the heart of many problems in Algebra and Geometry is the difference between these powers. The long-term goals of the proposal are to develop approaches to understand the properties of and measure the differences between these powers and to explore applications.An especially important family of ideals is the monomial ideals. These non-trivial ideals provide the base case for testing hypotheses and building arguments for more involved ideals. Also, there are tools that allow us to use monomial ideals to better understand properties of general ideals. I will use techniques from Graph Theory and Linear Programming to study invariants of symbolic powers of monomial ideals. I will also exploit a polyhedron to obtain data about the symbolic power of a monomial ideal. Underlying many conjectures of the relationships between regular and symbolic powers are ideals connected to geometric objects called fat points. Despite arising in numerous contexts, properties of these objects are difficult to compute. I will generalize a procedure that will use geometric information about the points to obtain algebraic information about the ideals.Work from this proposal will expand and deepen known tools while developing new techniques. Applications include coding theory, Hadamard products, statistics and physics. As such, the impact is anticipated to be wide-ranging.Highly qualified personnel (HQP) will master material related to the proposal, develop the ability to generate sound questions, learn computer skills via computer algebra systems, learn best practices in giving presentations, and strengthen writing skills by publishing. These skills will serve them well in a wide range of careers in fields such as mathematics, physics, statistics and computer science. Moreover, HQP will form collaborations by growing together within the research group and networking.

我的研究探讨了交换代数，组合和几何形状的交集。 这些领域的重叠使我们能够使用不同的观点解决问题。 将这些领域融合在一起，使我们能够开发新的技术和问题。 代数和几何形状的交点的一个简单示例是使用方程来描述平面中的线条。 同样，代数和图理论之间也存在重叠。 图是一个节点的集合，边缘连接了一些节点。 我们将一个变量分配给每个节点，并将变量的乘积分配给一个称为单一的变量。 这样做会导致对单一理想的研究，这是一系列单一组合。 当我们查看多项式评估为零的点时，我们需要捕获消失的复杂性。 例如，0是多项式函数的消失点f（x）= x and g（x）= x^2。 但是，G（x）= x^2处的消失被认为更加“复杂”。 为了研究复杂性，我们研究了称为理想及其常规和象征力量的多项式集合。 代数和几何形状中许多问题的核心是这些力量之间的差异。 该提案的长期目标是开发方法，以了解这些权力和探索应用之间的差异和衡量差异。理想的一个尤其重要的家族是单一理想。 这些非平凡的理想为测试假设和建立更多涉及理想的论点提供了基本案例。 此外，有一些工具使我们能够使用单一理想来更好地了解一般理想的特性。 我将使用图理论和线性编程中的技术来研究单一理想的符号能力的不变性。 我还将利用一个多面体来获得有关单一理想的符号能力的数据。 基本的许多猜想是常规力量和符号力之间的关系是与称为脂肪点的几何对象相关的理想。 尽管在许多情况下出现，但这些对象的属性很难计算。 我将概括一个程序，该程序将使用有关点的几何信息来获取有关理想的代数信息。该提案中的工作将在开发新技术的同时扩展和加深已知的工具。 应用包括编码理论，Hadamard产品，统计和物理学。 因此，预计影响将是广泛的。合格的人员（HQP）将掌握与该提案相关的材料，发展产生声音问题的能力，通过计算机代数系统学习计算机技能，学习演示文稿的最佳实践，并通过出版来增强写作技巧。 这些技能将在数学，物理，统计和计算机科学等领域的广泛职业中很好地为他们提供服务。 此外，HQP将通过在研究小组和网络中共同成长来形成合作。