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: $ 1.17万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=648045
• 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.

我的研究探索交换代数、组合学和几何学的交集。这些领域的重叠使我们能够用不同的观点来解决问题。将这些领域放在一起使我们能够开发新的技术和问题。代数和几何相交的一个简单例子是使用方程来描述平面中的直线。同样，代数和图论之间也有重叠之处。图是具有连接某些节点的边的节点的集合。我们给每个节点分配一个变量，给每条边分配一个称为单项式的变量乘积。这样做导致了对单项理想的研究，单项理想是单项组合的集合。*当我们看多项式计算为零的点时，我们需要捕捉到消失的复杂性。例如，0是多项式函数f(X)=x和g(X)=x^2的消失点。然而，在g(X)=x^2处的消失点被认为是更“复杂”的。为了研究复杂性，我们研究了被称为理想的多项式的集合，以及它们的正则幂和符号幂。代数和几何中许多问题的核心是这两个幂之间的区别。该提案的长期目标是开发方法，以了解这些权力的性质和衡量它们之间的差异，并探索其应用。这些非平凡的理想为检验假设和为更复杂的理想建立论据提供了基础。此外，还有一些工具可以让我们使用单项理想来更好地理解一般理想的性质。我将使用图论和线性规划的技巧来研究单项理想的符号幂的不变量。我还将利用多面体来获得关于单项式理想的象征力的数据。*许多关于正则幂和符号幂之间关系的猜想的基础是与称为肥点的几何对象有关的理想。尽管出现在许多环境中，但这些对象的性质很难计算。我将推广一个程序，它将使用点的几何信息来获得关于理想的代数信息。*在此建议的基础上所做的工作将在开发新技术的同时扩展和深化已知的工具。应用领域包括编码理论、Hadamard积、统计学和物理学。因此，预计影响将是广泛的。*高素质人员(HQP)将掌握与提案相关的材料，发展生成合理问题的能力，通过计算机代数系统学习计算机技能，学习演讲的最佳实践，并通过出版加强写作技能。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为他们提供很好的服务。此外，HQP将通过在研究小组和网络中共同成长来形成协作。