CAREER: Problems in Commutative and Homological algebra

职业：交换代数和同调代数问题

• 批准号: 2236983
• 负责人: Eloísa Grifo
• 金额: $ 42.5万
• 依托单位: University of Nebraska-Lincoln
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2028-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2236983&HistoricalAwards=false
• 关键词: CAREER; Problems; Commutative; Homological; algebra

This is a project in commutative algebra, with connections to algebraic geometry, combinatorics, and arithmetic geometry. Commutative algebra is a field of abstract algebra that aims to understand the solution sets of system of polynomial equations. It is a fundamental area of research with applications and connections to fields such as robotics, statistics, and physics. This project involves studying classical systems of polynomial equations in familiar settings such as the real or complex numbers, but also the less understood setting of mixed characteristic, which has connections with number theory and arithmetic geometry. This award will also support activities in collaboration with local elementary and middle schools and graduate student training, as well as the promotion of work by early-career researchers.The PI will pursue research projects in homological and commutative algebra relating to the study and applications of p-derivations in mixed characteristic commutative algebra, symbolic powers, and cohomological support varieties. Many techniques in commutative algebra and algebraic geometry only work for an algebra over a field, because of the use of resolution of singularities and vanishing theorems in characteristic zero and the homological properties of the Frobenius map in positive characteristic. In contrast, the mixed characteristic setting is often more delicate, and many questions remain open only in that setting. Recent developments have shown that p-derivations, a tool from arithmetic geometry, can be applied to solve problems arising in mixed characteristic commutative algebra, especially when applied together with differential operators. These provide new avenues of research that will further these applications. The project will also address homological questions motivated by recent major breakthroughs related to cohomological support varieties and applications of the homotopy Lie algebra of a ring; and symbolic powers, an algebraic tool that can be used to answer classical geometric questions. This project is jointly funded by the Algebra and Number Theory Program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这是一个交换代数的项目，与代数几何，组合和算术几何形状有连接。交换代数是一个抽象代数的领域，旨在了解多项式方程系统的解决方案集。它是研究的基本领域，其应用和与机器人技术，统计和物理学等领域的联系。该项目涉及研究在熟悉的环境（例如真实或复杂数字）中的多项式方程式的经典系统，但也较少了解混合特征的设置，该设置与数字理论和算术几何形状有联系。该奖项还将支持与当地中小学以及研究生培训以及早期研究人员促进工作的活动的活动。PI将在同源和交换代数方面从事与研究的研究项目，以及与研究以及有关研究的应用，以及有关研究的应用程序，以及有关研究的研究，以及针对研究的应用程序，混合特征交换代数，符号能力和共同体支持品种中的P衍生。由于使用奇异性的分辨率并消失了特征零的定理以及Frobenius Map的同源性能，因此，可交换代数和代数几何形状中的许多技术仅适用于田间的代数。相比之下，混合特征环境通常更加精致，许多问题仅在该环境中保持开放。最近的事态发展表明，P衍生物是一种来自算术几何形状的工具，可用于解决在混合特征交换代数中引起的问题，尤其是与差分运算符一起应用时。这些提供了新的研究途径，这些途径将进一步进一步应用。该项目还将解决与圆环同质性支持品种和应用相关的近期重大突破所激发的同源问题；和符号力量，这是一种代数工具，可用于回答经典的几何问题。该项目由代数和数字理论计划和启发竞争性研究的既定计划共同资助（EPSCOR）。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的审查标准的评估来获得支持的。