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.

这是一个交换代数项目，与代数几何、组合学和算术几何有关。交换代数是抽象代数的一个领域，旨在理解多项式方程组的解集。它是一个基础研究领域，具有与机器人、统计学和物理学等领域的应用和联系。该项目涉及在熟悉的设置（例如实数或复数）中研究多项式方程的经典系统，但也涉及不太了解的混合特征设置，这与数论和算术几何有关。该奖项还将支持与当地中小学和研究生培训合作的活动，以及促进早期职业研究人员的工作。PI将开展同调和交换代数方面的研究项目，涉及混合特征交换代数、符号幂和上同调支持簇中的p-导数的研究和应用。交换代数和代数几何中的许多技术仅适用于域上的代数，因为在特征零中使用奇点解析和消失定理以及在正特征中使用 Frobenius 映射的同调性质。相比之下，混合特征设置通常更加微妙，并且许多问题仅在该设置中才悬而未决。最近的发展表明，p 导数（算术几何的一种工具）可用于解决混合特征交换代数中出现的问题，特别是与微分算子一起应用时。这些提供了新的研究途径，将进一步推动这些应用。该项目还将解决由最近与上同调支持簇和环同伦李代数应用相关的重大突破所引发的同调问题；和符号幂，一种可用于回答经典几何问题的代数工具。该项目由代数和数论计划以及刺激竞争性研究既定计划 (EPSCoR) 联合资助。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。