CAREER: Problems in Commutative and Homological algebra

这是一个交换代数的项目，与代数几何、组合学和算术几何有联系。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，与机器人、统计学和物理学等领域有着应用和联系。这个项目包括在熟悉的环境中研究经典的多项式方程组，如实数或复数，但也包括不太了解的混合特征设置，它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将从事与混合特征交换代数、符号幂和上同调支持变体中p导的研究和应用有关的同调和交换代数的研究项目。交换代数和代数几何中的许多技术只适用于域上的代数，因为使用了特征零点的奇异分解和消失定理以及正特征上Frobenius映射的同调性质。相比之下，混合特征设置通常更微妙，许多问题只有在这种设置中才存在。最近的发展表明，p-导数作为算术几何中的一种工具，可以应用于解决混合特征交换代数中的问题，特别是当与微分算子一起应用时。这些提供了新的研究途径，将进一步这些应用。该项目还将解决由最近与上同调支持变体和环的同伦李代数的应用有关的重大突破所激发的同调问题；符号能力，一个代数工具，可以用来回答经典的几何问题。该项目由代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.