New Structures in Homological Commutative Algebra

求解多项式方程是数学中最古老、最基本的课题之一。人们可能会认为，随着变量数量的增加，方程会变得越来越复杂；事实上，这种现象被称为“维度诅咒”，在数学中非常常见。然而，Ananyan和Hochster最近的工作表明，这种情况并没有发生：在一定的制度下，求解方程的复杂性不会随着变量数量的增加而增加。事实上，在某些方面，这个问题甚至变得更简单了。在这个项目中，PI将尝试将Ananyan和Hochster的核心见解应用到新型方程中。这将加深我们对方程组结构的理解，具有理论和计算应用的潜力。斯蒂尔曼猜想的最新进展导致了代数中大量新的有界结果，其中许多是对希尔伯特经典结果的现代扭曲。在以前的工作中，PI构造了新的极限环，涉及逆极限和超积，并将这些极限环应用于交换代数中的同调问题。这导致了斯蒂尔曼猜想的两个新的证明。PI建议为正则局部环和射影空间上的凝聚层建立类似的新框架。这个项目的智力价值将主要来自于广泛的有界性结果，这将产生规则局部环和射影空间上凝聚层的上同调。该项目还将通过PI对Madison Math Circle的领导对K-12教育产生影响，Madison Math Circle是一个扩展项目，为高中和高级中学生提供令人兴奋的数学和科学想法。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Solving polynomial equations is one of the oldest and most fundamental topics in mathematics. One might expect that equations would grow ever more complex as the number of variables increases; in fact, this sort of phenomenon is known as the "curse of dimensionality" and it is very common in mathematics. However, recent work of Ananyan and Hochster has shown that this does not happen: under a certain regime, the complexity of solving equations does not increase as the number of variables increases. In fact in some ways, the problem even becomes simpler. In this project, the PI will try to carry the core insights of Ananyan and Hochster to new types of equations. This would sharpen our understanding of the structure of systems of equations, with the potential for both theoretical and computational applications.The recent progress on Stillman's Conjecture has led to a plethora of new bounded results in algebra, many of which are modern twists on classical results of Hilbert. In previous work, the PI had constructed new limit rings, involving inverse limits and ultraproducts, and applied these limit rings to homological questions in commutative algebra. This led to two new proofs of Stillman's Conjecture. The PI proposes developing similar new frameworks for regular local rings and for coherent sheaves on projective space. The intellectual merit of this project would primarily come through the broad array of boundedness results this would yield for regular local rings and for cohomology of coherent sheaves on projective space This project will also have impacts on K-12 education through the PI's leadership of the Madison Math Circle, an outreach program that provides a taste of exciting ideas in math and science to high school and advanced middle school students.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.