New Structures in Homological Commutative Algebra

同调交换代数的新结构

基本信息

批准号：
1902123
负责人：
Daniel Erman
金额：
$ 25.35万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902123&HistoricalAwards=false
关键词：
New Structures Homological Commutative Algebra

项目摘要

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.

求解多项式方程是数学中最古老和最基本的课题之一。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“维数诅咒”，在数学中很常见。然而，Ananyan 和 Hochster 最近的工作表明，这种情况不会发生：在某种情况下，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，问题甚至变得更简单。在这个项目中，PI 将尝试将 Ananyan 和 Hochster 的核心见解运用到新型方程中。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，其中许多是对经典结果的现代扭曲希尔伯特. 在之前的工作中，PI构造了新的极限环，涉及逆极限和超乘积，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证明。 PI 建议为常规局部环和投影空间上的相干滑轮开发类似的新框架。该项目的智力价值主要来自于广泛的有界结果，这将产生常规局部环和射影空间上相干滑轮的上同调。该项目还将通过 PI 对麦迪逊的领导对 K-12 教育产生影响Math Circle 是一项外展计划，旨在向高中生和高年级中学生提供数学和科学方面令人兴奋的想法。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进行评估，被认为值得支持审查标准。