New Structures in Homological Commutative Algebra

同调交换代数的新结构

• 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

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.

解决多项式方程是数学中最古老，最基本的主题之一。 人们可能会期望随着变量的增加，方程将变得越来越复杂。实际上，这种现象被称为“维度的诅咒”，并且在数学中非常普遍。 但是，Ananyan和Hochster的最新工作表明，这并没有发生：在某个制度下，求解方程的复杂性不会随着变量数量的增加而增加。 实际上，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的核心见解带入新的方程式。 这将使我们对方程系统结构的理解具有理论和计算应用的潜力。Stillman猜想的最新进展导致代数中有很多新的有界成果，其中许多是希尔伯特经典成果的现代曲折。 在先前的工作中，PI构建了新的极限环，涉及逆极限和超副作用，并将这些极限环应用于交换代数中的同源性问题。 这导致了斯蒂尔曼猜想的两个新证明。 PI建议为常规的本地环和投影空间上的连贯滑轮开发类似的新框架。 该项目的智力优点主要是通过各种各样的界限结果来实现的，这将带来定期的本地戒指以及在投射空间上相干吊羊的协同学，该项目还将通过PI通过PI的领导而对K-12教育产生影响，这是通过PI的领导层的领导才能，这是一项外展计划，该计划在数学和高级学生中提供了令人兴奋的宣传和高级宣传。通过基金会的智力优点和更广泛的影响评估标准通过评估来支持。

项目成果

Equivariant Hilbert Basis Theorem

等变希尔伯特基本定理

• 发表时间: 2020
• 期刊: Mathematical research letters
• 影响因子: 1
• 作者: Erman, Daniel;Sam, Steven;Snowden, Andrew
• 通讯作者: Snowden, Andrew

Virtual resolutions for a product of projective spaces

射影空间乘积的虚拟分辨率

• DOI: 10.14231/ag-2020-013
• 发表时间: 2020
• 期刊: Algebraic Geometry
• 影响因子: 1.5
• 作者: Erman, Daniel

Small projective spaces and Stillman uniformity for sheaves

• DOI: 10.14231/ag-2021-010
• 发表时间: 2021-05-01
• 期刊: ALGEBRAIC GEOMETRY
• 影响因子: 1.5
• 作者: Erman, Daniel;Sam, Steven, V;Snowden, Andrew
• 通讯作者: Snowden, Andrew

Big Polynomial Rings with Imperfect Coefficient Fields

具有不完美系数域的大多项式环

• DOI: 10.1307/mmj/1603353740
• 发表时间: 2021
• 期刊: Michigan Mathematical Journal
• 影响因子: 0.9
• 作者: Erman, Daniel;Sam, Steven V;Snowden, Andrew
• 通讯作者: Snowden, Andrew

Characteristic Dependence of Syzygies of Random Monomial Ideals

随机单项式理想的对称性的特征依赖性

• DOI: 10.1137/21m1392474
• 发表时间: 2022
• 期刊: SIAM Journal on Discrete Mathematics
• 影响因子: 0.8
• 作者: Booms-Peot, Caitlyn;Erman, Daniel;Yang, Jay
• 通讯作者: Yang, Jay