Commutative Algebra: Frobenius in Geometry and Combinatorics

交换代数：几何和组合学中的弗罗贝尼乌斯

• 批准号: 1501625
• 负责人: Karen Smith
• 金额: $ 30.36万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501625&HistoricalAwards=false
• 关键词: Commutative; Algebra; Frobenius; Geometry; Combinatorics

Polynomials are mathematical functions that describe many different kinds of behavior in mathematics, physics, and engineering. They are concrete enough to be easily manipulated by hand or computer, and easy to understand mathematically, but also flexible enough to model many different situations in nature and science. Commutative algebra and algebraic geometry are the large branches of mathematics concerned primarily with understanding polynomials and the geometric shapes they define. Their broader applications range from the computer aided design used in the automotive and entertainment industries to the error correcting codes used in many digital media. This project in commutative algebra and algebraic geometry focuses on basic research in these fields, while broadly training a diverse group of PhD and undergraduate students in commutative algebra and algebraic geometry.One specific type of algebra to be investigated are cluster algebras, which are commutative rings defined iteratively that have turned up in many branches of mathematics and physics, including the classical field of total positivity for matrices, the representation theory of lie algebras, number theory, Teichmuller theory, mirror symmetry, Poisson Geometry, discrete dynamical systems, string theory, wiring diagrams and networks, and more. The specific proposed research involves 1) developing the commutative algebra of cluster algebras, including basic notions such as localization and blowing up, as well as prime characteristic features such as test ideals; 2) reduction to prime characteristic and the iteration of the Frobenius map to understand singularities, cohomology and other features of algebraic varieties and commutative rings, studying invariants defined via Frobenius in prime characteristic, their relationship to cluster structures and their potential use in birational geometry; and 3) settling a question of Kollar on the behavior of local cohomology under base change (the main application of which is to reduce questions about maps between local Picard groups of complex varieties to characteristic p). The PI's role as a senior researcher is an integral part of the project: training the next generation of algebraists is woven into the very fabric of the research and one of the project's main objectives.

多项式是描述数学、物理和工程中许多不同类型行为的数学函数。它们足够具体，可以很容易地用手或计算机操作，并且很容易在数学上理解，但也足够灵活，可以模拟自然和科学中的许多不同情况。交换代数和代数几何是数学的大分支，主要涉及理解多项式和它们定义的几何形状。其更广泛的应用范围从汽车和娱乐行业中使用的计算机辅助设计到许多数字媒体中使用的纠错码。 交换代数和代数几何的这个项目侧重于这些领域的基础研究，同时广泛培训交换代数和代数几何的各种博士和本科生。要研究的一种特定类型的代数是簇代数，它是迭代定义的交换环，出现在数学和物理的许多分支中，包括矩阵的全正性的经典领域，李代数的表示理论，数论，Teichmuller理论，镜像对称，泊松几何，离散动力系统，弦理论，接线图和网络，等等。 具体的研究内容包括：1）发展簇代数的交换代数，包括局部化和爆破等基本概念，以及测试理想等主要特征; 2）还原到素特征和Frobenius映射的迭代，以理解代数簇和交换环的奇点，上同调和其他特征，研究由Frobenius定义的素特征不变量，它们与簇结构的关系以及它们在双有理几何中的潜在用途;解决了Kollar关于局部上同调在基变化下的行为的一个问题（其主要应用是减少问题之间的地图局部皮卡集团复杂的品种，以特点p）。PI作为高级研究员的角色是该项目不可或缺的一部分：培训下一代代数学家是该研究的基础，也是该项目的主要目标之一。

项目成果

Uniform Harbourne–Huneke bounds via flat extensions

统一 Harbourne–Huneke 通过平坦延伸部分进行弹跳

• DOI: 10.1016/j.jalgebra.2018.08.024
• 发表时间: 2018
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Walker, Robert M.

Local cohomology and base change

局部上同调和基数变化

• DOI: 10.1016/j.jalgebra.2017.09.036
• 发表时间: 2018
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Smith, Karen E.

Uniform approximation of Abhyankar valuation ideals in function fields of prime characteristic

质数特征函数域中 Abhyankar 估值理想的均匀逼近

• DOI: 10.1090/tran/7917
• 发表时间: 2019
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Datta, Rankeya

Rational Singularities and Uniform Symbolic Topologies

有理奇点和统一符号拓扑

• 发表时间: 2016
• 期刊: Illinois journal of mathematics
• 影响因子: 0.6
• 作者: Walker, Robert M.

Non-commutative resolutions of toric varieties

环面簇的非交换解析

• DOI: 10.1016/j.aim.2019.04.021
• 发表时间: 2019
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Faber, Eleonore;Muller, Greg;Smith, Karen E.
• 通讯作者: Smith, Karen E.