Commutative Algebra: Frobenius in Geometry and Combinatorics

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电脑操作，也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及其所定义的几何形状。它们的广泛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项目侧重于这些领域的基础研究，同时广泛培养交换代数和代数几何领域的博士和本科生。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，在数学和物理的许多分支中出现，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。具体拟开展的研究包括：1)发展聚类代数的交换代数，包括局部化、爆破等基本概念，以及测试理想等基本特征；2)将Frobenius映射还原为素数特征，迭代Frobenius映射以理解代数变异和交换环的奇异性、上同调等特征，研究Frobenius在素数特征上定义的不变量及其与聚类结构的关系及其在两族几何中的潜在应用；3)解决了一个关于基变化下局部上同的行为的Kollar问题（其主要应用是将复变的局部Picard群之间的映射问题约简为特征p）。PI作为高级研究员的角色是该项目不可分割的一部分：培训下一代代数数学家是该研究的组成部分，也是该项目的主要目标之一。

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.