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计算机操作，很容易在数学上理解，但也足够灵活，可以模拟自然界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定义的几何形状。它们的应用范围更广，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。这个交换代数和代数几何的项目专注于这些领域的基础研究，同时广泛地培训交换代数和代数几何的不同博士生和本科生。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，已经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、泰希米勒理论、镜像对称性、泊松几何、离散动力系统、弦理论、接线图和网络等等。具体的研究包括：1)发展簇代数的交换代数，包括局部化、爆破等基本概念，以及测试理想等素数特征；2)素数特征的约化和Frobenius映射的迭代以了解代数簇和交换环的奇点、上同调等特征，研究Frobenius定义的素数特征中的不变量，它们与簇结构的关系以及它们在二元几何中的潜在用途；3)解决Kollar关于基变化下局部上同调行为的问题(其主要应用是归结关于复数簇的局部Picard群到特征p的映射的问题)。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研究的基础，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.