Studies in Commutative Algebra and Algebraic Geometry

代数几何研究多项式方程族的解。人们可以用方程图形的高维相似来几何地研究解集，或者通过研究形成所谓交换环的几何解集上的函数的行为来代数地研究解集。这提供了一个有价值的双重视角。该项目将对这一领域的核心、基本概念有更深入的理解，其中几个概念最近已被用于在长期猜测的基础上取得进展。结果将提供关于方程解集性质的定量和定性信息，并有望在一些长期存在的问题上取得重大进展。该项目是多方面的，将提供许多与研究生和博士后教师合作的机会，这将促进他们作为研究人员的成长。该项目的某些部分将适合培养本科生进行研究。该项目将探索多项式的强度的应用。这个强度的概念是最近由PI与一位合作者联合提出的，并被证明是他们证明Stillman猜想的关键要素。该项目将使用多项式强度来回答剩下的几个问题，例如，关于获得初级分解的界限--与变量的数量无关--尽可能尖锐。另一个方向是使用完美几何的思想来构造一个在所有特征中都有效的紧闭包理论，它既具有持久性，又具有令人满意的测试元素理论。Perfectoid技术已经在这一领域取得了很大进展。PI还将继续研究Lim Cohen-Macaulay模的理论，旨在解决关于交集多重性行为的一个长期猜测。其他方向包括：局部上同调模的极小素数的有限性的研究；可用于研究强F-正则环的局部上同调的滤子定理；Eisenbud，Green和Harris关于多项式环上的理想的Hilbert函数的行为的长期猜想；以及理想的普通幂和符号幂的一致比较。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic geometry studies solutions of families of polynomial equations. One can study the solution set geometrically, using higher dimensional analogues of the graph of an equation, or algebraically, by investigating the behavior of functions on the geometric solution set that form what is called a commutative ring. This provides a valuable dual perspective. The project will yield a deeper understanding of central, fundamental notions in this area, several of which have been used recently to make progress on long-standing conjectures. The results will give both quantitative and qualitative information about the nature of the solution sets of equations and are expected to lead to significant progress on a number of long-standing questions. The project is multi-faceted and will provide many opportunities for collaboration with graduate students and postdoctoral faculty that will foster their growth as researchers. Some portions of the project will be suitable for training undergraduates to do research.The project will explore applications of the strength of a polynomial. This notion of strength was recently introduced by the PI, in joint work with a collaborator, and proved to be a critical element in their proof of Stillman's conjecture. The project will use polynomial strength to answer several remaining questions, for example, about obtaining bounds for primary decomposition--independent of the number of variables--that are as sharp as possible. Another direction is to use ideas from perfectoid geometry to construct a tight closure theory, valid in all characteristics, that has both persistence and a satisfactory theory of test elements. Perfectoid techniques have already led to a great deal of progress in this area. The PI will also continue to work on the theory of lim Cohen-Macaulay modules, aimed at resolving a long-standing conjecture about the behavior of intersection multiplicities. Other directions include the study of: finiteness of minimal primes of local cohomology modules; filtration theorems for local cohomology that can be used to investigate strongly F-regular rings; a long-standing conjecture of Eisenbud, Green, and Harris on the behavior of Hilbert functions of ideals in polynomial rings; and the uniform comparison of ordinary and symbolic powers of ideals.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.