Studies in Commutative Algebra and Algebraic Geometry

代数几何研究多项式方程族的解。 人们可以用几何学的方法来研究解集，使用方程图的高维类似物，或者用代数学的方法，通过研究形成所谓交换环的几何解集上的函数的行为。这提供了一个有价值的双重视角。 该项目将使人们更深入地了解这一领域的核心基本概念，其中几个概念最近已被用于在长期存在的问题上取得进展。 结果将提供有关方程组解的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，并将提供与研究生和博士后教师，这将促进他们作为研究人员的成长合作的许多机会。本专题的部分内容将适合于培养本科生进行研究。本专题将探讨多项式强度的应用。 这个强度的概念是最近由PI与合作者联合提出的，并被证明是他们证明斯蒂尔曼猜想的关键因素。该项目将使用多项式强度来回答几个剩余的问题，例如，关于获得初级分解的界限-独立于变量的数量-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紧闭包理论，在所有特征中都有效，既有持久性又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的进展。PI还将继续研究lim Cohen-Macaulay模理论，旨在解决关于交叉多重性行为的长期猜想。其他方向包括研究：局部上同调模的极小素数的有限性;局部上同调的过滤定理，可用于研究强F-正则环; Eisenbud，绿色和Harris关于多项式环中理想的Hilbert函数的行为的长期猜想;这个奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准。

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.