Studies in Commutative Algebra and Algebraic Geometry

首席研究员将研究交换代数和代数几何中的几个问题。代数几何研究许多多项式方程在许多变量下的解。理解这个问题对许多科学、工程和其他学科都至关重要。通常很难确定是期望得到任意解、有限多解还是无限多解：在最后一种情况下，人们想知道在描述解时有多少个自由度。人们可以从几何上或代数上研究解，通过研究解空间上形成所谓交换环的某些函数。这种对偶视角在交换代数和代数几何之间建立了非常有价值的紧密联系。将要研究的问题是长期存在的，具有根本的、中心的重要性。获得的结果将通过期刊和书籍出版物、会议和讲习班讲座以及通过互联网传播。有很强的教育成分。首席研究员将继续通过他多年来建立的几个项目指导研究生、博士后和高中生，特别关注那些在数学科学领域代表性不足的学生。首席研究员将继续探索诺埃尔环理论中几个长期存在的问题。Tigran Ananyan和PI已经在所有的特征上证明了Stillman的猜想，但是在射影维和其他性质上的界限，比如与初等分解相关的理想生成元的个数，当然还可以改进，这个问题将会被研究。PI还将研究在混合特征中是否存在具有某些标准性质的紧闭的类似物：通常的冒号捕获和Dietz版本的该性质，正则环的所有理想都是闭合的性质，持久性和测试元素理论。最后两项似乎是最困难的。预计来自完美曲面几何的新方法将是所需的工具之一。与Sema Gunturkun一起，PI正在研究长期存在的二次型艾森巴德-格林-哈里斯猜想，该猜想预测了某些理想的希尔伯特函数的最小值。PI将与Bhargav Bhatt和Linquan Ma合作，继续研究模的lim Cohen-Macaulay序列的存在性和性质。这项工作，以及其他一些渐近方法，可能解决长期开放的问题，是否Serre交叉多重是正的，一般来说，在混合特征正则环的情况下。另一个研究方向包括支持的有限性和局部上同调的其他性质，如在给定理想的有支持的局部域上，最高不灭局部上同调模的忠实性。PI与杰克·杰弗里斯（Jack Jeffries）正在进行一个项目，探索后一个问题。其中一些问题是为了与研究生和博士后教员合作解决的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The Principal Investigator will work on several questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of many polynomial equations in many variables. Understanding this problem is of fundamental importance in many sciences, in engineering, and in other disciplines as well. It is often difficult to determine whether to expect any solution, finitely many solutions, or infinitely many: in the last case one wants to know how many degrees of freedom one has in describing the solutions. One can study the solutions geometrically, or algebraically, by investigating certain functions on the solution space that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry that is very valuable. The problems that will be studied are long standing and of fundamental, central importance. The results obtained will be disseminated by journal and book publications, lectures at conferences and workshops, and via the internet. There is a strong educational component. The principal investigator will continue to mentor graduate students, postdocs, high school students through several programs that he has established over the years with a particular attention to attracting students that underrepresented in the mathematical sciences.The Principal Investigator will continue exploring several long standing questions in the theory of Noetherian rings. Tigran Ananyan and the PI have proved Stillman's conjecture in all characteristics, but the bounds on projective dimension and on other properties, such as numbers of generators of ideals associated with primary decompositions, can certainly be improved, and this problem will be studied. The PI will also study whether there is an analogue of tight closure in mixed characteristic that has the certain standard properties: both the usual colon-capturing and Dietz's version of this property, the property that all ideals of regular rings are closed, persistence, and a test element theory. It is the final two that appear to be most difficult. It is expected that new methods from perfectoid geometry will be among the tools needed. Jointly with Sema Gunturkun, the PI is studying the long standing Eisenbud-Green-Harris conjecture on quadratic forms, which predicts minimum values for the Hilbert functions of certain ideals. Jointly with Bhargav Bhatt and Linquan Ma, the PI will continue to study the existence and properties of lim Cohen-Macaulay sequences of modules. This work, as well as some other asymptotic approaches, may resolve the long open question of whether Serre intersection multiplicities are positive, in general, in the case of mixed characteristic regular rings. Another direction for research includes the finiteness of support and other properties of local cohomology, such as the faithfulness of the highest non-vanishing local cohomology module over a local domain with support in a specified ideal. The PI has a continuing project with Jack Jeffries exploring the latter problem. A number of these problems are intended for collaboration with graduate students and postdoctoral faculty.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.