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Studies in Commutative Algebra and Algebraic Geometry

交换代数和代数几何研究

基本信息

批准号：
1902116
负责人：
Melvin Hochster
金额：
$ 26.98万
依托单位：
Regents of the University of Michigan - Ann Arbor
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-15 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902116&HistoricalAwards=false
关键词：
Studies Commutative Algebra Algebraic Geometry

项目摘要

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.

主要研究者将研究交换代数和代数几何中的几个问题。代数几何学研究多变量多项式方程的解。理解这个问题在许多科学、工程和其他学科中具有根本的重要性。通常很难确定是否期望有任何解、无穷多个解或无穷多个解：在最后一种情况下，人们想知道在描述解时有多少自由度。人们可以通过研究形成所谓交换环的解空间上的某些函数来几何地或代数地研究解。这种双重视角在交换代数和代数几何之间建立了非常有价值的密切联系。要研究的问题是长期存在的，具有根本性和中心重要性。所取得的成果将通过期刊和书籍出版物、会议和讲习班上的讲座以及通过互联网传播。有很强的教育成分。首席研究员将继续指导研究生，博士后，高中生通过几个项目，他已经建立了多年来，特别注意吸引学生，在数学科学的代表性不足.首席研究员将继续探索诺特环理论中的几个长期存在的问题. Tigran Ananyan和PI已经证明了Stillman猜想的所有特征，但是投射维数和其他属性的界限，例如与初等分解相关的理想的生成元的数量，肯定可以改进，并且将研究这个问题。 PI还将研究是否存在具有某些标准性质的混合特征紧闭的类似物：通常的冒号捕获和Dietz版本的此属性，正则环的所有理想都是封闭的属性，持久性和测试元素理论。最后两个似乎是最困难的。预计，从perfectoid几何的新方法将是所需的工具之一。与Sema Gunturkun一起，PI正在研究关于二次型的长期存在的Eisenbud-Green-Harris猜想，该猜想预测了某些理想的希尔伯特函数的最小值。与Bhargav Bhatt和Linquan Ma一起，PI将继续研究模的lim Cohen-Macaulay序列的存在性和性质。这项工作，以及其他一些渐近的方法，可能会解决长期开放的问题，是否塞尔交叉多重性是积极的，在一般情况下，在混合特征正则环。另一个研究方向包括支撑的有限性和局部上同调的其他性质，例如在特定理想中支撑的局部区域上的最高非零局部上同调模的忠实性。PI有一个持续的项目与杰克杰弗里斯探索后一个问题。该奖项反映了NSF的法定使命，并被认为值得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来支持。