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:2200501
- 负责人:
- 金额:$ 64万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2022
- 资助国家:美国
- 起止时间:2022-07-01 至 2028-06-30
- 项目状态:未结题
- 来源:
- 关键词:
项目摘要
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.
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。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Karen Smith其他文献
Evaluation of the Multi-Test device for immediate hypersensitivity skin testing.
用于立即过敏皮肤测试的多重测试装置的评估。
- DOI:
10.1016/0091-6749(92)90471-d - 发表时间:
1992 - 期刊:
- 影响因子:0
- 作者:
Robert B. Berkowitz;David G. Tinkelman;Cheryl Lutz;Angela Crummie;Karen Smith - 通讯作者:
Karen Smith
Critical discourse analysis and higher education research
批判性话语分析与高等教育研究
- DOI:
10.1108/s1479-3628(2013)0000009007 - 发表时间:
2013 - 期刊:
- 影响因子:0
- 作者:
Karen Smith - 通讯作者:
Karen Smith
Factors associated with emergency medical service delays in suspected ST‐elevation myocardial infarction in Victoria, Australia: A retrospective study
澳大利亚维多利亚州疑似 ST 段抬高型心肌梗死患者紧急医疗服务延误的相关因素:一项回顾性研究
- DOI:
- 发表时间:
2020 - 期刊:
- 影响因子:2.3
- 作者:
Ahmad Alrawashdeh;Z. Nehme;B. Williams;Karen Smith;M. Stephenson;S. Bernard;P. Cameron;D. Stub - 通讯作者:
D. Stub
Residential aged care homes: Why do they call ‘000’? A study of the emergency prehospital care of older people living in residential aged care homes
居家养老院:为何将其称为“000”?对居家养老院老年人的院前紧急护理的研究
- DOI:
10.1111/1742-6723.13650 - 发表时间:
2020 - 期刊:
- 影响因子:2.3
- 作者:
R. Dwyer;B. Gabbe;T. Tran;Karen Smith;J. Lowthian - 通讯作者:
J. Lowthian
Differentiation of confirmed major trauma patients and potential major trauma patients using pre-hospital trauma triage criteria.
使用院前创伤分诊标准区分已确诊的重大创伤患者和潜在的重大创伤患者。
- DOI:
10.1016/j.injury.2010.03.035 - 发表时间:
2011 - 期刊:
- 影响因子:2.5
- 作者:
S. Cox;Karen Smith;A. Currell;L. Harriss;B. Barger;P. Cameron - 通讯作者:
P. Cameron
Karen Smith的其他文献
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{{ truncateString('Karen Smith', 18)}}的其他基金
Commutative Algebra: Extremal Singularities in Prime Characteristic
交换代数:素数特征中的极值奇点
- 批准号:
2101075 - 财政年份:2021
- 资助金额:
$ 64万 - 项目类别:
Continuing Grant
FRG: Collaborative Research: Algebraic Geometry and Singularities in Positive and Mixed Characteristic
FRG:合作研究:代数几何和正特征和混合特征中的奇点
- 批准号:
1952399 - 财政年份:2020
- 资助金额:
$ 64万 - 项目类别:
Continuing Grant
Commutative Algebra: F-Regularity in Algebraic Geometry and Non-Commutative Algebra
交换代数:代数几何和非交换代数中的 F 正则性
- 批准号:
1801697 - 财政年份:2018
- 资助金额:
$ 64万 - 项目类别:
Continuing Grant
Algorithm Development For Reconstruction Of Design Elements
设计元素重构的算法开发
- 批准号:
1658987 - 财政年份:2017
- 资助金额:
$ 64万 - 项目类别:
Standard Grant
The Impact of the Stratosphere on Arctic Climate
平流层对北极气候的影响
- 批准号:
1603350 - 财政年份:2016
- 资助金额:
$ 64万 - 项目类别:
Standard Grant
Commutative Algebra: Frobenius in Geometry and Combinatorics
交换代数:几何和组合学中的弗罗贝尼乌斯
- 批准号:
1501625 - 财政年份:2015
- 资助金额:
$ 64万 - 项目类别:
Continuing Grant
EMSW21-RTG: Developing American Research Leadership in Algebraic Geometry and its Boundaries
EMSW21-RTG:发展美国在代数几何及其边界方面的研究领导地位
- 批准号:
0943832 - 财政年份:2010
- 资助金额:
$ 64万 - 项目类别:
Continuing Grant
Bringing Frobenius to Bear on Birational Algebraic Geometry
将弗罗贝尼乌斯应用于双有理代数几何
- 批准号:
1001764 - 财政年份:2010
- 资助金额:
$ 64万 - 项目类别:
Continuing Grant
Commutative Algebra and its Interactions, July 31 - August 3, 2008
交换代数及其相互作用,2008年7月31日至8月3日
- 批准号:
0810844 - 财政年份:2008
- 资助金额:
$ 64万 - 项目类别:
Standard Grant
Noncommutative Geometry and Cherednik Algebras
非交换几何和切里德尼克代数
- 批准号:
0555750 - 财政年份:2006
- 资助金额:
$ 64万 - 项目类别:
Continuing Grant
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Pan American Advanced Studies Institute: Commutative Algebra and Its Interactions with Algebraic Geometry, Representation Theory, and Physics; Guanajuato, Mexico; May 14-25, 2012
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1123059 - 财政年份:2012
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