Studies in Commutative Algebra and Algebraic Geometry

交换代数和代数几何研究

基本信息

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。

项目成果

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Karen Smith其他文献

『功利主義と政策思想の展開』第3章「シジウィック・ムーア・ピグー-功利主義・正義の観点から-」(音無通宏編)
《功利主义与政策思想的发展》第3章《西奇威克、摩尔和庇古——从功利主义和正义的视角》(音无道弘主编)
  • DOI:
  • 发表时间:
    2011
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Chulhee Kang;Femida Handy;Lesley Hustinx;Ram Cnaan;Jeffrey L.Brudney;Debbie Haski-Leventhal;Kirsten Holmes;Lucas Meijs;Anne Birgitta Pessi;Bhagyashree Ranade;Karen Smith;Naoto Yamauchi;Sinisa Zrinscak;山崎聡
  • 通讯作者:
    山崎聡
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
Testing for Drugs of Abuse in Children and Adolescents: Addendum—Testing in Schools and at Home
儿童和青少年滥用药物检测:附录 — 在学校和家庭进行检测
  • DOI:
    10.1542/peds.2006-3688
  • 发表时间:
    2007
  • 期刊:
  • 影响因子:
    8
  • 作者:
    Alain Joffe;Marylou Behnke;J. Knight;P. Kokotailo;Tammy H. Sims;Janet Williams;J. Kulig;Deborah Simkin;Linn Goldberg;Sharon Levy;Karen Smith;Robert D. Murray;B. L. Frankowski;R. Gereige;C. Mears;Michele M. Roland;Thomas L. Young;Linda M. Grant;Daniel Hyman;Harold Magalnick;George J. Monteverdi;Evan G. Pattishall;Nancy LaCursia;Donna Mazyck;Mary E. Vernon;Robin Wallace;Madra Guinn
  • 通讯作者:
    Madra Guinn
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
Energy in Schools: Empowering Children to Deliver Behavioural Change for Sustainability
学校活力:赋予儿童行为改变以实现可持续发展的能力

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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交换代数和代数几何研究
  • 批准号:
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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
泛美高等研究院:交换代数及其与代数几何、表示论和物理学的相互作用;
  • 批准号:
    1123059
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    2012
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    $ 64万
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  • 批准号:
    1259142
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  • 批准号:
    0901145
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    2009
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