Commutative Algebra: Extremal Singularities in Prime Characteristic

交换代数:素数特征中的极值奇点

基本信息

项目摘要

The project is in algebraic geometry, the science of understanding geometric shapes that can be described by polynomial equations. Polynomials are ubiquitous in mathematics because they can be easily manipulated by hand or by machine, yet they exhibit a large range of behavior which can model nearly any shape, from simple circles and lines to complicated images which could appear in medical imaging or an animated movie. The project seeks to understand the singularities of algebraic varieties---places where the shape is pinched or folded over on itself. We will devise tools to measure how "bad" these singularities are, and attempt to classify the "worst" ones. The project will be carried out by the PI with a team of trainees, including undergraduate students, graduate students, post-docs, and other collaborators.The project investigates lower bounds on the F-pure threshold of polynomials over an algebraically closed field of characteristic p. The goal is to find sharp lower bounds, classify the singularities achieving that lower bound, and apply these results to open problems in the field. More specifically, a first direction is to prove general lower bounds on F-pure threshold in terms of other invariants, such as multiplicity, and then identify the varieties—“extremal singularities”— achieving those bounds. Next, the aim is to classify Frobenius forms, which are the “extremal singularities” for a certain lower bound on F-pure threshold in the homogeneous case. A third direction is to advance progress on a conjecture of Kleiman and Piene characterizing hypersurfaces whose Gauss maps are extremal in certain ways, using properties of Frobenius forms. The final direction is an investigation into whether Frobenius forms may define hypersurfaces admitting non-commutative resolutions of singularities, contrary to previous speculations about mildness of singularities for varieties with non-commutative resolutions.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.
该项目是在代数几何,科学的理解几何形状,可以描述的多项式方程。多项式在数学中无处不在,因为它们可以很容易地用手或机器操作,但它们表现出广泛的行为,可以模拟几乎任何形状,从简单的圆形和线条到可能出现在医学成像或动画电影中的复杂图像。该项目旨在理解代数簇的奇点-形状被挤压或折叠的地方。我们将设计工具来衡量这些奇点有多“坏”,并试图对“最坏”的奇点进行分类。该项目将由PI与包括本科生、研究生、博士后和其他合作者在内的受训者团队共同开展。该项目研究特征为p的代数闭域上多项式的F纯阈值的下界。目标是找到尖锐的下界,对达到该下界的奇点进行分类,并将这些结果应用于该领域的公开问题。更具体地说,第一个方向是证明F-纯阈值在其他不变量,如多重性方面的一般下界,然后确定品种-“极值奇点”-实现这些界限。接下来,我们的目标是分类Frobenius形式,这是“极值奇点”的F-纯阈值的某个下界在齐次的情况下。 第三个方向是推进Kleiman和Piene猜想的进展,该猜想描述了高斯映射在某些方面是极值的超曲面,使用Frobenius形式的性质。 最后一个方向是调查弗罗贝纽斯形式是否可以定义超曲面承认非交换决议的奇异性,相反,以前的猜测温和的奇异性品种与非交换决议。这个奖项反映了NSF的法定使命,并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

项目成果

期刊论文数量(2)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Reducedness of formally unramified algebras over fields
域上形式无枝代数的约简
  • DOI:
    10.1016/j.jalgebra.2021.03.002
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    0.9
  • 作者:
    Mukhopadhyay, Alapan;Smith, Karen E.
  • 通讯作者:
    Smith, Karen E.
Cubic surfaces of characteristic two
特征二的立方表面
  • DOI:
    10.1090/tran/8341
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    1.3
  • 作者:
    Kadyrsizova, Zhibek;Kenkel, Jennifer;Page, Janet;Singh, Jyoti;Smith, Karen E.;Vraciu, Adela;Witt, Emily E.
  • 通讯作者:
    Witt, Emily E.
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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

Studies in Commutative Algebra and Algebraic Geometry
交换代数和代数几何研究
  • 批准号:
    2200501
  • 财政年份:
    2022
  • 资助金额:
    $ 39万
  • 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:
    $ 39万
  • 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:
    $ 39万
  • 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:
    $ 39万
  • 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:
    $ 39万
  • 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:
    $ 39万
  • 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:
    $ 39万
  • 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:
    $ 39万
  • 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:
    $ 39万
  • 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:
    $ 39万
  • 项目类别:
    Continuing Grant

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REU Site: Research Experiences for Undergraduates in Algebra and Discrete Mathematics at Auburn University
REU 网站:奥本大学代数和离散数学本科生的研究经验
  • 批准号:
    2349684
  • 财政年份:
    2024
  • 资助金额:
    $ 39万
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Conference: Underrepresented Students in Algebra and Topology Research Symposium (USTARS)
会议:代数和拓扑研究研讨会(USTARS)中代表性不足的学生
  • 批准号:
    2400006
  • 财政年份:
    2024
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    $ 39万
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    Standard Grant
Positive and Mixed Characteristic Birational Geometry and its Connections with Commutative Algebra and Arithmetic Geometry
正混合特征双有理几何及其与交换代数和算术几何的联系
  • 批准号:
    2401360
  • 财政年份:
    2024
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    $ 39万
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Studies in Categorical Algebra
分类代数研究
  • 批准号:
    2348833
  • 财政年份:
    2024
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    $ 39万
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    Continuing Grant
On combinatorics, the algebra, topology, and geometry of a new class of graphs that generalize ordinary and ribbon graphs
关于组合学、一类新图的代数、拓扑和几何,概括了普通图和带状图
  • 批准号:
    24K06659
  • 财政年份:
    2024
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    $ 39万
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    Grant-in-Aid for Scientific Research (C)
RTG: Applied Algebra at the University of South Florida
RTG:南佛罗里达大学应用代数
  • 批准号:
    2342254
  • 财政年份:
    2024
  • 资助金额:
    $ 39万
  • 项目类别:
    Continuing Grant
Conference: Research School: Bridges between Algebra and Combinatorics
会议:研究学院:代数与组合学之间的桥梁
  • 批准号:
    2416063
  • 财政年份:
    2024
  • 资助金额:
    $ 39万
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Conference: Fairfax Algebra Days 2024
会议:2024 年费尔法克斯代数日
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    2337178
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    2024
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    $ 39万
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CAREER: Leveraging Randomization and Structure in Computational Linear Algebra for Data Science
职业:利用计算线性代数中的随机化和结构进行数据科学
  • 批准号:
    2338655
  • 财政年份:
    2024
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    $ 39万
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    Continuing Grant
Stable Homotopy Theory in Algebra, Topology, and Geometry
代数、拓扑和几何中的稳定同伦理论
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    2414922
  • 财政年份:
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  • 资助金额:
    $ 39万
  • 项目类别:
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