Cohomology, Representations, and Coverings of Curves

曲线的上同调、表示和覆盖

基本信息

  • 批准号:
    1600056
  • 负责人:
  • 金额:
    $ 19.5万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2016
  • 资助国家:
    美国
  • 起止时间:
    2016-07-15 至 2019-10-31
  • 项目状态:
    已结题

项目摘要

The mathematical concept of a group, a set together with an operation for combining two elements to produce another element, plays a central role in mathematics and its applications because it encodes intuitive notions of symmetry in a precise manner. The classification of the collection of finite groups known as finite simple groups is a monumental achievement of modern mathematics that has led to a revolution in using group theory to study other fields. The utility of group theory has been greatly expanded by advances in computation. One goal of the project is to describe useful presentations of finite simple groups that lead to more computational efficiency. This will lead to solutions of fundamental problems in number theory. Another goal of the project is to study a group-theoretical problem that will lead will lead to results showing the existence and construction of expander graphs, which have been of great importance in computer science.In more detail, one goal of the project is to prove a generalization of the fact that if H is a finitely generated Zariski dense subgroup of a semisimple algebraic group, then H contains a strongly dense subgroup. This will give some new results about superstrong approximation in number theory and results on expander graphs. A second goal of the project is to prove the conjecture that every finite simple group has a presentation with two generations and at most four relations. This will lead to advances in computational number theory. A third goal of the project is to classify generic stabilizers for simple algebraic groups in irreducible linear representations. This will fit into the program of Bhargava to solve classification problems of algebraic families. A fourth goal of the project is to classify monodromy groups of coverings of low genus Riemann surfaces.
群的数学概念是一个集合,以及组合两个元素以产生另一个元素的运算,它在数学及其应用中发挥着核心作用,因为它以一种精确的方式对对称的直观概念进行了编码。对称为有限单群的有限群集合的分类是现代数学的一项不朽成就,它导致了使用群论来研究其他领域的一场革命。群论在计算方面的进步大大扩展了它的用途。该项目的一个目标是描述有限单群的有用表示,从而导致更高的计算效率。这将导致数论中基本问题的解决。这个项目的另一个目标是研究一个群论问题,这个问题将导致证明扩张图的存在和构造的结果,这在计算机科学中一直是非常重要的。更详细地说,这个项目的一个目标是证明了如果H是半单代数群的有限生成的Zariski稠密子群,那么H包含一个强稠密子群。这将给出数论中关于超强逼近的一些新结果和关于扩展图的一些结果。该项目的第二个目标是证明一个猜想,即每个有限单群都有一个有两代且至多有四个关系的表示。这将导致计算数论的进步。该项目的第三个目标是在不可约的线性表示中对简单代数群的通用稳定器进行分类。这将适用于Bhargava解决代数族分类问题的程序。该项目的第四个目标是对低亏格黎曼曲面覆盖的单色群进行分类。

项目成果

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会议论文数量(0)
专利数量(0)

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Robert Guralnick其他文献

Modular characters, hall subgroups, and normal complements
Reimagining species on the move across space and time
  • DOI:
    10.1016/j.tree.2025.03.015
  • 发表时间:
    2025-07-01
  • 期刊:
  • 影响因子:
    17.300
  • 作者:
    Alexa L. Fredston;Morgan W. Tingley;Montague H.C. Neate-Clegg;Luke J. Evans;Laura H. Antão;Natalie C. Ban;I-Ching Chen;Yi-Wen Chen;Lise Comte;David P. Edwards;Birgitta Evengard;Belen Fadrique;Sophie H. Falkeis;Robert Guralnick;David H. Klinges;Jonas J. Lembrechts;Jonathan Lenoir;Juliano Palacios-Abrantes;Aníbal Pauchard;Gretta Pecl;Brett R. Scheffers
  • 通讯作者:
    Brett R. Scheffers
Primitive monodromy groups of genus at most two
  • DOI:
    10.1016/j.jalgebra.2014.06.020
  • 发表时间:
    2014-11-01
  • 期刊:
  • 影响因子:
  • 作者:
    Daniel Frohardt;Robert Guralnick;Kay Magaard
  • 通讯作者:
    Kay Magaard
On rational and concise words
  • DOI:
    10.1016/j.jalgebra.2015.02.003
  • 发表时间:
    2015-05-01
  • 期刊:
  • 影响因子:
  • 作者:
    Robert Guralnick;Pavel Shumyatsky
  • 通讯作者:
    Pavel Shumyatsky
The automorphism groups of a family of maximal curves
  • DOI:
    10.1016/j.jalgebra.2012.03.036
  • 发表时间:
    2012-07-01
  • 期刊:
  • 影响因子:
  • 作者:
    Robert Guralnick;Beth Malmskog;Rachel Pries
  • 通讯作者:
    Rachel Pries

Robert Guralnick的其他文献

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{{ truncateString('Robert Guralnick', 18)}}的其他基金

IntBIO Collaborative Research: Assessing drivers of the nitrogen-fixing symbiosis at continental scales
IntBIO 合作研究:评估大陆尺度固氮共生的驱动因素
  • 批准号:
    2316267
  • 财政年份:
    2023
  • 资助金额:
    $ 19.5万
  • 项目类别:
    Standard Grant
Collaborative Research: Ranges: Building Capacity to Extend Mammal Specimens from Western North America
合作研究:范围:建设能力以扩展北美西部的哺乳动物标本
  • 批准号:
    2228392
  • 财政年份:
    2023
  • 资助金额:
    $ 19.5万
  • 项目类别:
    Continuing Grant
Collaborative Research: Phenobase: Community, infrastructure, and data for global-scale analyses of plant phenology
合作研究:Phenobase:用于全球范围植物物候分析的社区、基础设施和数据
  • 批准号:
    2223512
  • 财政年份:
    2022
  • 资助金额:
    $ 19.5万
  • 项目类别:
    Continuing Grant
Collaborative Research: CIBR: Leaping the Specimen Digitization Gap: Connecting Novel Tools, Machine Learning and Public Participation to Label Digitization Efforts
合作研究:CIBR:跨越标本数字化差距:将新工具、机器学习和公众参与与标签数字化工作联系起来
  • 批准号:
    2027234
  • 财政年份:
    2021
  • 资助金额:
    $ 19.5万
  • 项目类别:
    Standard Grant
Collaborative Research: LightningBug, An Integrated Pipeline to Overcome The Biodiversity Digitization Gap
合作研究:LightningBug,克服生物多样性数字化差距的综合管道
  • 批准号:
    2104152
  • 财政年份:
    2021
  • 资助金额:
    $ 19.5万
  • 项目类别:
    Continuing Grant
Collaborative Research: Origins and drivers of extinction of Caribbean Avifauna
合作研究:加勒比鸟类灭绝的起源和驱动因素
  • 批准号:
    2033905
  • 财政年份:
    2021
  • 资助金额:
    $ 19.5万
  • 项目类别:
    Continuing Grant
Collaborative Research: Genealogy of Odonata (GEODE): Dispersal and color as drivers of 300 million years of global dragonfly evolution
合作研究:蜻蜓目 (GEODE) 谱系:传播和颜色是 3 亿年全球蜻蜓进化的驱动力
  • 批准号:
    2002457
  • 财政年份:
    2020
  • 资助金额:
    $ 19.5万
  • 项目类别:
    Continuing Grant
IIBR RoL: Collaborative Research: A Rules Of Life Engine (RoLE) Model to Uncover Fundamental Processes Governing Biodiversity
IIBR RoL:协作研究:揭示生物多样性基本过程的生命规则引擎 (RoLE) 模型
  • 批准号:
    1927286
  • 财政年份:
    2019
  • 资助金额:
    $ 19.5万
  • 项目类别:
    Standard Grant
Cohomology and Representations of Finite and Algebraic Groups with Applications
有限代数群的上同调和表示及其应用
  • 批准号:
    1901595
  • 财政年份:
    2019
  • 资助金额:
    $ 19.5万
  • 项目类别:
    Continuing Grant
Collaborative Research: ABI Innovation: FuTRES, an Ontology-Based Functional Trait Resource for Paleo- and Neo-biologists
合作研究:ABI 创新:FuTRES,为古生物学家和新生物学家提供的基于本体的功能性状资源
  • 批准号:
    1759898
  • 财政年份:
    2018
  • 资助金额:
    $ 19.5万
  • 项目类别:
    Standard Grant

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