Recognition Theorems of Group Theory with Applications to Modular Galois Theory and Desingularization

群论的识别定理及其在模伽罗瓦理论和去奇异化中的应用

基本信息

  • 批准号:
    9988166
  • 负责人:
  • 金额:
    $ 9.36万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2000
  • 资助国家:
    美国
  • 起止时间:
    2000-06-15 至 2004-11-30
  • 项目状态:
    已结题

项目摘要

Jung's method of complex desingularization does not adopt to nonzero characteristic because in the complex case the local fundamental group above a normal crossing of the branch locus is Abelian whereas in nonzero characteristic it need not even be solvable. This is shown by constructing an unsolvable surface covering of degree six in characteristic five. Taking a plane section of this unsolvable surface covering, leads to a conjecture about the structure of the algebraic fundamental group of an affine algebraic curve over an algebraically closed ground field of nonzero characteristic. This Affine Curve Conjecture was settled affirmatively by Harbater and Raynaud. Various Recognition Theorems of group theory, which played a crucial role in the initial explorations of this Conjecture, have lead to some progress in a refined version of this Conjecture over finite ground fields. These Recognition Theorems, have also lead to some progress in Conjectures about Local and Global algebraic fundamental groups above normal crossings of branch loci of higher dimensional algebraic varieties. Another fertile application area for the Recognition Theorems has been in the direction of Permutation Polynomials and Exceptional Polynomials. This area, together with Guralnick's very recent work on genus zero coverings, as well as the theory of Moore-Carlitz-Drinfeld Modules is a rich source for finding explicit equations with prescribed Galois groups. The proposer intends to continue his investigations into the application of the Recognition into these areas of Galois theory. This research is in the combination of the fields of algebraic geometry and group theory. Although they are amongst the oldest parts of modern mathematics, both these fields have had a revolutionary flowering in the past fifty years. In its origin, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials. Likewise, group theory had its origin in the study of symmetries. Nowadays, both these fields make use of methods not only from algebra, but also from analysis and topology, and conversely they are finding applications in those fields as well as in physics, theoretical computer science and robotics. Moreover, interplay between algebraic geometry and group theory continues to enrich both these disciplines.
荣格的复去奇异化方法不适用于非零特征,因为在复的情况下,分支轨迹的法向交叉之上的局部基本群是阿贝尔群,而在非零特征中,它甚至不需要是可解的。这是通过构造一个特征为5的6度不可解曲面覆盖来证明的。取这个不可解曲面覆盖的一个平面截面,得到一个关于非零特征的代数闭基域上仿射代数曲线的代数基本群结构的猜想。Harbater和Raynaud肯定地解决了这个仿射曲线猜想。群论中的各种识别定理在这个猜想的最初探索中起了关键作用,并导致了有限基域上这个猜想的改进版本的一些进展。这些识别定理也导致了高维代数簇的分支轨迹的正规交叉上的局部和整体代数基本群猜想的某些进展。识别定理的另一个丰富的应用领域是排列多项式和例外多项式。这一领域,连同Guralnick最近的工作属零覆盖,以及理论的摩尔-Carlitz-Drinfeld模块是一个丰富的来源,寻找明确的方程与规定的伽罗瓦群。提议者打算继续他的研究,以应用到这些领域的伽罗瓦理论的承认。 本研究是代数几何与群论相结合的研究。虽然它们是现代数学中最古老的部分之一,但这两个领域在过去的五十年里都有了革命性的发展。在其起源中,代数几何处理的图形可以在平面上由最简单的方程(即多项式)定义。同样,群论也起源于对称性的研究。如今,这两个领域不仅使用代数方法,而且还使用分析和拓扑学方法,相反,它们在这些领域以及物理学,理论计算机科学和机器人学中找到了应用。此外,代数几何和群论之间的相互作用继续丰富这两个学科。

项目成果

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Shreeram Abhyankar其他文献

Reduction to multiplicity less thanp in ap-cyclic extension of a two dimensional regular local ring (p = characteristic of the residue field)
  • DOI:
    10.1007/bf01360724
  • 发表时间:
    1964-02-01
  • 期刊:
  • 影响因子:
    1.400
  • 作者:
    Shreeram Abhyankar
  • 通讯作者:
    Shreeram Abhyankar
Über die endliche Erzeugung der Fundamentalgruppe einer komplex-algebraischen Mannigfaltigkeit
  • DOI:
    10.1007/bf01352262
  • 发表时间:
    1960-08-01
  • 期刊:
  • 影响因子:
    1.400
  • 作者:
    Shreeram Abhyankar
  • 通讯作者:
    Shreeram Abhyankar
Concepts of order and rank on a complex space, and a condition for normality
  • DOI:
    10.1007/bf01360171
  • 发表时间:
    1960-04-01
  • 期刊:
  • 影响因子:
    1.400
  • 作者:
    Shreeram Abhyankar
  • 通讯作者:
    Shreeram Abhyankar
Uniformization of jungian local domains
  • DOI:
    10.1007/bf01371613
  • 发表时间:
    1965-08-01
  • 期刊:
  • 影响因子:
    1.400
  • 作者:
    Shreeram Abhyankar
  • 通讯作者:
    Shreeram Abhyankar
Uniformization inp-cyclic extensions of algebraic surfaces over ground fields of characteristicp
  • DOI:
    10.1007/bf01361177
  • 发表时间:
    1964-04-01
  • 期刊:
  • 影响因子:
    1.400
  • 作者:
    Shreeram Abhyankar
  • 通讯作者:
    Shreeram Abhyankar

Shreeram Abhyankar的其他文献

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{{ truncateString('Shreeram Abhyankar', 18)}}的其他基金

Recognition Theorems of Group Theory and Various Inverse Galois Problems
群论和各种逆伽罗瓦问题的识别定理
  • 批准号:
    9732592
  • 财政年份:
    1998
  • 资助金额:
    $ 9.36万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Algorithmic Algebraic Geometry
数学科学:算法代数几何
  • 批准号:
    9101424
  • 财政年份:
    1991
  • 资助金额:
    $ 9.36万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Algorithmic Algebraic Geometry
数学科学:算法代数几何
  • 批准号:
    8816286
  • 财政年份:
    1989
  • 资助金额:
    $ 9.36万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Topics in Algebra and Algebraic Geometry
数学科学:代数和代数几何主题
  • 批准号:
    8500491
  • 财政年份:
    1985
  • 资助金额:
    $ 9.36万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Topics in Algebra and Algebraic Geometry
数学科学:代数和代数几何主题
  • 批准号:
    8002900
  • 财政年份:
    1980
  • 资助金额:
    $ 9.36万
  • 项目类别:
    Continuing Grant
Topics in Algebra and Algebraic Geometry
代数和代数几何专题
  • 批准号:
    7800947
  • 财政年份:
    1978
  • 资助金额:
    $ 9.36万
  • 项目类别:
    Standard Grant
Topics in Algebra and Algebraic Geometry
代数和代数几何专题
  • 批准号:
    7509090
  • 财政年份:
    1975
  • 资助金额:
    $ 9.36万
  • 项目类别:
    Continuing Grant

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