Affine algebraic geometry in dimensions two and three

二维和三维仿射代数几何

基本信息

  • 批准号:
    RGPIN-2015-04539
  • 负责人:
  • 金额:
    $ 1.02万
  • 依托单位:
  • 依托单位国家:
    加拿大
  • 项目类别:
    Discovery Grants Program - Individual
  • 财政年份:
    2018
  • 资助国家:
    加拿大
  • 起止时间:
    2018-01-01 至 2019-12-31
  • 项目状态:
    已结题

项目摘要

In Algebra, a set of objects that can be added and multiplied is called a "ring". For instance, consider the set of all polynomials in the three variables x,y,z. This is a set of objects (the objects being the polynomials), and these objects can be added and multiplied (we learn in highschool how to add and multiply two polynomials), so this set of polynomials is an example of a ring; it is called the ring of polynomials in three variables, and I shall denote it by R[3]. Similarly, if n is any positive integer, one can consider the ring of polynomials in n variables, which I denote by R[n].***Because we first learn about polynomials in highschool, one might get the impression that polynomial rings are relatively simple. However, this is not at all the case. Over the last 50 years, mathematicians have been trying to elucidate the structure of the ring R[3]; some progress has been made, and is still being made, but that ring is still largely a mystery. The ring R[2] is better understood than R[3], but still entails many unanswered questions.  The ring R[1] is very well understood.  The larger the value of n, the more complicated the ring R[n].******Rings of polynomials are of fundamental importance in all of algebra and geometry, and in the sciences that use advanced algebraic theories (theoretical physics, for instance).  In certain research areas, progress is slowed down because of our insufficient understanding of rings of polynomials. ******In the study of R[n], encouraging results have been obtained in recent years by using a tool called "locally nilpotent derivations". This is a tool that can be used for studying all rings, not just rings of polynomials. ******This research proposal consists of two components. The first component is to develop the general theory of locally nilpotent derivations, and to apply that theory to investigate the structure of R[3]. ******Before describing the other component of this research proposal I must first say that, given any ring R, one can consider the set E(R) of "ring endomorphisms" of R. It would be long to explain what an endomorphism is, so I will simply say, metaphorically, that studying E(R) is like studying how R interacts with itself, whatever that might mean; what the reader should keep in mind is the idea that understanding the structure of E(R) is as important as understanding R itself. In the case where R is a ring of polynomials, it turns out that E(R) is in some sense more complicated than R. For instance, the ring R[1] is very well understood, but E( R[1] ) has been the subject of intensive research for the last 90 years, and is still a lively research area with applications in several fields, including cryptography. The second component of my research proposal is an investigation of E( R[2] ). Very little is known about E( R[2] ), and in view of the importance that the study of E( R[1] ) has had, one can anticipate that progress in understanding E( R[2] ) would have a significant impact.**
在代数中,一组可以相加和相乘的对象称为“环”。 例如,考虑三个变量 x、y、z 中所有多项式的集合。 这是一组对象(对象是多项式),这些对象可以相加和相乘(我们在高中学习如何将两个多项式相加和相乘),所以这组多项式是一个环的例子;它被称为三变量多项式环,我将用 R[3] 来表示它。 类似地,如果n是任何正整数,我们可以考虑n个变量的多项式环,我用R[n]表示。***因为我们在高中第一次学习多项式,人们可能会觉得多项式环相对简单。 然而,事实并非如此。 过去 50 年来,数学家们一直试图阐明环 R 的结构[3];已经取得了一些进展,并且仍在取得进展,但那枚戒指在很大程度上仍然是一个谜。 环 R[2] 比 R[3] 更容易理解,但仍然存在许多未解答的问题。  环R[1]很好理解。  n 的值越大,环 R[n] 就越复杂。****** 多项式环在所有代数和几何中以及使用高级代数理论的科学(例如理论物理学)中都具有根本重要性。  在某些研究领域,由于我们对多项式环的理解不够,进展缓慢。 ******在R[n]的研究中,近年来通过使用一种称为“局部幂幂求导”的工具取得了令人鼓舞的结果。 这是一个可用于研究所有环的工具,而不仅仅是多项式环。 ******本研究计划由两个部分组成。 第一个组成部分是发展局部幂零导数的一般理论,并应用该理论来研究 R[3] 的结构。 ******在描述本研究提案的其他组成部分之前,我必须首先说,给定任何环 R,我们可以考虑 R 的“环自同态”集合 E(R)。要解释什么是自同态会很长,所以我会简单地说,隐喻性地,研究 E(R) 就像研究 R 如何与自身相互作用,无论这意味着什么;读者应该记住的是,理解 E(R) 的结构与理解 R 本身同样重要。 在 R 是多项式环的情况下,事实证明 E(R) 在某种意义上比 R 更复杂。例如,环 R[1] 很好理解,但 E( R[1] ) 在过去 90 年来一直是深入研究的主题,并且仍然是一个活跃的研究领域,在包括密码学在内的多个领域都有应用。 我的研究计划的第二个组成部分是对 E( R[2] ) 的调查。 人们对 E( R[2] ) 知之甚少,鉴于 E( R[1] ) 研究的重要性,可以预见,理解 E( R[2] ) 方面的进展将产生重大影响。**

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)

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ monograph.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ sciAawards.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ conferencePapers.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ patent.updateTime }}

Daigle, Daniel其他文献

Morphological Processing and Learning to Read: The Case of Deaf Children
  • DOI:
    10.1179/1557069x14y.0000000036
  • 发表时间:
    2014-01-01
  • 期刊:
  • 影响因子:
    1.4
  • 作者:
    Berthiaume, Rachel;Daigle, Daniel
  • 通讯作者:
    Daigle, Daniel
What Do Spelling Errors Tell Us about Deaf Learners of French?

Daigle, Daniel的其他文献

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

{{ truncateString('Daigle, Daniel', 18)}}的其他基金

Affine algebraic geometry in dimensions two and three
二维和三维仿射代数几何
  • 批准号:
    RGPIN-2015-04539
  • 财政年份:
    2019
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Affine algebraic geometry in dimensions two and three
二维和三维仿射代数几何
  • 批准号:
    RGPIN-2015-04539
  • 财政年份:
    2017
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Affine algebraic geometry in dimensions two and three
二维和三维仿射代数几何
  • 批准号:
    RGPIN-2015-04539
  • 财政年份:
    2016
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Affine algebraic geometry in dimensions two and three
二维和三维仿射代数几何
  • 批准号:
    RGPIN-2015-04539
  • 财政年份:
    2015
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Affine algebraic varieties which admit many Ga-actions
允许许多 Ga 作用的仿射代数簇
  • 批准号:
    104976-2010
  • 财政年份:
    2014
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Affine algebraic varieties which admit many Ga-actions
允许许多 Ga 作用的仿射代数簇
  • 批准号:
    104976-2010
  • 财政年份:
    2013
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Affine algebraic varieties which admit many Ga-actions
允许许多 Ga 作用的仿射代数簇
  • 批准号:
    104976-2010
  • 财政年份:
    2012
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Affine algebraic varieties which admit many Ga-actions
允许许多 Ga 作用的仿射代数簇
  • 批准号:
    104976-2010
  • 财政年份:
    2011
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Affine algebraic varieties which admit many Ga-actions
允许许多 Ga 作用的仿射代数簇
  • 批准号:
    104976-2010
  • 财政年份:
    2010
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Low dimensional affine algebraic geometry
低维仿射代数几何
  • 批准号:
    104976-2005
  • 财政年份:
    2009
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual

相似国自然基金

Lienard系统的不变代数曲线、可积性与极限环问题研究
  • 批准号:
    12301200
  • 批准年份:
    2023
  • 资助金额:
    30.00 万元
  • 项目类别:
    青年科学基金项目
对RS和AG码新型软判决代数译码的研究
  • 批准号:
    61671486
  • 批准年份:
    2016
  • 资助金额:
    60.0 万元
  • 项目类别:
    面上项目
同伦和Hodge理论的方法在Algebraic Cycle中的应用
  • 批准号:
    11171234
  • 批准年份:
    2011
  • 资助金额:
    40.0 万元
  • 项目类别:
    面上项目

相似海外基金

Affine algebraic geometry in dimensions two and three
二维和三维仿射代数几何
  • 批准号:
    RGPIN-2015-04539
  • 财政年份:
    2019
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Affine algebraic geometry in dimensions two and three
二维和三维仿射代数几何
  • 批准号:
    RGPIN-2015-04539
  • 财政年份:
    2017
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Affine algebraic geometry in dimensions two and three
二维和三维仿射代数几何
  • 批准号:
    RGPIN-2015-04539
  • 财政年份:
    2016
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Affine algebraic geometry in dimensions two and three
二维和三维仿射代数几何
  • 批准号:
    RGPIN-2015-04539
  • 财政年份:
    2015
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Investigations in affine algebraic geometry
仿射代数几何研究
  • 批准号:
    194395-2011
  • 财政年份:
    2015
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Investigations in affine algebraic geometry
仿射代数几何研究
  • 批准号:
    194395-2011
  • 财政年份:
    2014
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Investigations in affine algebraic geometry
仿射代数几何研究
  • 批准号:
    194395-2011
  • 财政年份:
    2013
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Investigations in affine algebraic geometry
仿射代数几何研究
  • 批准号:
    194395-2011
  • 财政年份:
    2012
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
Applications of minimal model theory to higher dimensional affine algebraic geometry
最小模型理论在高维仿射代数几何中的应用
  • 批准号:
    24740003
  • 财政年份:
    2012
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Grant-in-Aid for Young Scientists (B)
Investigations in affine algebraic geometry
仿射代数几何研究
  • 批准号:
    194395-2011
  • 财政年份:
    2011
  • 资助金额:
    $ 1.02万
  • 项目类别:
    Discovery Grants Program - Individual
{{ showInfoDetail.title }}

作者:{{ showInfoDetail.author }}

知道了