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
• 负责人: Daigle, Daniel
• 金额: $ 1.02万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675379
• 关键词: Affine; algebraic; geometry; dimensions; two

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被称为“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（这些对象是多项式），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；它被称为三元多项式环，我用r[3]表示。类似地，如果n是任意正整数，可以考虑n个变量的多项式环，我用R[n]表示。因为我们第一次学习多项式是在高中，有人可能会觉得多项式环是相对简单的。然而，事实并非如此。在过去的50年里，数学家们一直在试图阐明环r[3]的结构；已经取得了一些进展，而且还在继续取得进展，但这个环在很大程度上仍然是一个谜。环r[2]比r[3]更容易理解，但仍有许多未解之谜。环r[1]是很容易理解的。n的值越大，环R[n]越复杂。******多项式环在所有代数和几何以及使用高级代数理论的科学（例如理论物理）中都具有重要的基础意义。在某些研究领域，由于我们对多项式环的理解不足，进展缓慢。******在R[n]的研究中，近年来利用一种称为“局部幂零导数”的工具获得了令人鼓舞的结果。这是一个可以用来研究所有环的工具，而不仅仅是多项式环。******本研究计划由两部分组成。第一部分是发展局部幂零导数的一般理论，并应用该理论研究r[3]的结构。******在描述这个研究计划的其他部分之前，我必须首先说，给定任何环R，人们可以考虑R的“环自同态”集合E(R)。解释什么是自同态是很长时间的，所以我将简单地说，隐喻地说，研究E(R)就像研究R如何与自身相互作用，不管这可能意味着什么；读者应该记住的是，理解E(R)的结构和理解R本身一样重要。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]被很好地理解，但E（R[1]）在过去的90年里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几个领域都有应用。我的研究计划的第二个组成部分是对E（R[2]）的调查。我们对E（R[2]）知之甚少，鉴于对E（R[1]）研究的重要性，我们可以预见，对E（R[2]）的理解将产生重大影响