Affine algebraic geometry in dimensions two and three

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的所有多项式的集合。这是一个对象的集合(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)，所以这组多项式是环的一个例子；它被称为三个变量的多项式环，我将用R[3]表示它。同样，如果n是任意正整数，则可以考虑n元多项式环，我用R[n]表示。*因为我们在高中时就开始学习多项式，所以人们可能会觉得多项式环相对简单。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，而且仍在取得进展，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，但仍有许多悬而未决的问题。环R[1]是非常好理解的。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在使用高级代数理论的科学(例如理论物理)中都具有基本的重要性。但在某些研究领域，由于我们对多项式环的了解不足，进展缓慢。在R[n]的研究中，近年来利用一种称为“局部幂零导子”的工具取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。*本研究提案由两部分组成。第一部分是发展局部幂零导子的一般理论，并应用该理论研究R[3]的结构。*在描述这个研究方案的另一个组成部分之前，我必须首先指出，给定任何环R，人们可以考虑R的“环自同态”的集合E(R)。要解释什么是自同态是很长的，所以我将简单地比喻地说，研究E(R)就像研究R如何与其自身相互作用，无论这可能意味着什么；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，E(R)在某种意义上比R更复杂。例如，环R[1]很容易被理解，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。我的研究建议的第二部分是对E(R[2])的调查。人们对E(R[2])知之甚少，鉴于对E(R[1])的研究具有的重要性，可以预见，在理解E(R[2])方面的进展将产生重大影响。

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.**