Affine algebraic geometry in dimensions two and three

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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.**