Mathematical Sciences: Topology, Symplectic Geometry and Equivariant Algebraic Geometry

[401858 . Petrie]本课题涉及两个一般问题，第一，给出具有代数群作用的仿射变量（复数）上的稳定平凡等变向量束集合的描述。以特殊情况为例，其中带有动作的品种是一种表示。在表示上是否存在非平凡等变向量束仍然是一个悬而未决的问题。答案取决于群体，有许多群体的答案是未知的。该项目将扩展已知答案的群列表，并且，当存在非平凡向量束时，它打算为具有基表示的束的同构类开发一个概念下界。这些结果将应用于仿射空间上的代数作用。另一个研究对象是Abyhankar-Moh问题：设f是n个复变量的多项式，其中f=1同构于复（n-1）空间。是否存在仿射n空间的自同构，当与f组合时，是一个线性函数？当n=2时，答案是已知的，而当n大于2时，答案是未知的。这里有一点动机，以及对上述问题之一——Abyhankar-Moh问题的外行人描述。它符合纯数学和应用数学的主要问题领域之一。给定两个n变量的函数，何时可以通过变量变换（n空间的自同构）将其中一个变换为另一个？为了解释这意味着什么并激发问题，考虑两个变量x和y的两个函数：f(x,y)=x和h(x,y)=x+yy。f(x,y)=1的点的集合，即x=1，以及h(x,y)=1的点的集合，即x+yy=1，都表示x，y平面上的无限线。第一个是直线，第二个是抛物线x+yy=1。这两个集合中的每一个都代表了平面上的无限大的线，这一事实在数学上是通过平面上的坐标变化来表达的，它把f变成了h。坐标变化x'=x+yy, y'=y把f变成了h，正如我们看到的，f(x',y')=f(x+yy,y)=x+yy=h（x,y）。如果一个电工必须通过切断电线来切断连接，那么在每种情况下他都只有一根电线可以切断。将其与g(x,y)=xx=1定义的集合进行对比。这个集合由两条无限的线组成，用直线x=1和x=-1表示。在这种情况下，电工想要切断集合g(x,y)=1所表示的连接，就必须切断两根电线；所以情况在物理上是不一样的。这可以用数学上的事实来解释，即二维空间的坐标不会改变，从而将f（或h）转换为g。Abyhankar-Moh问题适用于这种情况。在这些术语中，问题被表示为：给定两个函数f和h （n个变量），并且集合f=1和h=1是（n-1）空间的“同构”，是否有一个n空间的坐标变换将f变换为h？上面的讨论说明了n=2的情况，在这种情况下答案已知是“是”，但对于n的其他值则未知

9401858 Petrie This project is concerned with two general problems, first, to give a description of the set of stably trivial equivariant vector bundles over an affine variety (over the complex numbers) having an action of an algebraic group. Take the special case where the variety with action is a representation. It is still an open question whether there are non-trivial equivariant vector bundles over representations. The answer depends on the group, and there are many groups for which the answer is unknown. The project will extend the list of groups where the answer is known and, when non-trivial vector bundles exist, it is intended to develop a conceptual lower bound for the isomorphism classes of bundles with base a representation. Such results will have applications for algebraic actions on affine space. The other object of study is the Abyhankar-Moh Problem: Let f be a polynomial in n complex variables for which f=1 is isomorphic to complex (n-1)-space. Is there an automorphism of affine n-space which when composed with f, is a linear function? The answer is known for n=2 and unknown for higher n. Here is a little motivation and a layman's description of one of the above problems, the Abyhankar-Moh Problem. It fits into one of the main problem areas of pure and applied mathematics. Given two functions of n variables, when can one be transformed into the other by a change of variables (automorphism of n-space)? To explain what this means and motivate the problem, consider the two functions of two variables x and y: f(x,y)=x and h(x,y)=x+yy. The set of points where f(x,y)=1, i.e. x=1, and the set of points where h(x,y)=1, i.e. x+yy=1, both represent an infinite wire in the x,y-plane. The first is a line, and the second is the parabola x+yy=1. The fact that each of these two sets represents an infinite wire in the plane is expressed mathematically by the fact that there is a change of coordinates in the plane which transforms f into h. The change of coordinates x'=x+yy, y'=y transforms f to h, as one sees from f(x',y')=f(x+yy,y)=x+yy=h(x,y). An electrician who had to sever a connection by cutting the wires would in each case have one wire to cut. Contrast that to the set defined by g(x,y)=xx=1. That set consists of two infinite wires represented by the lines x=1 and x=-1. In this case an electrician wishing to sever the connection represented by the set g(x,y)=1 would have to cut two wires; so the situations are not physically the same. This is explained mathematically by the fact that there is no change of coordinates of two-space which transforms f (or h) to g. The Abyhankar-Moh Problem fits into this setting. In these terms the problem is expressed as: Given two functions f and h of n variables with the property that the sets f=1 and h=1 are ``isomorphic'' to (n-1)-space, is there a change of coordinates of n-space which transforms f to h? The above discussion illustrates the case n=2, and the answer is known to be "yes" in that case, but it is unknown for other values of n. ***