Mathematical Sciences: Topology, Symplectic Geometry and Equivariant Algebraic Geometry

9401858皮特里这个项目涉及两个一般问题，第一，给出具有代数群作用的仿射簇(复数上)上的稳定平凡等变向量丛的集合的描述。以带有动作的变化是表示的特殊情况为例。是否存在表示上的非平凡等变向量丛仍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，当存在非平凡向量丛时，它旨在开发具有基础a表示的丛的同构类的概念下界。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，且f=1同构于复(n-1)-空间。仿射n-空间是否存在自同构，当与f合成时，它是线性函数？对于n=2，答案是已知的，对于更高的n，答案是未知的。下面是对上述问题之一Abyhankar-Moh问题的一个小小的动机和外行的描述。它属于纯数学和应用数学的主要问题领域之一。给定n个变量的两个函数，什么时候可以通过变量的改变(n-空间的自同构)将其中一个转换成另一个？为了解释这意味着什么并激发问题的动机，考虑两个变量x和y的两个函数：f(x，y)=x和h(x，y)=x+yy。其中f(x，y)=1(即x=1)的点集和h(x，y)=1(即x+yy=1)的点集都表示x，y平面上的无限导线。第一个是直线，第二个是抛物线x+yy=1。这两个集合都表示平面上的一条无限长的线，这一事实在数学上是通过平面上的坐标变化来表示的，它将f变换为h。坐标x‘=x+yy，y’=y将f变换为h，正如从f(x‘，y’)=f(x+yy，y)=x+yy=h(x，y)所看到的那样。必须通过切断电线来切断连接的电工，在每种情况下都有一根电线需要切断。这与由g(x，y)=xx=1定义的集合形成对比。该集合由x=1和x=-1这两条线表示的两条无限导线组成。在这种情况下，希望切断由集合g(x，y)=1表示的连接的电工将不得不切断两条导线；因此，物理上的情况不同。从数学上解释这一点的原因是，将f(或h)变换为g的两个空间的坐标不变。Abyhankar-Moh问题就符合这种情况。在这些条件下，问题被表示为：给定n个变量的两个函数f和h，其性质是集合f=1和h=1与(n-1)-空间“同构”，是否存在将f变换为h的n-空间的坐标变化？上面的讨论说明了n=2的情况，在这种情况下答案是已知的“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. ***