Mathematical Sciences: Topology, Symplectic Geometry and Equivariant Algebraic Geometry

数学科学：拓扑学、辛几何和等变代数几何

• 批准号: 9401858
• 负责人: Ted Petrie
• 金额: $ 13万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-02-01 至 1999-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401858&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topology; Symplectic; Geometry

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. ***

小行星9401858 该项目涉及两个一般问题，首先，给出具有代数群作用的仿射簇（复数）上的稳定平凡等变向量丛集的描述。 举一个特殊的例子，有行动的多样性是一种表象。 表示上是否存在非平凡等变向量丛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。 该项目将扩展的答案是已知的，当非平凡的向量束存在的组的列表，它的目的是开发一个概念上的下界的同构类的束与基地的表示。 这样的结果将在仿射空间上的代数作用有应用。 另一个研究对象是Abyhankar-Moh问题：设f是n个复变的多项式，其中f=1同构于复（n-1）-空间。 仿射n-空间是否存在一个自同构，当它与f构成时，是一个线性函数？ 对于n=2，答案是已知的，对于更高的n，答案是未知的。 这里是一个小动机和一个门外汉的描述上述问题之一，Abyhankar-Moh问题。 它适合于纯数学和应用数学的主要问题领域之一。 给定两个n元函数，什么时候一个可以通过变量的变换（n空间的自同构）转化为另一个？为了解释这意味着什么和激发这个问题，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，和h（x，y）=1的点的集合，即x+yy=1，都表示x，y平面中的无限线。 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表示平面中的无限导线，这一事实在数学上可以用平面中坐标的变化将f变换为h来表示。 坐标x '= x+yy，y'= y的变化将f变换为h，如从f（x '，y'）=f（x+yy，y）=x+yy=h（x，y）所见。 一个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要切断。 与g（x，y）=xx=1定义的集合形成对比。 这个集合由两条无限长的线组成，分别用线x=1和x=-1表示。 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根电线;因此情况在物理上是不相同的。 这在数学上可以解释为，将f（或h）变换为g的二维空间的坐标没有变化。 Abyhankar-Moh问题就属于这种情况。 在这些术语的问题表示为：给定两个函数f和h的n个变量的性质，即集f=1和h=1是"同构“的（n-1）-空间，是否有一个变化的坐标的n-空间的转换f到h？ 上面的讨论说明了n=2的情况，在这种情况下答案是“是”，但对于其他n值是未知的。 ***