Mathematical Sciences: Topology, Symplectic Geometry and Equivariant Algebraic Geometry

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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. ***