Algebraic Automorphisms of Affine Space

University Of Southern Indiana, Evansville IN

Investigators

Abstract

The investigator studies the general affine groups and their related geometry (affine algebraic geometry). These groups are the algebraic automorphisms of affine spaces over a field. Group actions and their associated rings of invariants play a central role in this investigation. Of particular interest are actions of the additive group of the underlying field, or equivalently, locally nilpotent derivations of polynomial rings. These automorphisms are related to several deep problems of algebraic geometry, such as the Jacobian Conjecture, the Nagata Conjecture, the Affine Cancellation Problem, and the Embedding Problem. One of the simplest and most important mathematical functions is a polynomial. A polynomial can be defined using any number of unknowns (variables). The branch of mathematics known as algebraic geometry studies the geometric objects naturally associated with polynomials: a polynomial in 2 variables defines a curve in the plane, a polynomial in 3 variables defines a surface in 3-dimensional space, and so on. These objects can be remarkabley complex, and one wishes to understand their symmetries, their intrinsic properties, how they intersect, etc. While aspects of algebraic geometry were already studied in classical Greece, it is a mainstream field of modern mathematics, both in theory and application. Two areas where recent research has led to fruitful applications are in high-speed computing (solving large polynomial systems) and in cryptography (fast and secure coding schemes).