Algebraic Automorphisms of Affine Space

研究者研究一般的仿射群及其相关的几何学（仿射代数几何学）。 这些群是域上仿射空间的代数自同构。 群作用及其相关的不变量环在这项研究中发挥着核心作用。特别感兴趣的是基础域的加法群的作用，或者等价地，多项式环的局部幂零导子。 这些自同构与代数几何的几个深层次问题有关，如雅可比猜想、Nagata猜想、仿射消去问题和嵌入问题。最简单和最重要的数学函数之一是多项式。 一个多项式可以用任意数量的未知数（变量）来定义。 数学的分支被称为代数几何研究的几何对象自然与多项式：2个变量的多项式定义了平面中的曲线，3个变量的多项式定义了三维空间中的曲面，等等。这些对象可以是非常复杂的，人们希望了解它们的对称性，它们的内在属性，它们如何相交，虽然代数几何的各个方面在古希腊已经被研究过了，但它在理论和应用上都是现代数学的主流领域。 最近的研究已经导致富有成效的应用的两个领域是在高速计算（解决大型多项式系统）和密码学（快速和安全的编码方案）。

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).