Investigations in affine algebraic geometry

在“代数”几何中，我们研究由多项式定义的对象(与 更简单的，例如线性的，或更复杂的，例如可微的)方程。在此背景下 “仿射空间”，数学家日常生活中的线性空间，是人们所熟悉的。 主要是作为我们真正感兴趣的物体的友好庇护所，三维空间中的球体， 然而，当为它们自己而研究时，它们很快就开始摆出令人惊讶的姿势 棘手且在很大程度上尚未解决的问题：它们具有哪些特征？何谓 他们的对称性？有什么方法可以将其中一个嵌入到另一个中？这些是一些 关于“仿射代数几何”的基本问题。 这项建议中要研究的一些相关问题是：在多大程度上 仿射空间上的群作用线性？非常接近仿射平面的曲面的研究 在某种意义上，它们具有非常简单的拓扑，例如，是可收缩的。的研究。 平面有理曲线及其在仿射平面中的嵌入。 这项研究是对以前工作的延续，是一个大型、生动的 这一领域的国际活动。

In "algebraic" geometry we study objects defined by polynomial (as opposed to simpler, e.g., linear, or more complicated, e.g. differentiable) equations. In this context "affine spaces", the linear spaces in the every day life of a mathematician, are familiar mainly as friendly shelters for objects that really interest us, a sphere in 3-space, for instance.Yet, when studied for their own sake, they soon begin to pose surprisingly thorny and largely unresolved questions: Which properties characterize them? What are their symmetries? What are the ways to embed one of them in another? These are some of the basic questions of "affine algebraic geometry". Some of the interrelated problems to be studied in this proposal are: To what extent are group actions on affine spaces linear? The study of surfaces very close to the affine plane in the sense that they have a very simple topology, e.g., are contractible. The study of plane rational curves and their embeddings in the affine plane. This research is a continuation of previous work and forms part of a large, lively international activity in this area.