Investigations in affine algebraic geometry

仿射代数几何研究

• 批准号: 194395-2011
• 负责人: Russell, Peter
• 金额: $ 0.95万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569776
• 关键词: Investigations; affine; algebraic; geometry

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.

在“代数”几何中，我们研究由多项式定义的对象（与 更简单，例如，线性的或更复杂的，例如可微分的）方程。在这方面 “仿射空间”，数学家日常生活中的线性空间，是我们所熟悉的 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，当研究它们自身时，它们很快就开始摆出令人惊讶的姿势 棘手的和基本上未解决的问题：哪些属性表征它们？是什么 它们的对称性有什么方法可以将其中一个嵌入另一个？这些都是一些 仿射代数几何的基本问题。 本建议中要研究的一些相互关联的问题是： 线性仿射空间上的群作用对非常接近仿射平面的曲面的研究 从它们具有非常简单的拓扑的意义上来说，例如，是可以收缩的研究 平面有理曲线及其在仿射平面上的嵌入。 这项研究是以前工作的延续，是一个大型的，生动的 这方面的国际活动。