Algebraic Groups and Galois Geometries

代数群和伽罗瓦几何

• 批准号: 137522-2012
• 负责人: Wehlau, David
• 金额: $ 1.82万
• 依托单位: Royal Military College of Canada
• 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=569292
• 关键词: Algebraic; Groups; Galois; Geometries

I pursue research in two separate areas of mathematics. Invariant theory is concerned with the study of symmetry. The collection of symmetries that an object possesses can be collected together into a mathematical object known as a group. Properties of the object can be deduced from the structure of its group of symmetries. One of the earliest sources of invariant theory was attempts to understand perspective. Invariant theory has many modern applications, including applications to computer vision, satellite and outer space navigation, fingerprint identificaton and materials sciences. I also study Galois geometries. This is the study of geometry in spaces which contain only finitely many points. I am particularly concerned with caps. These are sets of points for which no three lie on the same line. Caps are equivalent to certain codes. These codes are used to encode information compactly and in a form which minimizes the errors in transmission. Indeed these compact forms provide a method for correcting any transmission errors that might occur. Invariant theory also has applications to coding theory.

我在两个不同的数学领域进行研究。 不变理论涉及对称性的研究。一个对象所具有的对称性集合可以被集中到一个称为群的数学对象中。物体的性质可以从其对称群的结构中推导出来。不变理论最早的来源之一是尝试理解透视。不变理论有许多现代应用，包括计算机视觉、卫星和外层空间导航、指纹识别和材料科学。 我还研究伽罗瓦几何。这是对仅包含有限多个点的空间中的几何的研究。我特别关心帽子。这些点的集合中没有三个点位于同一条线上。 Caps 相当于某些代码。这些代码用于以最小化传输错误的形式对信息进行紧凑编码。 事实上，这些紧凑的形式提供了一种纠正可能发生的任何传输错误的方法。不变理论也适用于编码理论。