Algebraic Groups and Galois Geometries

我在两个不同的数学领域进行研究。 不变量理论是关于对称性的研究。一个对象所具有的对称性的集合可以被集合在一起成为一个数学对象，称为群。物体的性质可以从它的对称群的结构中推导出来。不变量理论最早的来源之一是试图理解透视。不变量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。 我也研究伽罗瓦几何。这是一门研究空间中的几何学的学科，空间中只包含1000个点。我特别关心的是帽子。这些是没有三个点位于同一条线上的点集。 大写字母等同于某些代码。这些代码用于对信息进行编码，使信息传输中的错误最小化。 实际上，这些紧凑的形式提供了一种用于校正可能发生的任何传输错误的方法。不变量理论也可以应用于编码理论。

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.