Algebraic Transformation Groups

在数学中，一个物体或结构的对称性是通过一组保持它的对称变换来体现的。在基本的层面上，这是从旋转、平移和它们的组合开始的，这些都保留了墙纸的图案。在更高级的层面上，这导致了化学中晶体的对称群，或者生物学中蛋白质结构的对称群，或者例如狭义相对论的要求，即运动定律在洛伦兹变换下必须不变。

In mathematics, the symmetries of an object or structure is captured by the group of symmetry transformations that preserve it. At an elementary level this starts with the rotations, translations, and combinations thereof, which preserve a wallpaper pattern. On a more advanced level this leads to the symmetry groups of crystals in chemistry, or of protein structures in biology, or for instance the requirements of special relativity that the laws of motion must be invariant under Lorentz transformations. In short, transformation groups (that is, very literally, groups of transformation that "act" on a geometric object) appear everywhere, and their understanding is crucial for many areas of mathematics and science in general. In this project, we study several aspects of this theory. In many applications one needs to understand sets of solutions of a system of polynomial equations, so called varieties. The most fundamental variety is arguably projective space which has in particular the property that it looks everywhere the same (for any two points there is a symmetry mapping one to the other). Homogeneous spaces are natural generalizations of this concept. Here we study varieties that are important at getting a better understanding of these. A second focus point is to study the rank of tensors. Loosely speaking, the rank of a tensor is a measure for its complexity and usually very hard to determine, but nevertheless important in actual applications, which range from algebraic statistics (in particular phylogenetics) to complexity theory. A final part concerns itself with invariant theory, a classical topic that has been important to the development of modern algebra.