Algebraic Transformation Groups

在数学中，一个物体或结构的对称性是由保持它的一组对称变换来捕捉的。在初级水平上，这从旋转、平移和它们的组合开始，它们保留了墙纸图案。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是不变的。 简而言之，变换群(也就是，从字面上讲，就是“作用于”几何对象的变换群)随处可见，它们的理解对于数学和科学的许多领域都是至关重要的。 在这个项目中，我们对这一理论的几个方面进行了研究。在许多应用中，人们需要了解多项式方程组的解的集合，即所谓的变种。可以说，最基本的变化是射影空间，它有一个特别的性质，即它看起来到处都是一样的(对于任何两个点，都有一个对称映射从一个到另一个)。齐性空间是这一概念的自然推广。在这里，我们研究对更好地了解这些品种很重要的品种。第二个焦点是研究张量的等级。不严格地说，张量的等级是对其复杂性的衡量，通常很难确定，但在从代数统计学(特别是系统发育学)到复杂性理论的实际应用中仍然很重要。最后一部分涉及不变理论，这是一个对现代代数发展很重要的经典话题。

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.