Algebraic Transformation Groups

在数学中，物体或结构的对称性是由保持它的对称性变换组来捕获的。在初级水平上，这从旋转，平移及其组合开始，它们保持墙纸图案。在更高的层次上，这导致了化学中晶体的对称群，或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（也就是字面上的“作用于”几何对象的变换群）到处都是，对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，我们研究了这个理论的几个方面。在许多应用中，人们需要了解多项式方程组的解的集合，所谓的簇。 最基本的变体可以说是射影空间，它特别具有看起来处处相同的性质（对于任何两个点，都存在一个到另一个的对称映射）。齐性空间是这个概念的自然推广。在这里，我们研究的品种是重要的，在获得更好的理解这些。第二个焦点是研究张量的秩。不严格地说，张量的秩是其复杂性的度量，通常很难确定，但在实际应用中很重要，从代数统计（特别是遗传学）到复杂性理论。最后一部分关注本身与不变的理论，一个经典的主题，一直是重要的发展现代代数。

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.