Algebraic Transformation Groups

代数变换群

• 批准号: 341744-2012
• 负责人: Kuttler, Jochen
• 金额: $ 1.09万
• 依托单位: University of Alberta
• 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=572353
• 关键词: Algebraic; Transformation; Groups

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.

在数学中，物体或结构的对称性是通过保留它的一组对称变换来捕获的。在初级层面上，这从旋转、平移及其组合开始，从而保留壁纸图案。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保持不变。 简而言之，变换群（即，从字面上看，“作用”在几何对象上的变换群）无处不在，并且它们的理解对于数学和科学的许多领域至关重要。 在这个项目中，我们研究了该理论的几个方面。在许多应用中，我们需要理解多项式方程组的解集，即所谓的簇。 最基本的变化可以说是射影空间，它特别具有这样的属性：它看起来到处都相同（对于任意两点，存在彼此映射的对称性）。均质空间是这个概念的自然概括。在这里，我们研究对于更好地了解这些非常重要的品种。第二个重点是研究张量的秩。宽松地说，张量的秩是对其复杂性的度量，通常很难确定，但在实际应用中很重要，从代数统计（特别是系统发育学）到复杂性理论。最后一部分涉及不变理论，这是一个对现代代数发展非常重要的经典主题。