Algebraic Transformation Groups

代数变换群

• 批准号: RGPIN-2017-05405
• 负责人: Kuttler, Jochen
• 金额: $ 1.17万
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675706
• 关键词: Algebraic; Transformation; Groups

In a general sense, this program proposes to investigate symmetries of certain mathematical objects. ******Symmetries are at the heart of mathematical investigation. These can be symmetries of geometric objects (triangles, squares, but also significantly more intricate things, in mathematical language referred to as manifolds, or their algebraic generalizations, varieties). ***Such symmetries are usually described by structures called groups. This builds on the elementary but fundamental insight that if an object of whatever nature has two symmetries, then usually applying one of these symmetries after the other again results in a symmetry (for example, a square can be rotated by 90 degrees. Performing two such rotations consequently results in another symmetry, namely rotation by 180 degrees). This generalizes to much more complicated objects. ******The purpose of this program is, in a broad sense, to understand the symmetries of complicated algebraic varieties, and use these to describe their singularities. Here, a "singularity" in geometric terms is roughly speaking an "exception" in the general shape of the thing; for example, in the case of a triangle, the three corners would be considered singularities, as they look significantly different from the rest of the triangle. It is not easy to define precisely what "singularity" means without resorting to technical jargon, but understanding such singularities has widespread mathematical applications. ******The second part of the project deals with an abstract concept, called "derivations." Derivations (not all, but the most important ones) arise as approximations of the kind of symmetries alluded to above. For reasonable classes of symmetries, there exists a class of derivations that describes these symmetries "infinitesimally" (for examples a circle admits as symmetries arbitrary rotations; an infinitesimal approximation of such a rotation would the a (very "infinitesimally" short) translation along a straight tangent of the circle). To understand the original symmetries, one should first understand their approximations in the form of derivations. ******Understanding symmetries and related objects (Transformation Groups and Lie Algebras) has wide spread applications both within Mathematics, but also in Physics and other sciences. For example the theory of transformation groups can help in analysing phylogenetic trees (describing hereditary relationships between species) in Biology. **

在一般意义上，该计划旨在研究某些数学对象的对称性。** 对称性是数学研究的核心。这些可以是几何对象的对称性（三角形，正方形，但也可以是更复杂的东西，在数学语言中称为流形，或它们的代数推广，变种）。* 这种对称性通常用称为群的结构来描述。这是建立在一个基本但基本的见解上的，如果一个物体有两个对称性，那么通常在另一个对称性之后再次应用其中一个对称性会导致对称性（例如，正方形可以旋转90度）。执行两次这样的旋转因此导致另一种对称，即旋转180度）。这适用于更复杂的对象。** 这个程序的目的是，在广义上，理解复杂代数簇的对称性，并使用这些来描述它们的奇点。在这里，几何术语中的“奇点”粗略地说是事物一般形状中的“例外”;例如，在三角形的情况下，三个角将被认为是奇点，因为它们看起来与三角形的其余部分明显不同。要准确定义“奇点”的含义而不诉诸技术术语并不容易，但理解这种奇点具有广泛的数学应用。** 项目的第二部分涉及一个抽象的概念，称为“派生”。“导子（不是所有的，但最重要的）是作为上面提到的那类对称的近似而出现的。对于合理的对称类，存在一类“无穷小”地描述这些对称的导子（例如，一个圆允许任意旋转作为对称;这种旋转的无穷小近似是沿着圆的直切线的（非常“无穷小”短的）平移）。 要理解原始的对称性，首先应该理解它们以导子形式的近似。** 理解对称性和相关对象（变换群和李代数）在数学中有广泛的应用，但也在物理学和其他科学中。例如，转换群理论可以帮助分析生物学中的系统发育树（描述物种之间的遗传关系）。 **