Algebraic Transformation Groups

在一般意义上，这个程序建议研究某些数学对象的对称性。*对称性是数学研究的核心。它们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，或者它们的代数推广，变种)。*这种对称性通常由称为群的结构描述。这建立在一个基本但基本的洞察力之上，即如果任何性质的对象都有两个对称性，那么通常将其中一个对称性应用于另一个对称性会再次产生对称性(例如，一个正方形可以旋转90度。因此，执行两次这样的旋转会导致另一种对称，即旋转180度)。这适用于更复杂的对象。*本程序的目的是在广义上理解复杂代数簇的对称性，并用这些对称性来描述它们的奇点。在这里，几何术语中的“奇点”粗略地说是事物一般形状中的一个“例外”；例如，在一个三角形的情况下，三个角将被认为是奇点，因为它们看起来与三角形的其他部分有显著的不同。不借助专业术语就很难准确定义“奇点”是什么意思，但理解这种奇点在数学上有着广泛的应用。*项目的第二部分涉及一个抽象的概念，称为“派生”。导数(不是所有的，但最重要的)是作为上面提到的那种对称的近似出现的。对于合理的对称类，存在一类导子，它“无限小”地描述这些对称(例如，一个圆被认为是对称的任意旋转；这种旋转的无穷小近似将沿着该圆的一条直线平移(非常“无限小”地短))。要理解最初的对称性，首先应该理解它们以导数的形式进行的近似。*了解对称及其相关对象(变换群和李代数)在数学中有着广泛的应用，在物理学和其他科学中也有广泛的应用。例如，转化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)。**

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. **