On Cherlin's conjecture for finite binary primitive permutation groups

本研究关注数学对象的“局部对称性”和“全局对称性”之间的联系。为了理解这意味着什么，让我们考虑一个简单的数学对象的对称性：一个正六边形。如果我随机选择这个六边形的两条不同的边，那么很明显它们“看起来是一样的”——这只是因为六边形是正形的，所以所有的边都有相同的长度。这是一个“局部对称”的例子——这是一个数学术语，指的是一个数学对象的两个部分看起来一样。另一方面，因为六边形是规则的，所以有许多“全局对称性”——这些是我的对象的转换，所以在转换之后它仍然“看起来一样”。例如，在正六边形的情况下，我可以通过一条连接两个相对角的线来反射六边形，得到的物体将与我开始时的物体相同。同样地，我可以把六边形旋转60度的任意倍数同样成立——这些都是全局对称的例子。数学家们在研究任何给定的数学对象时自然想知道的一个问题是“局部对称性的存在何时意味着全局对称性的存在？”例如，在上面的例子中，很明显，给定六边形上的任意一对边，我可以找到将第一条边移动到第二条边的全局对称（例如旋转）。因此我们可以说，这里的局部对称只是全局对称的结果。事实上，这是相当不寻常的：大多数数学对象将有许多局部对称性，而不是一些全局对称性的结果。所有的局部对称性都来源于全局对称性的数学对象是非常特殊的，它被称为齐次。本项目研究的是同质关系结构。关系结构只是网络的一种特殊泛化：取一堆“节点”并将它们与“边”连接起来，这样你就构成了一个网络（想象一下由道路连接的城市，或者在现实生活中通过电线连接的计算机）。事实上，六边形可以看作是一个网络——每个角都可以看作是一个节点（有6个这样的节点），然后有6条边连接这些节点。我们想知道哪些网络是同质的。为了充分理解这个问题，我们需要对如何定义网络的“对称性”概念稍微小心一点，这里没有时间做这个。作为一个提示，让我们提一下，六边形给出的网络不是齐次的，而五边形给出的网络是齐次的！最后，让我们说一下我们的方法：当一个人研究数学中的对称性时，他实际上是在研究群论。任何数学对象的对称集合（比如正六边形的反射和旋转）被称为与该对象相关的群。人们可以“抽象地”研究这个群体，即不需要真正研究与之相关的对象。例如，如果我做一个特定的反射，然后对我的六边形做一个特定的旋转，我最终会得到一个新的六边形对称（事实上，它将是另一个反射），我可以把这看作是我的对称的一种“乘法”：我把两个对称“相乘”在一起，结果是第三个对称。为了完整地描述我的六边形的对称群，我只需要写下所有对称对的“乘法表”。人们对不同群体的结构了解甚多。事实上，最著名和最重要的数学定理之一被称为有限单群的分类，它描述了一类重要群的结构。在本研究中，我们将使用这个定理来研究齐次关系结构；我们的目标是对这些对象的一个重要子类进行分类。

This research concerns the connection between the "local symmetry" and "global symmetry" of a mathematical object. To understand what this means let us consider the symmetries of a simple mathematical object: a regular hexagon. If I randomly pick two different sides of this hexagon, then it is clear that they "look the same" -- this just follows from the fact that the hexagon is regular and so all sides have the same length. This is an example of a "local symmetry" -- that's the mathematical terminology for a situation where two portions of a mathematical object that look the same. On the other hand, because the hexagon is regular there are many "global symmetries" -- these are transformations of my object so that after the transformation it still "looks the same". In the case of the regular hexagon, for instance, I can reflect the hexagon through a line connecting two opposite corners, and the resulting object will be the same as the one I started with. Likewise, I can rotate my hexagon by any multiple of 60 degrees and the same will be true -- these are all examples of global symmetries.Now a natural question that mathematicians want to know when they study any given mathematical object is "when does the presence of a local symmetry imply the presence of a global symmetry?". For instance in the example above, it is clear that given any pair of edges on my hexagon, I can find a global symmetry (a rotation, for example) that moves the first edge to the second edge. Thus we could say that the local symmetry here is just a consequence of the global symmetry. In fact this is rather unusual: most mathematical objects will have many local symmetries that are NOT consequences of some global symmetry. A mathematical object for which all local symmetries derive from global symmetries is very special, and is called HOMOGENEOUS.The research in this project concerns homogeneous RELATIONAL STRUCTURES. A relational structure is just a particular generalization of a network: take a bunch of "nodes" and connect them with "edges" and you have made yourself a network (think of cities connected by roads, or computers connected by wires for real-life examples). Indeed the hexagon can be thought of as a network -- each corner can be thought of as a node (there are 6 of these), and then there are 6 edges connecting the nodes. We would like to know which networks are homogeneous. To fully understand this question, one needs to be a bit careful about how we define the notion of "symmetry" for a network and there is not time to do this here. As a teaser, though, let us mention that the network given by a hexagon is NOT homogeneous, whereas the network given by a pentagon IS!Finally, let us say a word about our methods: whenever one studies symmetry in mathematics, one is effectively doing GROUP THEORY. The set of symmetries of any mathematical object (say the reflections and rotations of our regular hexagon), is called the GROUP associated to the object. One can study this group "in the abstract", i.e. without really needing to study the object it is associated with. For instance, if I perform a particular reflection and then a particular rotation of my hexagon, I will end up with a new symmetry of the hexagon (in fact it will be another of the reflections) and I can think of this as a type of "multiplication" of my symmetries: I've "multiplied" two symmetries together and the result is a third symmetry. To fully describe the symmetry group of my hexagon I just need to write down the "multiplication table" of all pairs of symmetries.A great deal is known about the structure of different groups. Indeed one of the most famous and important mathematical theorems is called THE CLASSIFICATION OF FINITE SIMPLE GROUPS and it describes the structure of an important class of groups. In this research we will use this theorem to study homogeneous relational structures; our aim is to classify an important subclass of these objects.