Classical forms of homogeneity in continuum theory

拓扑空间是一种可以弯曲和变形的形状。齐次拓扑空间是一个完全对称的空间，像一个圆或球，所以它的所有点在拓扑上是不可区分的。这是数学应用中模型的共同特征，例如在物理学中，研究对象生活在齐次流形中。许多同质空间就像一个“宇宙”，其中存在着大量的空间。例如，著名的门格尔立方体是一条齐次曲线（一维空间），它包含所有其他曲线的副本。由于这类宇宙在拓扑学及其应用上的实用性，寻找和分类齐次空间成为人们感兴趣的问题。本研究旨在提出一种新的齐次曲线分类方法。有一些迷人的曲线具有非常复杂的分形结构，称为“遗传不可分解”。遗传不可分解曲线理论的最新发展在齐次曲线的分类上取得了突破，即在二维平面上只包含三条齐次曲线：圆，称为“伪弧”的遗传不可分解曲线，以及这两者的混合称为“伪弧的圆”。本研究的下一步是对三维空间中的所有齐次曲线进行分类。这项研究将有助于这种分类，从遗传上不可分解的曲线开始。具体来说，我们打算确定是否还有其他类似伪弧的显著曲线尚未被发现。一类特殊的曲线，由树和广义树组成，被称为“树突”，在计算机科学和逻辑中经常出现。一棵完全对称的树（或树突）不可能是齐次的，因为它总是有不止一种类型的点：例如，它的端点在拓扑结构上与其他点不同。最近的研究表明，一种新的同质性概念，称为“同质度3”（意思是有3种不同类型的点），抓住了树状体拓扑对称的适当概念。特别是，均匀度为3的树状体往往具有与均匀空间相同的普适性。本研究将推进均匀度为3的树状体的分类，以期发掘新的通用树状体，并探索其应用前景。该研究项目将在尼皮辛大学创造最多3个博士后职位，这将加强加拿大拓扑学研究的活动和全球形象，并吸引更多的顶尖研究人员访问尼皮辛，加强我们社区的科学交流。该项目还将资助五个学生研究助理职位，为我们的学生提供高等数学培训和研究经验，为他们在学术界和工业界的富有成效的职业生涯做好准备。

A topological space is a shape which may be bent and deformed. A homogeneous topological space is one which is totally symmetric, like a circle or sphere, so that all of its points are topologically indistinguishable. This is a common feature of models in applications of mathematics, e.g. in physics, where objects under study live in a homogeneous manifold. Many homogeneous spaces serve as a "universe" in which a large group of spaces reside. For example, the famous Menger cube is a homogeneous curve (1-dimensional space) which contains a copy of every other curve. Due to the practicality of such universes in topology and its applications, it is of interest to seek out and classify homogeneous spaces. This research aims to contribute new classifications of homogeneous curves. There are some fascinating curves which have a very complex, fractal-like structure, called "hereditarily indecomposable". Recent developments in the theory of hereditarily indecomposable curves have produced a breakthrough in the classification of homogeneous curves, namely that there are only three homogeneous curves contained in a 2-dimensional plane: the circle, a hereditarily indecomposable curve called the "pseudo-arc", and a hybrid of these two called the "circle of pseudo-arcs". The next step in this study is to classify all homogeneous curves in 3-dimensional space. This research will contribute towards this classification, beginning with hereditarily indecomposable curves. Specifically, we intend to determine whether there are any other remarkable curves like the pseudo-arc which are yet to be discovered. A special class of curves, consisting of trees and generalized trees called "dendroids", features regularly in computer science and logic. A totally symmetric tree (or dendroid) cannot be homogeneous because it will always have more than one type of point: its endpoints are topologically different from its other points, for example. Recent work suggests that a newer notion of homogeneity, called "homogeneity degree 3" (meaning there are exactly 3 different types of points), captures an appropriate concept of topological symmetry for dendroids. In particular, the dendroids with homogeneity degree 3 tend to be universal in the same way that homogeneous spaces often are. This research will advance the classification of dendroids of homogeneity degree 3, with the goal of unearthing new universal dendroids, and exploring their applications. The proposed research project will create up to three postdoctoral fellow positions at Nipissing University, which will bolster the activity and global profile of Canadian topology research and attract more top researchers to visit Nipissing, enhancing the scientific exchange taking place in our community. The project will also fund five student research assistant positions, providing our students with advanced mathematics training and research experience, which will prepare them for productive careers in academia and industry.