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，就像圆或球面一样，使得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，所研究的对象生活在齐次流形中。*许多齐次空间充当居住着一大群空间的“宇宙”。例如，著名的门格立方体是一条齐次曲线(一维空间)，它包含每条其他曲线的副本。由于这类空间在拓扑学及其应用上的实用性，寻找和分类齐性空间是很有意义的。这项研究旨在为齐次曲线的新分类做出贡献。*有一些迷人的曲线，它们具有非常复杂的、类似于分形的结构，被称为“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，即在二维平面上只有三条齐次曲线：圆，称为“伪弧”的遗传不可分解曲线，以及这两者的混合体，称为“伪弧圆”。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。具体地说，我们打算确定是否还有其他像伪弧这样的值得注意的曲线尚未被发现。*一类特殊的曲线，由树和广义树组成，称为“树形”，经常出现在计算机科学和逻辑中。完全对称的树(或树形)不可能是齐次的，因为它总是有多种类型的点：例如，它的端点在拓扑上与它的其他点不同。最近的工作表明，一个更新的同质性概念，称为“同质性程度3”(意思是恰好有3种不同类型的点)，捕捉到了树突的适当的拓扑对称性概念。特别地，齐次度为3的树突往往是普适的，这与齐次空间通常是相同的。这项研究将推进同质性程度为3的树突的分类，目标是发现新的通用树突，并探索它们的应用。*拟议的研究项目将在尼皮辛大学创造最多三个博士后研究员职位，这将增强加拿大拓扑学研究的活跃度和全球知名度，并吸引更多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级数学培训和研究经验，这将为他们在学术界和工业界从事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.