Grant: Discovery Grants Program - Individual

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.