Mathematics and Statistics
Topological Manifolds are abstract spaces that locally resemble Euclidean space. For example, consider a round globe and a flat map. The map is a 2-dimensional representation of a 3-dimensional space. Given any point on the globe we can find a corresponding position on the map, and vice versa. This correspondence is called a chart. With a sufficient number of charts, we can describe the whole space. Such a collection of charts is called an Atlas. It is possible to construct different Atlases for the same space, allowing us to move from one chart, to the space, to another chart. This process is called a transition map. The areas of focus for this project include several examples of manifolds such as curves, n-spheres, and the torus. We explore and illustrate different approaches to charts on these manifolds, the properties of a manifold, examples of spaces that fail to meet these requirements, and the derivation of transition maps.
Wilson, Grant, "Topological Manifolds" (2015). Thinking Matters Symposium Archive. 39.