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Nitya Nigam If you keep up with the latest developments in mathematical research, you may have heard the term “topology” being thrown around, but might not know exactly what it is. Topology is a branch of maths that studies the properties of spaces that stay the same after continuous deformation. In topology, objects can be stretched and squeezed like rubber, but they cannot be broken, so it is sometimes called “rubber-sheet geometry”. Under these rules, a triangle can be deformed into a circle, but the number 8 cannot, as it has two holes in it. So, circles and triangles are topologically equivalent, but are distinct from figure 8s. In honour of Maryam Mirzakhani’s birthday (the first woman to win a Field’s Medal, awarded to her for her groundbreaking research in topology), I thought I would write an article about one of current mathematics’ most active research fields. Links to relevant external resources are provided throughout the article in case you would like to extend your knowledge. You may wonder why topology is relevant. It has only emerged as a distinct mathematical field relatively recently; most topological research has been done after 1900. However, graph theory, which studies the properties of spaces built up from networks of vertices, edges and faces, is a form of topology. Spaces in graph theory are considered identical if all the vertices are connected up in the same way, regardless of their layout; graphs which are structurally identical but are laid out differently are called isomorphic, and are topologically equivalent. Graph theory has a wide range of applications in computer science, such as modelling computer networks, but it can also be used to optimise road networks and analyse linguistic trends. Nonetheless, the modern topological research goes far beyond graph theory, and has applications in branches of physics like vector fields and string theory. One main subfield of topology is point set topology. It analyses the local properties of spaces, and is closely related to calculus. It generalises the concept of continuity from calculus to define topological spaces, so the limits of sequences can be considered. If distances can be defined in these spaces, they are called metric spaces. In some cases, the distance cannot be defined - if a space maintains its continuity after a deformation is applied, it is still fundamentally the same space, but its “size” will have changed, so the concept of distance makes no sense. Another area of topology is algebraic topology, which instead considers the global properties of spaces. It answers topological questions by converting topological spaces into algebraic objects such as groups and rings. For example, topological spaces like the torus and the Klein bottle, pictured below, can be distinguished from each other because they have different homology groups (an algebraic concept based on integrating surfaces). Similarly, other algebraic concepts can be used to analyse and research topological spaces. The final area of topology discussed here is differential topology, which studies spaces with some kind of smoothness associated with each point. In this field, the triangle and circle would not be equivalent to each other in terms of smoothness - the triangle has hard corners, whereas the circle has a continuously curved edge. Differential topology is particularly relevant in vector field physics, and is therefore used to study things like magnetic and electric fields. It is also helpful in describing the 4-dimensional space-time structure of our universe.
This is just an overview of the immensely complex and constantly evolving field of topology. If this article sparked your interest, you can follow the latest mathematical research explained simply at ScienceDaily. Let me know in the comments what parts of topology you find particularly intriguing, and what you’d like to read about next!
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