Graph-Encoded Manifolds

A manifold is a space which locally resembles Euclidean space at each point. Surfaces (two-dimensional manifolds), such as the two-dimensional sphere, can be studied via embedding them in three-dimensional space, but this is a much more challenging task for manifolds of higher dimension. However, any manifold can be represented by a graph with coloured edges, such that the graph and its edge colourings encode all of the information about the manifold. These graph-encoded manifolds (gems) can always be drawn in two dimensions, and can be analysed using purely combinatorial techniques to uncover information about the original manifold. For example, the question of whether a manifold is orientable can be answered by determining whether the corresponding gem is a bipartite graph.

However, many general properties of gems are as yet unknown. This project aims to better understand the nature of gems, by finding large families of gems representing certain manifolds, such as the 3-sphere. To achieve this, techniques from combinatorics and algorithm design will be used.

Steven Condell

The University of Sydney

Steven is a third-year student studying mathematics and computer science at the University of Sydney. He has always been passionate about mathematics and its ability to affect so much of the world around us, and through university has also realised the power of computer programming. Steven is particularly interested in the ways in which mathematics and computer science can intersect to allow for new insights into problems. Some of his favourite courses at university so far have been those on number theory and algorithms. Outside of uni, Steven enjoys listening to music, solving puzzles, and spending time with friends and family.

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