Topology and Data
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Today there is an immense production of data, and the need for better methods to analyze data is ever increasing. Topology has many features and good ideas which seem favourable in analyzing certain datasets where statistics is starting to have problems. For example, we see this in datasets originating from microarray experiments. However, topological methods cannot be directly applied on finite point sets coming from such data, or atleast it will not say anything interesting. So, we have to modify the data sets in some way such that we can work on them with the topological machinery. This way of applying topology may be viewed as a kind of discrete version of topology. In this thesis we present some ways to construct simplicial complexes from a finite point cloud, in an attempt to model the underlying space. Together with simplicial homology and persistent homology and barcodes, we obtain a tool to uncover topological features in finite point clouds. This theory is tested with a Java software package called JPlex, which is an implementation of these ideas. Lastly, a method called Mapper is covered. This is also a method for creating simplicial complexes from a finite point cloud. However, Mapper is mostly used to create low dimensional simplicial complexes that can be easily visualized, and structures are then detected this way. An implementation of the Mapper method is also tested on a self made data set.