Description
When seeing the picture above, you probably immediately recognize a doughtnut (or in the mathematical lingo, a two-dimensional torus) rotating in three dimensional space. Your brain has "filled in the gaps", and associated a familiar topological shape to the cloud of points. If we could teach a computer to find such topological features of cloud points this would be very valuable, since computers have no trouble repeating the calculations in much higher dimensional spaces, where human visualization and pattern recognition capabilities fail.
One important approach to this problem, that you will explore during this project, is persistent homology. This technique has a very wide range of applicability, including such disparate fields as the evolution of viruses, the spread of diseases in the London Transit Network, fractal structures, cosmology and even string theory.