The charm of the Čech complex

We finish the term with a small amount of snow and temperatures just below zero. In total, this has been the eighth term here. We have covered a wide range of interesting mathematics and I have come across fascinating new ideas. For example, I have become intrigued by the peculiar area that is topological data analysis. It uses topological techniques to study data sets and does this in a way which is robust to noise.

We start, for instance, with a set of points in a space. We want to introduce a structure on the data. The simplest way to do this is to construct the Čech complex, named after Eduard Čech, who was a Czech mathematician in the first half of the last century and focused on geometry and topology.

Given a fixed radius, we draw a ball of this radius around each point. These balls provide some meat for the data set as a geometric structure. To construct the Čech complex for this radius, we introduce a segment between two points, if the corresponding two balls intersect; a triangle, if three intersect, and so on. In fact, the Čech complex looks topologically the same as the union of the balls.

As the radius increases, certain features of the data set, for example cycles or rings, are realized and later they are filled in and assimilated. In other words, they are born and then they perish. The lifespan gives us a notion of their significance as a distinctive feature of the data set and it changes continuously under continuous change of the data. This area draws on theory which I learned last term as a part of one of my favourite courses. I find these applications surprising and astonishing and I hope to learn more about this area, for example over the summer.

So much for talking about mathematics. On a completely different note, I will be spending a part of my Easter break in Oxford and that means that I will have to cook for myself not only over the weekend but over the week as well. I am therefore interested in trying out new recipes in the kitchen.





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