Topology and Complexity

Topology, the “rubber sheet geometry,” studies the properties of objects that do not change when they are pulled and stretched. Accepting somewhat fuzzy input, it is the part of mathematics that is typically applied when qualitative conclusions are reached. However, it has a — fascinating and not very well understood — quantitative aspect that is important in understanding singularities, and potentially, high-dimensional noisy data as well as aspects of large-scale geometry of networks. The talk will be a series of vignettes that display a number of different phenomena that arise or are illuminated when one keeps track of the complexity of geometric constructions.

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