Computational Geometry and Topology

This is a very new and exciting area in mathematics which crosses over from the pure to the applied world --and hopefully makes an impact! Generally speaking, this area intends to adapt and develop techniques inspired by Geometry and Topology and turn them into computational tools that are useful to solve difficult problems related to Data Analysis. This is particularly relevant in the era of Big Data, where specialists strive to make sense of the huge amounts of data that are produced on a daily basis.

An interesting example of how this can be done is, for example, the software Mapper, which was developed by the Applied and Computational Algebraic Topology Group at Stanford.

The image below was created using Mapper on the MNIST database during my Math 462 class in the Spring of 2014.

Geometric Analysis

This area of mathematics is very wide and very deep! Problems in geometric analysis usually are written in the language of differential geometry, and use tools from nonlinear PDEs, topology, and so on, to make way. Results in this area concern the topology of black holes, minimal surfaces, Ricci flow, and many more (rather delicate) topics.