Polar Varieties and Multiple Point Spaces

This project concerns Singularity Theory - a branch of mathematics invented by Isaac Newton.In some sense the purpose of this research is simple. I wish to take a collection of objects and decide whether the objects are alike. An everyday example could be that the collection of objects is a group of school kids and I will say that the kids are alike if they are the same age.In a more mathematical setting I am interested in collections of curves, (but I can also work with really weird objects that live in higher dimensions). I want to take some measurements from each curve in the collection and from those measurements decide whether all the curves are alike. In other words I am looking for what are the important defining features.In the example of the group of school kids there is an obvious measurement to make: ask each one their age!I can take many measurements from my curves but what I really want is to make the smallest possible number of measurements for each curve. After all, there is no point in doing extra work!The techniques developed in this project can later be used in other subjects such as computer vision and robotics.