Goodwillie Calculus and Applied Topology

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My proposal is concerned with persistent homology and Goodwillie calculus.******Imagine we have a bunch of points S in space. We can think of them as samples from a data set. We can try to understand them by computing traditional statistical quantities like their mean (or average). However if the points were all selected from a circle, how can we see the circle? This type of question is what persistent homology was designed to answer. ******The points themselves are just discrete dots, completely disconnected. We can slowly make them bigger, turning them into spherical blobs like water droplets. At first they will all stay separate, but as we make them bigger they will start to join together. If the points are arranged nicely enough, since we started with points from a circle, a (thick) circle will eventually appear. If we keep making the blobs bigger eventually the center of the circle will be filled in and we end up with just one big blob. ******Persistent homology can be used to understand the way this circle appears and then disappears. In particular if a circle appears when each blob around a point is size and disappears when each blob is size , then we get a “bar” starting at and ending at . The length of the bar says how long the circle lasts as the blobs increase in radius. Doing this for a general data set S can give rise to many different bars. Together these are called the bar code of the data set S, and tell us about the geometry of the data set. Bars that persist for longer (ie if - is large) represent more significant geometric features. We can also use this method to understand higher dimensional geometric features of data sets.******This way of looking at data has been around for over 20 years, and has been undergoing continuous development. We propose to study some specific aspects of persistent homology. First we will study metrics (or distances) between bar codes. This can be used to compare the geometry of two data sets. The points in S can also be changing in time, and this gives two parameter persistence. The structures that arise are much more complicated and we will also study them. ******The Goodwillie calculus part of the proposal is also concerned with geometry. If we consider say the possible positions of a robot arm, the possible positions of the balls on a pool table, or the possible positions of all the stars in the galaxy, we get something called a manifold. We try to understand the manifold by applying certain constructions to it (called functors) and seeing what we get. The problem is that the functors are often complicated, and so we resolve them into their so called Taylor tower. This is analogous to taking a function in calculus and replacing it by its Taylor polynomials. A general function can be very complicated, but polynomials are much easier to understand. We are mostly interested in what possible towers can occur, in other words what are the polynomials in this context.**