Another Word For It

Persistent Homology Talk at UIC: Slides by Jeremy Kun.

From the post:

Today I gave a twenty-minute talk at UI Chicago as part of the first annual Chicago Area Student SIAM Conference. My talk was titled “Recent Developments in Persistent Homology,” and it foreshadows the theoretical foundations and computational implementations we’ll be laying out on this blog in the coming months. Here’s the abstract:

Persistent homology is a recently developed technique for analyzing the topology of data sets. We will give a rough overview of the technique and sample successful applications to areas such as natural image analysis & texture classification, breast and liver cancer classification, molecular dynamical systems, and others.

The talk was received very well — mostly, I believe, because I waved my hands on the theoretical aspects and spent most of my time talking about the applications.

The slides.

Jeremy doesn’t hold out much hope the slides will be useful sans the lecture surrounding them.

He does includes references so see what you think of the slides + references.

Homology Theory — A Primer by Jeremy Kun.

From the post:

This series on topology has been long and hard, but we’re are quickly approaching the topics where we can actually write programs. For this and the next post on homology, the most important background we will need is a solid foundation in linear algebra, specifically in row-reducing matrices (and the interpretation of row-reduction as a change of basis of a linear operator).

Last time we engaged in a whirlwind tour of the fundamental group and homotopy theory. And we mean “whirlwind” as it sounds; it was all over the place in terms of organization. The most important fact that one should take away from that discussion is the idea that we can compute, algebraically, some qualitative features about a topological space related to “n-dimensional holes.” For one-dimensional things, a hole would look like a circle, and for two dimensional things, it would look like a hollow sphere, etc. More importantly, we saw that this algebraic data, which we called the fundamental group, is a topological invariant. That is, if two topological spaces have different fundamental groups, then they are “fundamentally” different under the topological lens (they are not homeomorphic, and not even homotopy equivalent).

Unfortunately the main difficulty of homotopy theory (and part of what makes it so interesting) is that these “holes” interact with each other in elusive and convoluted ways, and the algebra reflects it almost too well. Part of the problem with the fundamental group is that it deftly eludes our domain of interest: we don’t know a general method to compute the damn things!

Jeremy continues his series on topology and promises programs are not far ahead!