Minicourse: Biological Applications of Computational Topology

Why take this course: It's an exciting time in science in mathematics as biology has been added to the list of areas where there are deep and interesting mathematical problems. Traditionally, analytic approaches have been the focus of most applied math, but today geometry and topology are making their entry. This course will illustrate a variety of ways that topological thinking and methods can be used to study very applied problems.

Specifically we will look at applications of computational topology to:

1) Characterization of Gene Expression
2) Plant root shape determination
3) Image segmentation

The course will begin with the definition of persistent homology and discuss how it can be used to measure global features in datasets and images. Then we'll study each of the areas above in detail, showing how topological methods can be used in each case to capture features on importance. Along the way we'll cover local homology, computational Morse theory, elevation and several other topological tools of use.

Prerequisites: Familiarity with beginning topology and multivariable calculus are the only clear prerequisites, as we will assume familiarity with homology, talk about gradient flows of functions, etc.