At the University of Arizona, I worked in the field of Topological Data Analysis under the advisement of Professor Josh Levine. Our published work was ondesigning a discriminative, stable distance on merge trees.
Topological Data analysis (TDA) is the application of the pure mathematics field of topology to data analysis. The goal is to understand how the shape and structure of a dataset can uncover other information that cannot be captured by standard analyses. One of the main useful properties of TDA is that of "persistence simplification". TDA provides us a way to "rank" features dependent on their size. Once these are ranked, features can be removed dependent on this rank -- drastically simplifying the data structure. A common pipeline is to start by taking some two or three dimensional dataset and computing a derived topological structures, such as a Reeb graph or a Merge tree. From here, simplification may be used to make the graph into a less complex structure. Then, we can use these simplified structures for exploratory analysis or for comparison to other data sets.
My focus was specifically on how we compare the Reeb graphs or merge trees to one another. Since these are true graphs, the well-defined metrics that we construct on these graphs are often NP-hard. The published work was specifically understanding what the usefulness of these metrics are in the case of merge trees when compared to other, less complex distances. We employed an A* search algorithm to find the best matching between possible branch decomposition trees: topological structures derived from the original merge tree.