I am currently a PhD Student in the Applied Mathematics Program at the University of Arizona. My research interests are in the applications of topology to real-word problems. My passions outside of mathematics include Music and Web-Development/Design.
Analyzing Scalar Fields through Topological Similarity
I am currently being advised by Professor Joshua Levine of the Computer Science department on a project intended to study large-scale, multifaceted data sets using topological methods. The goal is to examine existing tools that are used to measure distance between topological structures (bottleneck distance for Persistence Diagrams and interleaving distance of Reeb Graphs, for example) with the intent to apply these metrics or modify them in a way that they become useful to the study of these data sets.
Currently, we are interested in creating an emperical study in which we create a comprehensive review of all the metrics that are currently being used for comparison between topological structures. The goal of this project is to develop concrete evidence where certain metrics provide more useful information than others. Furthermore, we would like to understand all of the computational complexities that are found in each metric. For example, the interleaving distance on Reeb graphs has a very strong, mathematical foundation. However, the process of computing this metric is known to be NP-hard.
In addition to this line of work, we'd like to investigate the possibility of leveraging machine learning to (1) create similarity metrics through supervised training, (2) create similarity metrics through unsupervised training, and (3) aid in the computation of computationally complex metric such as interleaving distance and functional distortion distance on Reeb graphs.
MuView is an in-development visualization tool to aid in the exploratory analysis of music review data. It pulls review data from multiple music publication websites to create a dynamic, interactive experience for users to navigate the multitude of reviews.
Many current review aggregation sites have no way to make complex queries on the data (such as unions and intersections of sets). Even finding simple lists of the top rated albums by specific publications are difficult. This is an attempt to allow users to navigate the lanscape of review data in a more versatile way. The features include choosing your own publications to pull data from, sort by ratings, peform unions and intersections of multiple sets (groups) of albums, view year-end ranks for each album, and more
MuView is built in React.js and utilizes D3.js for dynamic data interatcions. There will (eventually) be a live-version of this available to the public. At that time, I will most likely almost make the source code available.