My research interests are primarily in geometric and topological data analysis, as well as computational harmonic analysis. I am interested in consistency problems, namely the convergence of graphs and graph Laplacians to an underlying continuum domain and Laplacian, and their application to:
vasculature growth and morphology,
gerrymandering and redistricting, and
sampling and triangulation of manifolds.
A list of my papers and pre-prints can be found here.
Topological data analysis
I am interested in using TDA to 1) quantify the topological structure of a data object, and then 2) compare the structures of different data objects. One such project, joint with J.S. Marron (UNC) and T. Hamelryck (Copenhhagen), has been to study the structure of proteins and "protein space". The geometry we're imposing on "protein space" is via the Wasserstein metric on protein barcodes. I am also using TDA in a project with B. Walker (UNC) to detect and track looping behaviour in Hi-C chromosome data. In a sense we are doing statistics on (1+1)-parameter persistence modules (1 dimension for time, 1 dimension for a VR filtration) to compare and analyze simulated Hi-C data dynamics.
I am working on problems involving graph Laplacians (plus other operators) with my advisor J. Marzuola (UNC). In particular, we are looking at convergence properties of graph operators as the graph converges to (in an appropriate sense) a continuous domain; one natural setting is a geometric graph sampled from a domain in R^n. A recent project, joint with H.T. Wu (Duke) and J. Marzuola, examines convergence properties of the spectral flow studied by J. Marzuola, G. Cox (Memorial), and G. Berkolaiko (TAMU) and applications of this spectral flow to analyzing data.
I am working on aspects of deep learning for image segmentation, joint with researchers at LifeOmic and J. Toledo, J. Glazier (Indiana University). In particular I am working to produce synthetic retinal vasculature, with the goal of improving existing deep learning-based image segmentation techniques to be used in clinical applications. I have used two kinds of models, 1) from plant biology, and 2) inspired by branched optimal transport, to generate synthetic fundus images. I'm also working to quantify and do statistics on these images; the two approaches I have implemented include diffusion-based and topological descriptor-based representations, which can be vectorized and statistically analyzed.
Bayesian Models for Disease Progression
For my i4 project, I worked alongside digital health professionals (DHIT), machine learning experts (Tanjo.ai), and statisticians (Analytical Partners Consulting LLC) to develop Commerce with Confidence indeces to aid in the decision making process for NC communities planning their economic recovery during COVID. In particular, I am adapting Bayesian approaches to estimating effective reproductive numbers for NC counties; the model and code are adapted from rt.live/ .