My research interests broadly include applied and computational topology, topological data analysis (TDA), dynamical systems, network science, and opinion dynamics. Below are a few example of projects/topics I'm currently interested in.
Inspired by the work of Hickok et al. 2022 studying the spatial coverage of polling sites, we are working to extend the method to other resources such as parks. While all voting sites are (theoretically) the same, with resources like parks there is also a notion of quality. Using persistent homology, we identify holes in coverage of resources with heterogeneous quality. See a preprint here. This is joint work with Gillian Grindstaff, Abigail Hickok, Jiajie Luo, and Mason A. Porter.
Erin O'Neil (UCLA '24), a Queen's Road Foundation Undergraduate Fellow at UCLA, and I studied the coverage of cooling centers in different cities using persistent homology. With rising temperatures, it is important to identify regions where residents are most at risk for heat-based mortality. This work constituted Erin's senior research project. You can find this work in the SIAM Journal on Applied Mathematics here.
During Summer 2024, Mason Porter, Sidhanth Raman (UCI), and I co-mentored five undergraduate students (Makenna Greenwalt, Kelvin Luu, Zachary Ojakli, Amy Somers, and Ahoora Tamizifar) at the UCLA Geometry+Topology REU. In one project, the students analyzed patterns in faculty hiring in mathematics using persistent homology of networks. Using data from the Mathematics Genealogy Project, they were able to identify different hiring patterns when separating the data by gender. Preliminary findings can be found in their final report here.
Individual's opinions can change based on who they surround themselves with. We can model a very basic version of this phenomena as a network of agents (represented as nodes) and each agent has a continuous-valued opinion in [0,1]. Agents then update their opinions based on the opinions of their neighbors. Bounded-confidence models (BCMs) are a specific class of models where agents only change their opinion if their neighbor's opinion is within a "confidence bound" of their own. Questions often asked in this field relate to the steady state of the model (consensus vs. polarization), the time to convergence, or the effect of various choices (network topology, confidence bound, update rule, initial distribution of opinions, etc.).
During Summer 2023, Mason Porter and I co-mentored four undergraduate students (Max Collins, Emily Gong, Arie Ogranovich, and Nicholas White) on a project at the UCLA Computational and Applied Mathematics (CAM) REU. Our group worked on a bounded-confidence model of opinion dynamics with heterogeneous confidence bounds. Specifically, they developed a generalization of the standard Deffuant-Weisbuch model where an agent's confidence bound is a function of the agent's current opinion. They studied theoretical guarantees of their model (e.g. convergence to a limit state) and ran a variety of numerical experiments.
Most opinion models including BCMs only consider dyadic interactions between agents. In reality, conversations and interactions happen between any number of people. Adapting the BCM models to higher-order networks can allow for more realistic simulation of real-world behavior. In Spring 2026, my J.D. Power Center Research Associates, Abby Harito and Patrick Jackson, began exploring the different types of behavior you see in a BCM on higher-order networks. They built off of work by Hickok et al. to explore a synchronous model. Abby Harito is continuing to work on this in Fall 2026.
Persistent homology, a tool from TDA, has shown success in various application areas; one ever growing area of study in this field is time series analysis. Nonlinear time series analysis is a research field in and of itself with tools to capture structure in time series data. Persistent homology comes with a solid theoretical framework, is robust to noise, and quantifies the same type of structure as often appears in time series data. Thus combining tools from time series analysis and TDA provides a new approach to analyze and quantify behavior in time series data.
This was the basis of my dissertation research with Liz Munch at Michigan State University.
If you're an undergraduate at Holy Cross interested in research with me, fill out this interest form then I'll reach out to you to schedule a meeting.