Research

My Research Interests

My research interests broadly include applied and computational topology, topological data analysis (TDA), dynamical systems, network science, opinion dynamics, and machine learning. Below are a few example of projects/topics I'm currently interested in.

TDA and Applications to Social Systems

Coverage of Heterogeneous Resources

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. We expect to have a preprint in 2024. This is joint work with Gillian Grindstaff, Abigail Hickok, Jiajie Luo, and Mason A. Porter.

Cooling Center Coverage

Erin O'Neil, a Queen's Road Foundation Undergraduate Fellow at UCLA, and I are working on studying 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 constitutes Erin's senior research project. See her work on arXiv here.

Patterns in Faculty Hiring in Mathematics

During Summer 2024, Mason Porter, Sidhanth Raman (UCI), and I co-mentored five undergraduate students 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. This is ongoing work, but preliminary findings can be found in their final report here. 

Group Members:

Microbial Interaction Networks

I am interested in how spatial structure (such as network topology) and environmental factors (such as resources and toxins) impact interactions between agents in biological systems. Models of population dynamics often simplify dynamics by ignoring environmental impacts. Microbial communities are particularly effected by their environment. Microbes do not interact directly, instead, they impact each other through the production of resources and toxins.

One approach to analyzing microbial systems is through spatial models where one assumes that microbes live on some underlying structure. For simplicity, these models often assume that microbes live on a lattice. The microbiome then evolves based on the effects of the local neighborhood on each microbe.

In reality, microbes do not live on a lattice, so I am exploring the impact of the underlying structure on the dynamics of an existing spatial model. This is ongoing work in collaboration with Mason Porter (UCLA) and ecologists Elena Litchman and Christopher Klausmeier (both at Michigan State). 

Opinion Dynamics on Networks

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 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 on a project at the UCLA Computational and Applied Mathematics (CAM) REU. Our group worked on a bounded-confidence model of opinion dynamics with heterogenous 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. We expect to have a preprint in 2024.

Group Members:

To see more of what Max, Emily, Arie and Nick did, see their final report here.

Analyzing Temporal Data using TDA

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.