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.
Analyzing Spatial Data using TDA
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.
Microbial Interaction Networks
Coming soon :)
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.
Emily Gong (UCLA)
Arie Ogranovich (Rice University)
Nicholas White (University of Nebraska, Lincoln)
To see more of what Max, Emily, Arie and Nick were up to this summer, 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 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.