Interacting Particle Systems

I have been working with my advisor on hydrodynamic limits of interacting particle systems. Specifically, we are working to try and prove the hydrodynamic limit for the long range asymmetric simple exclusion process. Fraydoun Rezakhanlou proved a hydrodynamic limit for the asymmetric finite range case, and we have been working to extend his result. When the expected jump size for a particle is finite, the style of the proof is similar to Rezakhanlou's, but with added technicalities. When the expected jump size is infinite, a different proof is needed relying on techniques similar to those that Milton Jara used in the long range symmetric case. The proof has some neat tricks because of the complications of long range interactions that cannot be avoided in the asymmetric case. A preliminary overview of our work is summarized in the paper and slides I prepared for my comprehensive exam. Our final results can be found on arXiv.

Causal Diagrams

Starting in my last year of high school, I began conducting research with my father, a professor of epidemiology, on causal diagrams. Causal diagrams are directed acyclic graphs that depict causal relations, with the focus being to determine problems - and solutions to such problems - that might prevent accurate estimation of a causal effect. For example, any type of bias in a study can be encoded in a causal diagram. As part of our work, we have classified all biases into one of six types. We have developed results about the three previously known biases (confounding, colliding, and information bias); clarified misunderstandings and formally introduced two other biases (causal pathway and effect modification bias); and discovered a new type of bias (thought bias). Our work on these topics is summarized in the book chapter Causal diagrams and three pairs of biases. Recently, we wrote an article in which I proved a theorem showing precisely when colliding bias is created under a certain causal structure and provided a counterexample to a common claim present in the literature.

Opinion Dynamics
Opinion dynamics describes how members of a group change their opinions over time as they interact. Many of the models for opinion dynamics are deterministic in nature. My interest is in probabilistic models, which can nicely depict phenomena not present under deterministic models. Inspired by earlier work, I created two probabilistic models for opinion dynamics and have worked on them with a fellow graduate student. We have found conditions that guarantee consensus for both of our models. The proofs elegantly split into multiple cases depending on the interactions. We have also studied the behaviors of repulsing and overshooting opinions in relation to consensus. An overview of our results can be found in the following talk.

Axiomatic Set Theory
I am also interested in axiomatic set theory. I especially like Patrick Suppes' work on the topic. When time permits, I have been working to write what I hope will serve as a rigorous and intuitive introduction to the subject. Here is a sample of what I have written.