My research specialty surrounds the analysis of probability models---in particular, interacting particle systems and Laplacian growth models---on state spaces which are bounded in distance determined by electrical resistance. By studying explicit models and taking the right space-time scaling limits, I can rigorously derive (stochastic) partial differential equations or prove limit shape theorems on these spaces. These mathematical results capture various laws of nature, such as heat flow, wave propagation, charged particles in an electromagnetic field, and fluid dynamics.

My current focus project is to establish microscopic derivations of (non)linear (S)PDEs from interacting particle systems---the main model being the (weakly a)symmetric exclusion process with variable speed boundary---on spaces which are bounded in the resistance metric. The two tunable "knobs" are the bulk antisymmetric jump rates and the rates of the boundary birth-and-death dynamics. By tuning these knobs we obtain a large variety of (S)PDEs with appropriate boundary conditions, describing the macroscopic behavior of particles in and out of equilibrium. The underlying state spaces include trees, fractals, and random environments, and do not necessarily possess translational invariance like Euclidean spaces. See papers 9, 11, 16, 17, P1, P2, P3.

Other probability models I have studied include: internal diffusion-limited aggregation (13), rotor-router aggregation (14), abelian sandpiles (14), and Gaussian free fields (3). In all of these models I have established optimal or near-optimal results on the corresponding state spaces. For instance, paper 14 is the first to rigorously establish that abelian sandpile growth on a geometrically self-similar state space exhibits power law modulated by log-periodic oscillations.

In order to gain a deeper understanding of stochastic processes on resistance spaces, I have also studied spectral analysis of Laplacian and Schrödinger operators on self-similar fractal spaces, on which there are a number of peculiar features (4, 5, 15). The results have implications for wave propagation (7), and for the asymptotic complexity of combinatorial structures such as spanning trees (10) and cycle-rooted spanning forests (15).

My paper 9 is an example where these two aspects of my work (probability theory + spectral analysis) meet: spectral gap inequality + electric network reduction = moving particle lemma in the exclusion process.

While the models I study may seem disparate, my approach is ultimately grounded in the study of analysis on graphs (Laplacian, heat kernels, and spectral analysis), Markov processes (such as random walks, Markov chains, Poisson processes, and Brownian motions), and potential theory (using techniques such as Dirichlet forms and related functional inequalities).

Papers 1, 6, 8, 14, 15 involved notable contributions from undergraduate students in the form of numerical findings, conjectures, and lemmas, which were then turned into theorems by the senior author(s).

As of July 1, 2019, my research is supported in part by the National Science Foundation (NSF) DMS Probability program grant DMS-1855604, "Stochastic dynamics on large-scale networks" (2019-22).

I also acknowledge recent support from the Simons Foundation (via a Collaboration Grant for Mathematicians, 2017-19) and the Research Council of Colgate University (Major grant 2017-18, Picker Research Fellowship 2019-20).

2018-19: Research lectures at Bielefeld (Autumn 2018) and Lisboa (Winter 2019)
2017: Research tutorials & lectures at Cornell (June) and Bielefeld (July)

Spectral decimation of the magnetic Laplacian on the Sierpinski gasket (AMS Sectional Meeting @ Binghamton U, Oct '19, written and delivered by Ruoyu Guo '19).

Nonequilibrium fluctuations in the boundary-driven exclusion process on a resistance space (Stochastic Processes & their Applications conference @ Northwestern University, July '19).

Random walks, electric networks, moving particle lemma, and hydrodynamic limits (Leiden Most Informal Probability Seminar, May '19).

Laplacian growth and sandpiles on the Sierpinski gasket (UChicago Probability Seminar, Apr '19).

I work with select Colgate undergraduate students on challenging problems in probability and mathematical analysis that interface with physics (statistical physics, quantum physics) or theoretical computer science (randomized algorithms). While the topics look disparate, everything for me comes down to careful understanding of the Laplacian---generalization of the 2nd derivative in calculus, and more profoundly, the infinitesimal generator of a diffusion process---and the calculus thereof (e.g. integration by parts).

From experience, I look for students who not only have excelled in 300+-level MATH courses in the above-mentioned subjects, but also---more importantly---have the aptitude for independent problem-solving (analytical or numerical).

⚠ I am a mathematician, and my job is to come up with potential theorems and prove them logically.  Therefore I look for students whose mathematical ability is strong, whether or not they know how to program.  (Of course these two abilities are not mutually exclusive.) While I appreciate proficient programmers, the best outcomes occur with students who can contribute to theorem proving.

Please look below for research projects which resulted in arXiv preprints and then published in highly ranked journals.