I am broadly interested in applied probability, but I specialize in stochastic analysis and interacting particle systems, particularly limit theory and adjacent graph-theoretic concerns.
I have experience applying probabilistic techniques to machine learning, signal processing, mathematical biology, quantum networks, and quantum algorithms.
I give some highlights of some recent works below. For a complete and up-to-date list, see my Google Scholar.
Handwriting credit: Stanley Nicholson
My main work (jointly with my advisor Kavita Ramanan) considers stochastic processes whose interactions are governed by sparse graphs. Most graphs are sparse, but most limit theories only exist for dense ones. My goal is to establish the foundation for these systems under multiscale interactions, rare events, and control considerations. Keep an eye out for upcoming work!
Joint work with: Juan Diego Toscano, Vivek Oommen, Jerome Darbon, George Em Karniadakis
When trying to learn solutions (or solution maps) to PDEs, one minimizes a certain reweighted residual, scaled pointwise by the residual itself, to promote convergence uniformly in space---why can we do that? This work gives a general design principle for how to reweigh residuals at each collocation points and how the choice of weights translates to the primal optimization problem. The duality is derived from (generalized) Gibbs representation formula as well as asymptotics of variational forms.
Joint work with: Peter J. Thomas, Andrew W. Eckford
We communicate by encoding thoughts into words and sending them; cells communicate by modulating intensity of light, voltage, or signalling molecules that trigger downstream reactions. In this work, we formalize cellular signal transduction as a multistate-generalization of Kabanov's Poisson-type channel and study its capacity under general Markov inputs. We clarify the role of the structure of the receptor protein via variational formulas that upper and lower bounds the capacity,