Research

My current research interests lie in applied probability and mathematical biology. To be specific, I am interested in (1) approximation of measures, (2) chemical reaction network theory, (3) mean field theory of interacting particle systems on networks, as well as their applications in information theory, data sciences and deep learning, epidemiology, ecology, biochemistry, mechanics and social sciences. The mathematical tools I apply come from approximation theory, probability theory and stochastic processes, dynamical systems and ergodic theory, graph theory, functional analysis, and PDE. My professional goal is to continue to contribute to mathematical laws behind problems arising from sciences in general and to provide biological and/or their physical interpretations, with an emphasis on collaborative and interdisciplinary research.

1. Approximation of probability measures (also called quantization of probability distributions in information theory) stems from the theory of signal processing, namely to approximate a probability distribution by a discrete probability with a given number n of supporting atoms. This problem, rephrased as a partition problem of the underlying space, arises in various other contexts, for instance, in cluster analysis, machine learning, numerical integration, stochastic processes, mathematical finance, convex geometry, optimal transport, and kinetic theory.

In statistics, when dealing with empirical data sets, practical considerations may demand that all atoms have equal weights, or at least that they be integer multiples of one fixed unit weight. This leads to a problem on constrained best approximations of a given probability, with prescribed weights, or analogously, locations, by finitely supported probability measures. As a main part of my PhD thesis, we did an in-depth study of this problem in dimension one. Best uniform approximations are close analogues of support points in statistics, with wide applications in Bayesian statistics, the numerical solutions of SDEs, notably in mathematical finance, as well as mean field theory of interacting particle systems, e.g., in neuroscience.

2. Chemical Reaction Network Theory (CRNT), which dates back to 1970s and provides a coherent framework for describing the deterministic or stochastic dynamics of a large class of complex biological and cellular systems, known as chemical reaction networks (CRNs). The central topics in CRNT are to deduce dynamical properties, deterministic or stochastic, purely from the reaction graphs of a given network.

Typically, if the abundances of the constituent molecules of particles in a biochemical system are low, stochasticity comes into place in the interaction of different species or the evolution of a single species. These systems with intrinsic noise are usually modeled by a continuous time Markov chain (CTMC) on a countable state space.

My recent works in CRNT mainly focus on understanding long-term dynamics, structure, asymptotics and bifurcations of one-dimensional stochastic reaction networks. Here is a talk I recently presented on "Dynamics of One-Dimensional Stochastic Reaction Networks".

Here is a regular online seminar on CRNT:

3. Many science phenomena are described as interacting particle systems (IPS). How to characterize the limit dynamics of an IPS when the number of particles tends to infinity? In the simplest case where particles are indistinguishable and all-to-all coupled, it is known that as the number of particles tend to infinity, the empirical distribution supported on solutions of the IPS converge weakly to its mean field limit (MFL) given by the solution of a PDE, the Vlasov Equation (VE). Yet, many applications demand IPS coupled on networks/graphs which may be directed and heterogeneous. With the development of the limits of graphs, it is interesting to know, how the limit of a sequence of digraphs associated with the IPS influences the macroscopic MFL of the IPS. In a recent work, we studied this problem from a new measure-theoretic viewpoint. Furthermore, when the dynamics of the network coevolves with that on the network, we proposed a a new formulation to characterize the continuum limit as well as the mean field limit.