My initial research was focused on quantitative analysis of evolutionary PDEs (hyperbolic conservation laws, kinetic equations). Subsequently, I have also developed some skills in mathematical modeling, analysis, and simulations in various areas. I am currently interested in (but not limited to) the following research areas:
1. Kinetic theory of active particles: mathematical modeling and analysis
Active particle systems consider the interactions and collective behaviors of particles with activity. Different from the classical passive particles that interact with each other through collisions, the active particles can actively sense and respond to the environment, and interact with surrounding particles through specific moving strategies. Following Nicola Bellomo, the Kinetic Theory of Active Particles (KTAP) is developed by combining ideas from the classical kinetic theory with those of stochastic game theory, which has been widely applied to describe many behavioral (living) systems. We expect to build further mathematical modeling and analysis on this approach.
2. Mathematical modeling of crowd dynamics
Crowd dynamics has now been recognized as a multidisciplinary field that gained growing interest across many subjects such as mathematics, physics, civil engineering, computer sci- ence, safety & fire science, and transportation. The modeling approach of crowd dynamics can be developed at different scales, ranging from microscopic, mesoscopic, to macroscopic scales, based on the same principles and corresponding parameters. In recent years, the new trend in the study of human crowds is focused on incorporating the social behaviors of pedestrians into the crowd modeling, which is useful for describing heterogeneous behav- ioral features in crowds, especially, the dynamics of pedestrians with evolving psychological states. We intend to incorporate more sophisticated factors into crowd dynamics and expand the breadth of applications of the models.
3. Evolutionary problems in social science and behavioral economics
Many evolutionary problems in social dynamics and behavioral economics could be treated by new methods borrowed from generalized kinetic theory, through which interactions be- tween agents can be modeled by games that lead to collective dynamics, where the interaction rules can evolve over time within an artificial world that is also created by the contribution of the players, as the philosophy of Herbert A. Simon teaches us. Our main interest of study is mainly focused on a deeper understanding of how social dynamics can affect the economy and vice versa.
4. From swarm intelligence to artificial intelligence
In mathematical theory of behavioral swarms, the interacting entities, called active particles, follow the classical path from sensing and learning from others, to the decision-making of the subsequent dynamics, leading to collective behavior, called swarm intelligence. Based on this idea, we intend to train some mathematical models that could lead to artificial intelligence for possible real-world applications.