Turbulence is one of the fundamental systems with complicated nonlinear dynamics. Problem of turbulence is often referred to as the remaining hardest problem of the Classical Physics of the Century. I am interested in the theoretical description of turbulence and its computational modeling.

Nearly inviscid fluid flows are the most common known example of turbulence (e.g., air motions behind aircraft). The mathematical model is the well-known Navier-Stokes equations. Theoretically, one aims at finding appropriate statistical description of the solution to the Navier-Stokes equations. This has not yet been accomplished, due to the complexity of the matter. Various phenomenological models have been developed to address various aspects of the solution.

A particular aspect is the scaling of turbulence. The Hierarchical Structure model (or the She-Leveque model) give a remarkably accurate description of the scaling exponents.

According to the Hierarchical Structure model, turbulent fluctuations organize themselves into families which are parameterized by the length scale and by the amplitude. The two parameters are related, however, in the very same way as the space and time are related (a symmetry group underlies their transformation). What characterizes turbulence is the so-called most intermittent structures which are the rarest but statistically significant events. They are the "worst" singularities, mathematically speaking.

I am also interested in modeling (statistically and/or in a deterministic way) other complex systems such as the network of bio-molecular, population dynamics, human systems, etc. I am also fascinated by "the so-called information dynamics or phase dynamics" and its application to social psychology. I am interested in describing social phenomena using concepts developed in the modern study of nonlinear dynamics (e.g. phase, resonance).