1. Chaos
Although Poincaré is credited as the first to discover chaos in dynamical systems, it was not until Lorenz’s 1963 discovery of “deterministic and nonperiodic flow” in a simplified weather model that chaos theory started to blossom into what it is today. My interest lies in a particular class of ordinary differential equations called the Lorenz systems which have their roots in fluid and atmospheric dynamics and ultimately the Navier–Stokes equations. Together, we can think about obtaining new extensions of the existing nonlinear systems and/or investigating their chaotic properties.
2. Weather
As long as you are an earthling, you cannot avoid being affected by weather in one way or another. Modern-day weather forecasts require vast amounts of data, computation, and lots of tricks to get around problems, which in turn requires coordination of multiple government agencies and people. My interest lies in the predictability of weather and climate in a simulation practice known as numerical weather prediction (NWP). By simulating weather in the past, we can try answering questions like: is weather becoming more unpredictable due to climate change? What combination of physical schemes yield best results for a particular weather pattern seen in Southern Nevada? Which data assimilation method is best at blending the observations and simulation outputs?
3. Infectious Disease and ABM
While the COVID-19 pandemic is most fresh in our memory, we have to also remember that history is full of survival stories through a series of disease outbreaks, often with more than one strain of disease circulating simultaneously. Mathematical models can help us be better prepared by allowing us to experiment with different scenarios and predict likely scenarios days or months ahead. Models can also be used to help us collect the right kinds of data as we prepare for the next pandemic. Given enough computational resource, we can now carry out agent-based model (ABM) simulations of a disease outbreak, representing every individual’s pathway with respect to the disease (including the now-familiar Susceptible, Exposed, Infected, and Recovered states, and many other states and conditions we can think of). As a side quest, we can also explore other uses of ABM.