My research

Dynamical systems are present latently or explicitly in most phenomenon around us, such as in physical systems, signal processing, control, or even in the mechanism of numerical methods. Some of the oldest questions in this field are forecasting, reconstruction, and the identification of coherent patterns, as I have outlined in Figure 1. However, there are no methods that execute these tasks equally effectively across all systems, due to the varied nature and complexity of the dynamics. My interest as a mathematician is to appreciate the core dynamical concepts that govern the efficacy of various numerical techniques, and on the basis of this knowledge improve or innovate on these techniques.

Research_outline_NTU_2.pdf

Fig. 1. Computational tasks, real world applications, and their underlying mathematical issues. The second row contains examples of several real-world manifestations of dynamical systems and learning problems. The common investigations into these systems usually fall into one of the categories in the first row. The challenges and complexity of these tasks differ greatly across systems, and are governed by the more core concepts listed in the third row.

From my early training in dynamical systems theory, I have been gradually led to the field of data science and I see it today in a rich intersection of probability theory, geometry, operator theory and ergodic theory. It is a multi-faceted discipline of mathematics that allows creativity with building techniques, rigor in building the framework, and deeper insight into questions about the underlying assumptions. The objectives of this research project is a more thorough understanding into the theoretical and computational aspects of dynamical systems, by putting them in the context of these related theories. Figure 2 gives an outline of my research questions.

Research_outline_NTU.pdf

Fig. 2. Outline of research problems, and their inter-connections. The problems fall mainly into three categories - dynamical systems, kernel methods, and random dynamical systems. Some of the problems also have strong connections to geometry.

research_outline_3.pdf

Fig. 3. Outline of techniques and concepts, and their inter-connections. In this diagram I have focussed on the role of geometry in my research.

I have explained my research problems and goals in more details in the document below.

Research_master.pdf